Every large enterprise relies on a web of critical software systems. But what happens when the code at the heart of these systems was written decades ago, by experts who have long since retired?

We hear this story over and over:

• “It works… but don’t touch it.”
• “We’re one failure away from downtime.”
• “Our auditors don’t understand the code we’re running.”

From financial services to engineering and healthcare, unsupported numerical software is a growing risk. Not because businesses don’t care, but because fixing it is complex and costly.

The irony? Everyone knows modernization is the end goal—but few can afford to wait years for a total rebuild. What’s missing is the bridge. A practical path that gives enterprises confidence, continuity, and compliance while they figure out the future.

Our team is exploring what a bridge might look like: delivering new forms of support that ensure compliance, continuity, and structured transition without waiting years for a full rebuild. Early feedback has been eye-opening—some leaders only realized the scale of their risk once they saw the options mapped out.

In the months ahead, we’ll share insights to how enterprises can tackle this problem—what works, what doesn’t, and what’s next. What’s the biggest legacy challenge you’re facing right now?

Related reading:

• Mechanical Orchard reports that “maintaining those systems can be a costly burden, with an average of 60–80% of IT budgets allocated to keeping them running.” https://www.mechanical-orchard.com/insights/1-14-trillion-to-keep-the-lights-on-legacys-drag-on-productivity
• TechRadar Pro cites “a report from Dell estimates that organizations currently allocate 60‑80% of their IT budget to maintaining existing on‑site hardware and legacy apps.” https://www.techradar.com/pro/fix-it-even-if-it-aint-broke-the-price-of-legacy-technology  
• RecordPoint notes that in 2019, the U.S. federal government spent 80% of its IT budget on operations and maintenance, primarily involving legacy systems. https://www.recordpoint.com/blog/maintaining-legacy-systems-costs  
• Audacia similarly states: “Maintaining those systems can be a costly burden, with an average of 60‑80% of IT budgets allocated to keeping them running.” https://audacia.co.uk/blog/cost-of-legacy-systems  
• PSP‑IT says: “Research estimates that organizations spend anywhere between 60‑80% on keeping their older IT systems running.” https://psp-it.com/blogs/staggering-stats-that-show-how-outdated-technology-could-be-damaging-your-business/  

Non-Convex and Stochastic Optimization in 2025: The Engines of Real-World Intelligence

1  The Shape of Complexity in Modern Systems

Historically, optimization was largely confined to well-structured, convex problems — settings where theoretical guarantees and algorithmic efficiency aligned neatly. This made sense: algorithms for large-scale linear and convex programs, capable of handling millions of variables and constraints, have matured over decades. In contrast, non-convex problems — including those involving discrete or combinatorial structures — remained computationally intractable at scale. But by 2025, the landscape has shifted. Modern commercial systems are increasingly defined by complexity, scale, and uncertainty. As industries move from deterministic, rule-based frameworks to data-driven architectures infused with randomness, two methodological pillars have emerged as essential: non-convexity and stochasticity. These form the mathematical foundation for robust, adaptive optimization in the real world. 

• Non-convexity refers to problems where the objective function or constraints exhibit multiple local minima, flat plateaus, or discontinuities. Solving these problems requires escaping local optima and exploring globally.
• Stochasticity involves modeling randomness explicitly. This is crucial when data, inputs, or environments are noisy, incomplete, or changing—conditions that prevail in nearly all large-scale operational contexts. The marriage of optimization and stochastic processes is one of the most fruitful in all of applied mathematics.

These two paradigms — used alone or together — form the computational backbone of modern decision-making systems.

2  Where It Matters — Six Fronts of Transformation

2.1  AI and Machine Learning

Deep learning, the foundation of modern AI, inherently involves non-convex optimization landscapes. Training a neural network means minimizing a high-dimensional loss function riddled with saddle points and local minima. Gradient-based methods such as stochastic gradient descent (SGD) work well in practice, but newer methods like evolutionary strategies and Bayesian optimization are gaining ground for model tuning and hyperparameter search.

Additionally, neural architecture search (NAS) — where the architecture of the model is itself learned — requires solving a combinatorial, non-convex, and stochastic problem that blends learning with optimization. Reinforcement learning algorithms also depend heavily on stochasticity to explore state spaces and improve policies over time.

2.2  Distributed Systems

Modern distributed computing environments — from federated learning on edge devices to massive cloud clusters — face dynamic conditions that make optimization difficult. In federated learning, each client device has its own data distribution, leading to non-identical local losses. The global model must minimize a weighted sum of these heterogeneous objectives:

\[
\min_{w} \sum_{i=1}^N p_i \mathcal{L}_i(w)
\]

where \(p_i\) reflects the client importance or data volume. In cloud systems, tasks must be scheduled to optimize for latency, cost, and resource utilization—often with uncertain workloads and shifting resource availability. These conditions naturally introduce both non-convex costs and stochastic inputs.

2.3  Energy and Sustainability

Power systems are increasingly complex, integrating intermittent sources like wind and solar. Optimization in this domain often involves non-convex unit commitment problems and stochastic forecasts of supply and demand. Operators must ensure balance and stability while minimizing carbon emissions and cost.

2.4  Cloud Cost Management

Cloud computing introduces varied pricing models:

• On-demand: linear cost
• Reserved instances: fixed rate for long-term commitment
• Spot pricing: fluctuates with supply/demand

The total cost landscape is non-smooth and time-varying. AI-driven FinOps systems optimize over stochastic forecasts of future demand, taking advantage of market conditions while ensuring reliability.

2.5  Supply Chains & Logistics

Modern supply chain and logistics networks operate in highly dynamic environments shaped by real-time disruptions, demand variability, uncertain lead times, and complex geopolitical constraints. Traditional optimization approaches — such as shortest-path algorithms or deterministic linear programs — fall short when cost functions are non-linear, information is incomplete, or actions must be adapted sequentially over time.

Let us consider a canonical example: motion planning for autonomous logistics systems, such as drones, autonomous delivery vehicles, or robotic warehouse agents. These systems must determine a trajectory \( \{x_t\}_{t=1}^T \) over a planning horizon \( T \), where each \( x_t \in \mathbb{R}^d \) represents the system’s state (e.g., location, velocity) at time step \( t \). A general trajectory optimization problem can be formulated as:

Where:

• \( c(x_t, x_{t+1}) \) is a cost function capturing travel time, energy consumption, or risk exposure between consecutive states,
• \( \mathcal{F}_t \subset \mathbb{R}^d \) denotes the time-varying feasible region, accounting for traffic conditions, no-fly zones, terrain restrictions, or regulatory limits,
• The constraints may also include dynamic collision avoidance and resource usage limits (e.g., battery levels).

This formulation is inherently non-convex, due to:

• Non-linear dynamics or kinematic constraints (e.g., vehicle turning radii, acceleration limits),
• Piecewise or discontinuous cost structures (e.g., congestion pricing or toll thresholds),
• Obstacle avoidance modeled via non-convex spatial exclusions.

Due to the presence of uncertainty, factors such as stochastic travel times, real-time weather updates, or unexpected demand spikes lead to:

• Stochastic planning formulations, where travel cost or availability is scenario dependent,
• Online or receding-horizon control, continuously updating plans based on live sensor and market data.

In practice, solution methods include:

• Sampling-based motion planners (e.g., RRT*, PRM) for high-dimensional feasibility search,
• Mixed-integer nonlinear programming (MINLP) for incorporating discrete control logic (e.g., path segment selection),
• Reinforcement learning to learn adaptive policies under uncertainty,
• Heuristic or metaheuristic search (e.g., simulated annealing, genetic algorithms) for real-time route generation.

As logistics infrastructures scale and autonomy increases, non-convex and stochastic optimization frameworks become indispensable for enabling resilient, efficient, and real-time decision-making across global supply chains.

2.6  Modern Optimization Methods

To tackle such challenges, hybrid and heuristic approaches are common:

• Metaheuristics like genetic algorithms and particle swarm optimization explore rugged solution spaces.
• Reinforcement learning models problems as sequential decisions under uncertainty.
• Neural combinatorial optimization learns to solve optimization problems using neural networks.

These methods bypass assumptions of convexity or determinism, enabling robust optimization under real-world complexity.

3  Case Study — Modern Portfolio Optimization

3.1  Definitions and Setup

Let:

• \( \mathbf{w} \in \mathbb{R}^n \) be the portfolio weight vector, where each \( w_i \) is the fraction of capital allocated to asset \( i \)
• \( R_s \in \mathbb{R}^n \) be the vector of asset returns under scenario \( s \)
• \( p_s \in [0,1] \) be the probability associated with scenario \( s \), where \( \sum_s p_s = 1 \)
• \( L(\mathbf{w}, s) \) be the portfolio loss under scenario \( s \), defined as the negative return

3.2  Limitations of Classical Models

Traditional portfolio optimization uses the mean-variance framework:

\[
\min_{\mathbf{w}} \mathbf{w}^\top \Sigma \mathbf{w} \quad \text{subject to } \mathbf{w}^\top \mu \geq R, \quad \mathbf{w}^\top \mathbf{1} = 1, \quad \mathbf{w} \geq 0
\]

where:

• \( \mu \in \mathbb{R}^n \) is the vector of expected returns
• \item \( \Sigma \in \mathbb{R}^{n \times n} \) is the covariance matrix of returns
• \item \( R \) is the required minimum portfolio return

This formulation assumes normally distributed returns, convexity, and linear constraints — rarely satisfied in real markets, where returns usually follow leptokurtic and asymmetric distributions.

3.3  Tail Risk and CVaR Optimization

To handle asymmetric, heavy-tailed risks, we define:

\[
L(\mathbf{w}, s) = -\mathbf{w}^\top R_s
\]

and minimize the Conditional Value-at-Risk (CVaR) at level \( \alpha \in (0,1) \):

\[
\min_{\mathbf{w}, \nu, \xi_s} \quad \nu + \frac{1}{1 – \alpha} \sum_s p_s \xi_s
\]
\[
\text{subject to } \xi_s \geq L(\mathbf{w}, s) – \nu, \quad \xi_s \geq 0 \quad \forall s
\]

where:

• \( \nu \in \mathbb{R} \) is an auxiliary variable representing the Value-at-Risk (VaR), ie the quantile
• \( \xi_s \in \mathbb{R}_{\geq 0} \) captures the excess loss beyond \( \nu \) in each scenario

3.4  Non-Convex Constraints and Market Realism

The real-world feasible region includes:

• Transaction costs: Modeled as piecewise linear or nonlinear functions based on trade volume
• Liquidity constraints: \( w_i \leq \text{max tradable volume}_i \)
• Regulatory/ESG: Sector exposure bounds, carbon scores, or exclusion zones

These constraints often introduce non-convexities into the problem.

3.5  Solvers, Algorithms, and Deployment

Practical optimization pipelines often blend global and local strategies — genetic algorithms, simulated annealing, reinforcement learning, and hybrid methods — to navigate complex, nonconvex landscapes. Financial institutions operationalize these solutions using cloud compute for parallelism, GPUs for Monte Carlo acceleration, real-time data feeds for adaptive reoptimization, and rigorous stress testing for robustness.

4  Conclusion

Optimization today mirrors the complexity of the systems it governs. Non-convexity allows us to model the nonlinear, constraint-laden, and irregular nature of real systems, while stochastic methods embrace randomness as intrinsic to decision-making. Together, they form a unified framework for adaptive, scalable, and robust operations across AI, finance, energy, and logistics. In a world increasingly shaped by uncertainty and scale, these tools are no longer optional — they are foundational.

References

Shapiro, A., Dentcheva, D., & Ruszczynski, A. (2014). Lectures on Stochastic Programming: Modeling and Theory (2nd ed.). Society for Industrial and Applied Mathematics (SIAM). A comprehensive reference on stochastic programming, including CVaR, scenario modeling, and theoretical underpinnings.

Boyd, S., & Vandenberghe, L. (2004). Convex Optimization. Cambridge University Press. The foundational text on convex optimization, often cited to highlight where convex assumptions break down.

SKAO and nAG 

The SKA Observatory is an international endeavour to build the world’s largest radio telescopes, a €2 billion project aimed at transforming humanity’s understanding of the universe. Behind this monumental effort lies an equally groundbreaking challenge: processing petabytes of raw data every day into high-resolution astronomical images.

nAG’s High-Performance Computing (HPC) engineers play a pivotal role in making this possible. Their work ensures the seamless transformation of vast volumes of data into science-ready data products, supporting the SKAO’s mission to accelerate discovery in radio astronomy and gravitational wave science.

What part of the SKA telescopes are you focused on (e.g. data, processing, storage, networking, other)?

My focus is on science data processing — the stage of the pipeline where we run large-scale batch processing on HPC systems to transform raw telescope data into science-ready images and catalogues. This is where the heavy lifting happens, as we apply sophisticated algorithms to transform petabytes of data into images varying from 1 megapixel to over 17,000 megapixels in size.

While my primary role is within this area, I collaborate closely with colleagues across other critical parts of the SKAO software ecosystem. For example, the real-time processing team, who use FPGA arrays to hunt for pulsars in real-time before any data even touches a disk, and the SRCNet team — architects of a global network of data centres designed to receive, distribute, and further process the outputs from the Science Data Processor (SDP). Together, our efforts ensure that data flows smoothly from telescope to scientist, no matter where in the world the science happens.

Why is HPC critical to the operation of the SKA telescopes?

The concept for the SKA telescopes dates all the way back to the late 1980s But for decades, it remained an ambitious idea, waiting for computer hardware to catch up. Astronomers always knew that to unlock the SKA’s potential, we’d need computing systems fast enough — and affordable enough — to process the staggering volumes of data produced by its two massive telescopes.

That’s why the SKA telescopes are often described as “software telescopes”. Rather than relying on a single, gigantic dish, the SKA combines data from hundreds of dishes and thousands of antennas through software, creating images far superior to anything achievable with even the largest single-dish telescope.

Without HPC systems, the SKA telescopes simply wouldn’t be possible. HPC is the beating heart of the project, enabling us to turn torrents of raw signals into precise, high-resolution snapshots of the universe.

Image Copyright: SKAO · Collage of simulated images of future SKA-Low observations, showing what the telescope is expected to be able to produce as it grows in size. The images depict the same area of sky as that observed in the first image from a working version of the telescope, released in March 2025. Top left: By 2026/2027, SKA-Low will have more than 17,000 antennas and will become the most sensitive radio telescope of its kind in the world. It will be able to detect over 4,500 galaxies in this same patch of sky. Top right: By 2028/2029, SKA-Low will count over 78,000 antennas and be able to detect more than 23,000 galaxies in this field. Bottom: The full SKA-Low telescope will count more than 130,000 antennas spread over 74 km. Similar observations of this area will be able to detect some 43,000 galaxies, while deep surveys performed of this area of the sky from 2030 will be able to reveal up to 600,000 galaxies.

Describe your day-to-day involvement with the project?

Right now (May 2025), I’m focused on integrating MSv4 support into a key software tool called DP3 (pronounced “DP cubed”). DP3 plays a crucial role in the SKA data pipeline: it calibrates the raw signals we receive from the telescopes and flags corrupted data, ensuring the data is scientifically accurate. By adding MSv4 support, we’re enabling distributed calibration — allowing us to spread the workload across multiple computing nodes, with recent benchmarks showing 20x throughput compared to the most widely used MSv2 format.

Day to day, my work is hands-on software development. I spend much of my time writing and optimising code, but in a project of this scale and complexity, collaboration is just as critical as coding. With such a diverse software ecosystem, we work closely as a team to stay aligned and ensure we’re all moving towards our shared goals.

Beyond being part of the SDP software team, I’m also part of a collaborative working group that includes key contributors from the US National Radio Astronomy Observatory (NRAO) and the South African Radio Astronomy Observatory (SARAO). We meet weekly to align our software solutions, share technical insights, and ensure tight integration across our international efforts. This close collaboration is vital to ensure our tools work seamlessly across observatories and deliver the best possible outcomes for the global radio astronomy community.

And it’s not just about coding — I also work alongside teams contributing to the SDP roadmap and long-term vision. Together, we’re constantly refining our priorities and strategies to realise the ambition of building the SKA telescopes.

What are and have been the biggest technical challenges you’ve faced?

I’m a physicist by training — my PhD focused on writing software to simulate magnetic systems at the tiniest length scales. Interestingly, I was never particularly drawn to astronomy during my studies. But once I joined the SKA project, I quickly discovered that radio astronomy comes with its own rich history of specialised data processing techniques, as well as some truly unique challenges. One of the first hurdles I faced was the sheer breadth of new terminology and concepts specific to the field, especially those arising from the difficulties of observing the universe amidst a world saturated with wireless signals.

Once I found my footing in the landscape of radio astronomy, the real technical challenge emerged: squeezing every last ounce of performance from the hardware we have. This is because, despite being a €2 billion flagship project, the SKAO operates with a relatively modest budget for HPC infrastructure. My colleagues and I are deeply immersed in the fine details of both the data and the hardware, constantly engineering robust and scalable solutions to push the limits of what’s possible. It’s a challenge that demands not just technical skill, but creativity — one of the things that makes working on the SKA project so rewarding.

Image Credit and Copyright: SKAO/Cassandra Cavallaro · Author: Cassandra Cavallaro

The historic Lovell Telescope reflected in the window of SKAO Global HQ, UK. 

Were there any specific scale or performance issues that pushed the limits of what’s currently possible in HPC?

As part of the SDP development  we continually test our software and hardware against progressively larger volumes of data, all with the goal of hitting the required performance and scalability targets by the time the telescope comes online towards the end of the decade. Among all the challenges, the most demanding HPC bottleneck is unquestionably I/O performance. To realise the full SKA vision, our Science Processing Centres in Cape Town and Perth will need to sustain average read and write speeds of around 8 terabytes per second, 24 hours a day — a staggering figure.

While that might sound achievable in comparison to high-end data centres, the SKA project’s challenge lies in the complexity of our data access patterns. Unlike workloads that are “embarrassingly parallel”, where tasks can be distributed independently across compute nodes, our workflows require tightly coordinated data movement which is governed by the physics of radio astronomy. This means we have to choreograph data access across many nodes with precision. Achieving this level of orchestration is an ongoing frontier in HPC, and it’s pushing us right up against the limits of today’s hardware and software architectures.

Does the project require any novel approaches, tools, or technologies?

Wherever possible, we aim to use battle-tested, off-the-shelf solutions. This allows our developers to focus their efforts on the truly unique challenges of radio astronomy, rather than reinventing the wheel. Importantly, the software we build for the SKA telescopes isn’t just for internal use — it’s made available to the global radio astronomy community. That means there’s a strong emphasis on modernising and optimising existing tools to make them more robust, sustainable, and accessible for a wide range of users.

That said, we’re always keeping a close eye on advances in software, hardware, and infrastructure, and we actively adopt the latest technologies in data storage and numerical computing where they offer real benefits. Some of our most complex challenges centre around managing data dependencies efficiently. We work hard to minimise inter-process communication (IPC) and avoid data access contention, both of which are critical for scaling our software effectively across an entire HPC cluster. Solving these problems often requires creative, novel approaches to data orchestration — it’s at this intersection of proven technologies and innovative problem-solving where much of the SDP software progress happens.

Have you been doing anything that hasn’t been done before in this space?

I’ve been a strong advocate for integrating object storage technologies — widely adopted in cloud computing — into the SKAO’s Science Data Processor (SDP). This approach is still quite unconventional in HPC environments, but I believe it represents the start of a paradigm shift. As HPC increasingly converges with data science and “big data” workloads, the required performance characteristics of these systems are evolving. Our choice of underlying technologies needs to evolve too.

Object storage offers flexibility and scalability that align well with the data-intensive nature of the SKA telescopes. However, introducing new technologies into an established domain like HPC is never straightforward. From experience, I know it can be challenging to build consensus and overcome natural resistance to change. But, by demonstrating the tangible benefits of these approaches, I hope to help pave the way for a new generation of HPC systems that are better suited to the data challenges of modern scientific research.

Do you need to optimise code or hardware performance in a unique way?

Not all of my work is about writing code to make software faster — a significant part of my role involves running hardware optimisation experiments to guide critical infrastructure decisions. When you’re dealing with data at the scale of the SKA project, the amount of system memory (RAM) in each machine can have an outsized impact on performance.

In one case, my team and I observed a 4× performance boost by doubling the amount of RAM available. With the larger memory footprint, the operating system was able to cache all of data accessed by other processes in memory, dramatically reducing the number of read operations from high-latency storage like our Lustre partitions. Crucially, this also increases the bandwidth available for ingesting data from the telescope. What makes this particularly exciting is that it’s a performance improvement that doesn’t require huge investment. By carefully matching memory configurations to our projected data volumes, we can achieve significant gains cost-effectively.

How does your work contribute to the bigger mission of SKA — understanding the universe, detecting cosmic signals, etc.?

A huge part of a radio astronomer’s work today involves painstaking data processing — something the SKAO must fully automate because the vast data volumes are too large to transmit over the internet. By taking this burden off researchers, the SKAO will accelerate the pace of discovery, freeing scientists to focus on new ideas and groundbreaking research rather than manual data wrangling. It’s a transformation that will be a tremendous benefit to the global radio astronomy community.

The work I do directly enables the SKA telescopes to image structures in the universe with far greater sensitivity than ever before. By combining extreme sensitivity with unprecedented resolution, we’re opening a window into the faintest structures in the cosmos, allowing us to probe the physics of the distant past. With the SKA telescopes, we’ll be able to study the evolution of dark matter, investigate processes happening at the atomic scale in black hole jets, and even detect the very first light emitted in the universe.

One particularly exciting frontier is pulsar research. Pulsars — ancient, collapsed stars with the strongest magnetic fields known in the universe — are a relatively new focus in astrophysics. The SKA telescopes will vastly expand our ability to discover, characterise, and track pulsars — providing unparalleled opportunities to test the limits of Einstein’s theory of General Relativity.

Image: Pulsar neutron star. Source of radio emission in space.

Perhaps most thrilling of all, the SKA telescopes’ extraordinary sensitivity will play a crucial role in advancing gravitational wave astronomy. Experiments like LIGO, the most sensitive science instrument ever built, can detect ripples in space-time but struggle to differentiate between genuine gravitational waves and local interference — even something as mundane as a microwave being switched on in a local town. The SKA telescopes will act as a verification tool, confirming the astrophysical origins of LIGO’s signals. Once a gravitational wave is confirmed, the SKAO would be alerted. Then, its telescopes could be put into an emergency observation mode and rapidly point at the source. I’m convinced that when we witness the first direct observation of two black holes merging, it will be thanks to the SKAO.

What excites you most about working on the SKA project?

What excites me most about working on the SKA project is the chance to learn from, and collaborate with, some of the brightest minds in radio astronomy. I get a unique, behind-the-scenes view of how groundbreaking science is done, and I have the privilege of helping to develop new techniques that will directly contribute to published research and future discoveries.

There’s something incredibly rewarding about knowing that the tools and systems I’m helping to build could play a role in answering some of the biggest questions about our universe. And one day, when a Nobel Prize in Physics is awarded for discoveries made with SKA data, I’ll be able to say: I helped make that possible.

1.1 How MOO Differs from Single-Objective Optimization

In single-objective optimization, the focus rests on optimizing one measure of performance, often subject to constraints. By contrast, MOO considers several potentially conflicting objectives. Instead of a single best solution, MOO produces a Pareto front, capturing the principle that improving any one objective will inevitably compromise at least one other. This perspective highlights the necessity of balancing diverse objectives rather than maximizing a single metric in isolation.

1.2 Common Applications of MOO

Multi-objective optimization is broadly applicable across industry and research settings:

• Transportation & Logistics: Minimizing delivery costs while enhancing service efficiency.
• Finance: Balancing risk and return within regulatory constraints in portfolio optimization.
• Energy: Negotiating cost, efficiency, and environmental impact in power systems.
• Healthcare: Designing treatment plans that maximize efficacy while minimizing adverse side effects.
• Manufacturing & Engineering: Weighing material strength, production cost, and sustainability to optimize product design.

2 The Challenges of Competing Objectives

2.1 Understanding Trade-Offs in Optimization

A defining feature of MOO is that enhancement in one objective may necessitate compromises in another. For instance, in product design, boosting durability often requires higher-grade, and therefore more expensive, materials. In algorithmic trading, strategies that promise higher returns typically carry greater risk. Effective multi-objective decision-making thus hinges on recognizing and prioritizing such trade-offs in alignment with overarching goals.

2.2 The Pareto Principle and Pareto Optimality

A solution is said to be Pareto-optimal if no objective can be improved without adversely affecting at least one other objective. The set of all such non-dominated solutions forms the Pareto front, serving as a powerful visual aid for analyzing compromises. Formally, a solution \(x^*\) is Pareto-optimal if there is no other \(x\) such that:

\begin{equation}
f_i(x) \;\leq\; f_i\bigl(x^*\bigr) \quad \text{for all } i,
\text{ with at least one strict inequality}.
\end{equation} 

Decision-makers can examine the Pareto front to determine which specific blend of trade-offs best suits their operational constraints and organizational objectives.

3 Mathematical Foundations of MOO

A multi-objective optimization (MOO) problem involves the simultaneous minimization or maximization of multiple objectives:

\begin{equation}
\min_{x \in X} \quad F(x) = \bigl(f_1(x), f_2(x), \ldots, f_k(x)\bigr),
\end{equation}

where \( F(x) \) is a vector of \( k \) objective functions \( f_i(x) \), each mapping the decision space \( X \) to the objective space. In contrast to single-objective optimization, which yields a unique optimum, MOO problems typically result in a Pareto front where no single objective can be enhanced without worsening at least one other objective.

3.1 Constraints and Feasible Regions

Feasibility is governed by a set of constraints:

\begin{equation}
g_j(x) \;\leq\; 0, \quad h_l(x) \;=\; 0,
\end{equation}

which may represent:

• Inequality constraints (\( g_j(x) \leq 0 \)) that limit the permissible values of decision variables.
• Equality constraints (\( h_l(x) = 0 \)) that must be satisfied exactly, reflecting strict physical, budgetary, or regulatory demands.

Such constraints define the feasible region, ensuring proposed solutions respect real-world limits (e.g., resource availability, regulatory caps). Since MOO rarely offers a single global solution, tools such as Pareto-based ranking, weighted sum approaches, or constraint-handling techniques help decision-makers explore and compare trade-offs. Ultimately, the chosen solution depends on domain-specific priorities, decision-maker preferences, and realistic feasibility.

3.2 Common Methods for Constructing the MOO

Scholars and practitioners employ various methodologies to formulate the MOO problem, including:

• Weighted Sum Method: Aggregates objectives into a single function via weights. While straightforward, it may struggle with non-convex Pareto fronts.
• Epsilon-Constraint Method: Treats all but one objective as constraints, offering diverse Pareto solutions but requiring multiple runs.
• Goal Programming: Minimizes deviations from specified targets, well-suited for structured problems with clearly defined benchmarks.
• Pareto-Based Evolutionary Algorithms: Methods such as NSGA-II or MOEA/D iteratively evolve diverse Pareto-optimal solutions, though at a high computational cost.

4 Solution Techniques

4.1 Approaches for Solving MOO Problems

Selecting an appropriate technique for solving MOO problems depends on problem structure, computational resources, and objective complexity. Common approaches include:

• Mathematical Programming: Linear (LP) and Mixed-Integer Programming (MIP) are well-suited for problems with linear or piecewise-linear structures. Convex optimization methods can ensure global optimality in convex MOO tasks.
• Metaheuristic Algorithms: Techniques like Genetic Algorithms (NSGA-II, MOEA/D) and Particle Swarm Optimization (PSO) can handle non-convex or highly complex search spaces, though they often require substantial computational effort.
• Reinforcement Learning & AI: Adaptive, learning-based optimization that rebalances objectives in response to evolving system or market conditions.

These solution techniques, when combined with robust data handling and domain insights, guide practitioners toward identifying and refining trade-offs that satisfy the organizational, technical, and market dimensions of multi-objective problems.

5 Real-World Applications: Coffee Blending Optimization

The goal is to formulate a coffee blend that meets specific sensory and analytical targets while controlling cost. Table 1 compares the attributes of three coffee types—Colombian Supremo, Vietnamese Robusta, and Kenyan AA—against desired values for aftertaste, bitterness, sweetness, polyphenols, pH level, and lipid content. Each bean also has a distinct cost per unit. The challenge lies in determining the proportions of each coffee that balance these competing factors, thereby producing an overall blend that aligns with both quality and economic objectives.

5.1 Objective Function Formulation

The final objective function \( Q \) integrates three key criteria—sensory deviation (\( S \)),
analytical deviation (\( AN \)), and cost per unit (\( C \)). Each component is
weighted by an importance scalar, reflecting different strategic goals:

\begin{equation}
\boxed{
Q(S, AN, C) = w_S \cdot S \;+\; w_{AN} \cdot AN \;+\; w_C \cdot C
}
\end{equation}

Table 1: Sensory and analytical properties of the three blend components and the target profile. 

5.1.1 Sensory Deviation Component

Let \(\mathcal{S} = \{\text{Aftertaste (AT)}, \text{Bitterness (BT)}, \text{Sweetness (SW)}\}\) be the set of sensory attributes of interest. We define a target value \(\tau_i^*\) for each \(i \in \mathcal{S}\). The corresponding model prediction is \(\tau_i(\mathbf{x})\), which depends on the blend proportions \(\mathbf{x} = (x_1, x_2, x_3)\). Under an additive assumption, \(\tau_i(\mathbf{x})\) is a linear combination of the coffee types:

\begin{equation}
\tau_i(\mathbf{x}) \;=\; \alpha_{i1}\,x_1 + \alpha_{i2}\,x_2 + \alpha_{i3}\,x_3,
\end{equation}

where \(\alpha_{ij}\) represents the contribution of attribute \(i\) from coffee \(j\) (\(j \in \{\text{Colombian Supremo}, \text{Vietnamese Robusta}, \\ \text{Kenyan AA}\}\)). The sensory deviation \( S \) is then:

\begin{equation}
S \;=\; \sum_{i \in \mathcal{S}} \Big( \tau_i(\mathbf{x}) – \tau_i^* \Big)^2.
\end{equation}

For example, if \(\mathcal{S} = \{\text{AT}, \text{BT}, \text{SW}\}\) and the target values are \(\{\text{AT}^* = 6, \text{BT}^* = 4, \text{SW}^* = 4\}\), we have:
\begin{align*}
\tau_{\text{AT}}(\mathbf{x}) = 6x_1 + 4x_2 + 7x_3, \quad
\tau_{\text{BT}}(\mathbf{x}) = 4x_1 + 6x_2 + 3x_3, \quad
\tau_{\text{SW}}(\mathbf{x}) = 3x_1 + 2x_2 + 5x_3.
\end{align*}

5.1.2 Analytical Deviation Component

Similarly, let \(\mathcal{A} = \{\text{Polyphenols (PP)}, \text{Acidity / alkalinity level (pH)}, \text{Lipid Content (LIP)}\}\) be the set of analytical attributes. Each attribute \(a \in \mathcal{A}\) has a target value \(a^*\) and a predicted value \(a(\mathbf{x})\). The analytical deviation \( AN \) is:

\begin{equation}
AN \;=\; \sum_{a \in \mathcal{A}} \Big( a(\mathbf{x}) – a^* \Big)^2.
\end{equation}

As an example,
\[
\tau_{\text{PP}}(\mathbf{x}) = 2.2\,x_1 + 3.0\,x_2 + 4.1\,x_3, \quad
\tau_{\text{pH}}(\mathbf{x}) = 5.0\,x_1 + 4.8\,x_2 + 5.3\,x_3, \quad
\tau_{\text{LIP}}(\mathbf{x}) = 0.15\,x_1 + 0.20\,x_2 + 0.12\,x_3
\]

5.1.3 Cost Component

To incorporate economic considerations, we define a cost function \(C(\mathbf{x})\):
\begin{equation}
C(\mathbf{x}) \;=\; c_1\,x_1 + c_2\,x_2 + c_3\,x_3,
\end{equation}
where \(c_j\) is the unit cost associated with coffee \(j\).

5.1.4 Program Specification

Combining these components into a weighted sum framework yields:

\begin{alignat}{2}
\textbf{Minimize} \quad
& Q\bigl(\mathbf{x}\bigr)
= \underbrace{w_S \sum_{i \in \mathcal{S}} \bigl(\tau_i(\mathbf{x}) – \tau_i^*\bigr)^2 }_{\text{Sensory}}
\;+\; \underbrace{w_{AN} \sum_{a \in \mathcal{A}} \bigl(a(\mathbf{x}) – a^*\bigr)^2}_{\text{Analytical}}
\;+\;\underbrace{w_C \sum_{j=1}^{n} c_{i}x_{i}}_{\text{Cost}} \notag \\[6pt]
\textbf{subject to} \quad
& \sum_{j=1}^{3} x_j \;=\; 1, \notag \\
& 0 \;\leq\; x_j \;\leq\; 1, \quad j \in \{1,2,3\}.
\end{alignat}

Substituting all the input numbers and writing the optimization program explicitly:
\begin{alignat}{2}
\textbf{Minimize} \quad & Q(\mathbf{x}) =
w_S \Big[ \bigl(6x_1 + 4x_2 + 7x_3 – 6\bigr)^2 \;+\; \bigl(4x_1 + 6x_2 + 3x_3 – 4\bigr)^2
\;+\; \bigl(3x_1 + 2x_2 + 5x_3 – 4\bigr)^2 \Big] \notag \\[6pt]
&\quad +\, w_{AN} \Big[ \bigl(2.2x_1 + 3.0x_2 + 4.1x_3 – 3.0\bigr)^2
\;+\; \bigl(5.0x_1 + 4.8x_2 + 5.3x_3 – 5.1\bigr)^2
\; \notag \\[6pt]
&\quad +\, \bigl(0.15x_1 + 0.20x_2 + 0.12x_3 – 0.18\bigr)^2 \Big] \notag \\[6pt]
&\quad +w_C \Bigl(6x_1 + 2x_2 + 5x_3\Bigr), \notag \\[4pt]
\textbf{subject to:} \quad
&x_1 + x_2 + x_3 = 1, \notag \\
&0 \,\leq\, x_i \,\leq\, 1, \quad i \in \{1,2,3\}.
\end{alignat}

This comprehensive and flexible framework enables the integration of additional or alternative attributes by expanding the sets \(\mathcal{S}\) or \(\mathcal{A}\). By adjusting the weights \(w_S\), \(w_{AN}\), and \(w_C\), decision-makers can place greater emphasis on sensory, analytical, or economic objectives, thus aligning the solution with strategic goals and market demands.

5.1.5 Analytical Solution via Lagrange Multipliers 

When the sum of proportions equals 1, an analytical solution is obtainable via the
method of Lagrange multipliers. Introducing a Lagrange multiplier \( \lambda \):

\begin{equation}
\mathcal{L}(x_1, x_2, x_3, \lambda)
= Q(S, AN, C) \;-\; \lambda \left( \sum_{i=1}^{3} x_i – 1 \right).
\end{equation}

Setting
\[
\frac{\partial \mathcal{L}}{\partial x_i} = 0
\quad \text{and} \quad
\frac{\partial \mathcal{L}}{\partial \lambda} = 0
\]
for \( x_1, x_2, x_3, \lambda \) provides closed-form expressions (assuming polynomial or
similarly tractable objectives and constraints). In practical terms, this approach can guide blend decisions without resorting solely to numerical solvers.

5.2 Solving the Program: Scenario Analysis

In multi-objective optimization, balancing sensory quality, analytical composition, and cost constraints can markedly influence the optimal coffee blend formulation. The subsections below examine how varying objective weights or introducing cost limitations reshapes the solution space.

5.2.1 Scenario I: Omitting Cost (\( w_C = 0 \)) 

When cost considerations are excluded, the objective function emphasizes only sensory and analytical attributes. Mathematically, this is represented by:

• Sensory Priority: Increasing the importance of the sensory profile, \( w_{S} \), strongly favors Kenyan AA, and less so the Vietnamese beans.
• Analytical Priority: Setting \( w_{AN} = 1 \) shifts the solution toward
  Colombian beans, due to its analytical profile.

Figure 1: Bean distribution as a function of sensory and analytical weight \( w_{S}, w_{AN} \)

5.2.2 Scenario II: Imposing Cost Constraints 

Introducing a cost constraint reshapes the feasible region by restricting more expensive beans. In particular, imposing:
\[
4.1x_1 + 3.1x_2 + 2.6x_3 \leq 5
\]
steers the formulation toward lower-cost blends.

• Premium Beans: Although they offer superior flavor and analytical characteristics, higher-cost options may exceed budget limits.
• Lower-Cost Beans: These ingredients fit within the financial constraint, yet may compromise optimal flavor or analytical goals.

Figure 2: Bean allocation under a cost constraint as a function of the sensory weight \( w_{S}\).

Figure 3: Bean allocation under a cost constraint as a function of the analytical weight \(w_{AN} \).

Figure 4: Quality scores as a function of bean selection, with and without cost constraints.

Hence, cost considerations underscore critical trade-offs such as flavor excellence vs.\ cost efficiency and analytical precision vs.\ economic feasibility. The final solution must reconcile these competing factors in light of market strategy, customer preferences, and operational constraints.

6 Balancing Objectives

Organizations must align optimization priorities with overarching strategic goals. Sensitivity analysis is integral to this process, as it gauges how variations in parameters (e.g., cost limits, target attributes, or blending ratios) affect the resulting solutions. Monte Carlo simulations, gradient-based sensitivity methods, and Multi-Criteria Decision Analysis (MCDA) frameworks (e.g., the Analytic Hierarchy Process) can all provide systematic guidance in determining weight assignments. These approaches enhance robustness by revealing how the optimal blend responds to fluctuations in external conditions or changing objectives.

7 Future Trends and Challenges

7.1 Emerging Technologies in Optimization

Recent advances in AI-driven methods offer dynamic optimization strategies capable of adapting to evolving production environments and consumer demands. Concurrently, quantum computing demonstrates potential for tackling high-dimensional multi-objective optimization problems, promising exponential speed-ups relative to classical algorithms if key scalability hurdles can be surmounted.

7.2 Scalability, Computational Cost, and Uncertainty Handling

Large-scale multi-objective optimization frequently requires parallel computing and sophisticated heuristic or metaheuristic algorithms to mitigate computational demands. Uncertainty remains a further challenge: robust and stochastic optimization models address variations in market dynamics, resource availability, or quality-control metrics by explicitly accommodating parameter fluctuations.

8 Conclusion 

In modern decision-making contexts, multi-objective optimization enables a structured assessment of trade-offs among diverse objectives such as flavor, cost, and analytical attributes. By leveraging models that reflect both quantitative data and organizational priorities, stakeholders can develop solutions aligned with consumer preferences and financial constraints. As AI, quantum computing, and advanced metaheuristic techniques evolve, multi-objective optimization will remain a pivotal methodology for addressing the complexities of real-world problems.

References

1. Deb, K. (2001). Multi-Objective Optimization Using Evolutionary Algorithms. John Wiley & Sons.
2. Miettinen, K. (1999). Nonlinear Multiobjective Optimization. Springer.
3. Coello Coello, C. A., Lamont, G. B., & Van Veldhuizen, D. A. (2007).
   Evolutionary Algorithms for Solving Multi-Objective Problems. Springer.
4. Marler, R. T., & Arora, J. S. (2004). Survey of multi-objective optimization methods for engineering. Structural and Multidisciplinary Optimization, 26(6), 369-395.
5. Zitzler, E., Deb, K., & Thiele, L. (2000). Comparison of multiobjective evolutionary algorithms: Empirical results. Evolutionary Computation, 8(2), 173-195.

Before we dive in, let’s get the basics straight.

What is Machine Learning (ML)?

Machine Learning is an AI system that enables computers to learn from data and make predictions or decisions without being explicitly programmed.

• Traditional Programming: Follow fixed, predefined instructions.
• Machine Learning: Learn patterns from data and improve performance over time.

How It Works

1. Feed the model with data.
2. The model analyzes the data, identifies patterns, and generates insights.
3. As the model processes more data, it adjusts and improves its predictions.

Main Types of Machine Learning

ML is typically categorized into three types:

1. Supervised Learning

Definition: Models learn from labeled data (data with correct answers).

Examples:

• Regression: Predicts continuous values (e.g., housing prices).
• Classification: Predicts categories (e.g., spam vs. not spam).

2. Unsupervised Learning

Definition: Models analyze unlabeled data to find hidden patterns.

Example:

• Clustering: Grouping customers based on buying behavior.

3. Reinforcement Learning

Definition: Models learn by interacting with an environment and receiving rewards or penalties for their actions.

Example:

• Training robots to walk or optimizing household energy usage.

Why is Machine Learning Important?

ML is transforming industries by enhancing efficiency, decision-making, and problem-solving. Here are some examples:

• Manufacturing: Predict machine failures to enable preventive maintenance and reduce costs.
• Finance: Detect fraudulent transactions in real-time and personalize financial services, like tailored loans.
• Logistics: Optimize delivery routes to save time and fuel.
• Healthcare: Analyze medical data to assist with early diagnoses and personalized treatment plans.

What Are Optimization Techniques?

Optimization techniques are mathematical methods used to improve machine learning models. They find the best possible solution to a problem—often by minimizing errors (loss functions) or maximizing performance (accuracy).

Common Optimization Techniques

1. Gradient Descent

• What It Does: Adjusts a model’s parameters step-by-step to minimize errors.
• Why It Matters: It’s the backbone of training most ML models by ensuring models converge on the optimal solution.

2. Simulated Annealing

• How It Works: Inspired by the cooling of metals. It explores solutions, including some less-optimal ones initially, to avoid getting stuck in a local “best” solution.
• Best For: Complex problems with unpredictable solution spaces (e.g., scheduling, route optimization).

3. Bayesian Optimization

• What It Does: Uses probabilities to predict the quality of solutions before testing them.
• Key Benefit: Ideal for hyperparameter tuning, saving significant time and computational resources.

4. Genetic Algorithms

• How It Works: Mimics natural selection:
• Start with multiple solutions.
• Select the best.
• Combine them (crossover) and tweak (mutate) to improve results.
• Best For: Problems with many possible answers, like design optimization and scheduling.

5. Adagrad

• What It Does: Adjusts learning rates individually for each parameter based on past gradients.
• Why It’s Useful: Perfect for handling sparse data (e.g., text or image processing), as it focuses on less-frequent features for better learning.

Now that we’ve laid the foundation for ML and optimization, let’s dive deeper. In the following sections, we’ll explore how optimization techniques supercharge machine learning performance, address real-world challenges, and create tangible value across industries.

Fundamentals of Mathematical Optimization

What is Mathematical Optimization?

At its core, mathematical optimization is the process of finding the best possible solution for a given problem under a defined set of conditions or constraints. This involves:

1. Maximizing or minimizing an objective function: A mathematical formula representing the goal (e.g., minimize cost, maximize accuracy).
2. Satisfying constraints: Rules that restrict the possible solutions (e.g., limited resources or time).

Mathematical Expression

\(Optimize: f(x)subject\) \(to\) \(constraints: g(x)≤c\)

Where:

\(f(x)\): Objective function to maximize or minimize.
\(g(x)\): Constraints limiting \(x\).
\(c\): Boundaries of the constraints.

For instance, in ML, \(f(x)\) might represent the loss function (error rate), and the goal is to minimize it during model training.

Core Applications in Machine Learning

Mathematical optimization provides the foundation for many critical processes in ML. Let’s break down its primary applications:

1. Training Models: Minimizing Loss Functions

• What It Means: During training, ML models aim to minimize a loss function, which quantifies how far off predictions are from actual values. Optimization algorithms adjust parameters (e.g., weights and biases) to reduce this error step-by-step.
• Example: A neural network predicting housing prices uses optimization to minimize the difference between predicted and actual prices.

2. Hyperparameter Tuning: Finding Optimal Configurations

• What It Means: Hyperparameters are settings (e.g., learning rate, batch size) that control how ML models learn. Optimization techniques help find the best hyperparameters to maximize performance.
• Why It’s Critical: Poor hyperparameter settings can lead to models that overfit, underfit, or train too slowly.
• Example: Bayesian Optimization predicts the best combination of hyperparameters to reduce computational expense while boosting model accuracy.

3. Resource Allocation: Ensuring Computational Efficiency

• What It Means: ML models often require significant resources, like memory and GPU time. Optimization ensures resources are allocated efficiently to balance costs and performance.
• Example: Distributed ML systems optimize resource usage across multiple GPUs to reduce training time while staying within cost limits. Cloud platforms dynamically allocate server resources to balance loads and save energy.

Broader Perspective

While ML leverages optimization heavily, it’s important to recognize that mathematical optimization is a universal framework. Beyond ML, it’s used to:

• Optimize delivery routes in logistics.
• Allocate resources for financial portfolios.
• Streamline production schedules in manufacturing.

This versatility makes optimization an indispensable tool across industries, enabling smarter decision-making and improved efficiency.

Bringing It Together

Mathematical optimization provides the theoretical backbone for many ML tasks, from minimizing loss functions to efficiently allocating resources. Understanding these fundamentals is crucial to unlocking the full potential of machine learning models in solving real-world problems.

The Intersection of Optimization and AI/ML

Mathematical optimization and machine learning (ML) don’t just coexist—they fuel each other. Optimization serves as the engine driving ML’s efficiency and effectiveness, enabling smarter algorithms, faster decisions, and better results across industries. Let’s break this down.

How Optimization Techniques Empower ML

1. Training Efficiency

• What It Means: Optimization is the backbone of training ML models. By minimizing the loss function, it helps algorithms learn faster and with fewer computational resources.
• Example: Gradient Descent and its variants (e.g., Adam, RMSProp) iteratively refine model parameters, ensuring models converge to the best solution efficiently.
• Why It Matters: Faster convergence means reduced training time, lower computational costs, and quicker deployment of ML solutions.

2. Decision-Making Models

• What It Means: Optimization enables ML models to make decisions by maximizing or minimizing specific objectives under constraints.

Applications

• Dynamic Pricing: Optimization models help companies set prices in real-time, balancing supply, demand, and profit margins.
• Portfolio Management: Algorithms optimize asset allocation, considering risk and return.

• Example: Netflix uses optimization-powered recommendation systems to determine the best content to suggest for individual users.

3. Reinforcement Learning

• What It Means: Reinforcement learning (RL) leverages optimization to enable agents to learn optimal strategies by maximizing cumulative rewards over time.
• How It Works: RL problems are framed as Markov Decision Processes (MDPs), where optimization determines the best actions for an agent to take in a given state.
• Example: Autonomous vehicles use RL to optimize driving strategies, balancing speed, safety, and fuel efficiency.
• Why It Matters: Without optimization, RL agents would struggle to identify effective policies in complex, multi-step environments.

Further your optimization and AI/ML learning with direct insights straight to your inbox. Sign-up here. 

Case Study: How Machine Learning Slashed Costs and Delivered Faster

Disclaimer This fictional scenario illustrates the transformative potential of Machine Learning (ML) in logistics. It’s crafted for educational purposes and not based on a real account.

SwiftRoute was bleeding money, wasting time, and frustrating customers. Inefficient delivery routes, soaring fuel costs, and missed deadlines were tanking their profits and customer satisfaction.

• Inefficient Routes: Drivers spent 20% longer completing deliveries than competitors.
• Soaring Costs: Fuel consumption and vehicle maintenance cut deep into profit margins.
• Missed Deadlines: On-time delivery rates dropped below 75%, leading to angry customers and churn.

Faced with growing competition and shrinking margins, SwiftRoute needed a solution to regain control and rebuild trust.

The Solution: ML-Powered Logistics

SwiftRoute adopted a Machine Learning approach to optimize operations:

Data Cleaning:

• Consolidated years of messy GPS, delivery, and fleet data.
• Eliminated duplicates and errors to create a reliable dataset for analysis.

ML Model Development:

• Built predictive models to dynamically adjust delivery routes in real-time based on traffic, weather, and package loads.
• Applied linear programming and reinforcement learning for route optimization.

Pilot Testing:

• Rolled out the ML solution with a small fleet to identify weaknesses and fine-tune the system.

Company-Wide Deployment:

• Scaled the solution to the entire fleet and trained drivers using gamified tools to ensure buy-in and seamless adoption.

The Results: Tangible Wins

SwiftRoute transformed its logistics operations and achieved measurable success:

• Faster Deliveries: Average route times decreased considerably.
• Savings: Reduced fuel consumption and optimized vehicle usage resulted in decreased expenditure.
• Improved Customer Satisfaction: On-time deliveries improved.
• Total Operational Savings: Efficiency gains across the board.

The Obstacles and How They Overcame Them

SwiftRoute’s journey wasn’t without challenges, but strategic actions helped them overcome hurdles:

Messy Data: Time and effort were invested in cleaning and standardizing years of inconsistent data.

Driver Resistance: Gamified training programs incentivized drivers to embrace the new system and provided ongoing support.

Model Refinement: Iterative updates improved the model’s ability to handle real-time traffic and weather data.

Takeaways for Operations Researchers

SwiftRoute’s success offers actionable insights:

Start Small: Conduct pilot tests to validate ML solutions before scaling across operations.

Prioritize Data Quality: Clean, reliable data is the foundation of any successful ML initiative.

Invest in Team Training: Engage stakeholders early to ensure adoption and long-term success.

Measure KPIs Relentlessly: Track key metrics (e.g., delivery time, cost savings, customer satisfaction) to prove ROI and refine the solution.

Stay updated with optimization and AI/ML insights straight to your inbox. Sign-up here. 

When it comes to mathematical optimization, choosing the right solver is not just a technicality—it’s the key to unlocking efficiency, accuracy, and performance. Whether you’re tackling a sparse or dense problem, the solver you select profoundly impacts the resources you’ll need and the results you’ll achieve. This blog dives into the critical decision-making process of solver selection, why it matters, and how you can avoid common pitfalls.

Why Optimization Solver Choice is Essential

Optimization problems are as diverse as the industries they serve, from structural engineering to data science. But at their core, they all share a common challenge: efficiently balancing the computational cost with the scale and complexity of the problem. One of the most important questions in this process is: Is your problem sparse or dense? Answering this question early enables you to align your solver choice with the unique structure of your problem, saving time and computational resources.

Sparse vs. Dense Problems: Why It Matters

• Sparse Problems: These involve matrices where most elements are zero. Sparse solvers leverage this structure to avoid unnecessary computations, dramatically reducing memory and processing time.
• Dense Problems: These have many interdependencies among variables, requiring solvers designed to handle large, fully populated matrices efficiently.

The Hidden Value of Sparsity

For large-scale optimization, recognizing and exploiting sparsity can transform performance. Consider this example:

• A matrix with thousands of variables but a density below 1% enables sparse solvers to skip over zeros, cutting computational time and memory use significantly.
• Dense solvers, in contrast, struggle as problem size grows, often consuming exponentially more resources.

The results? Sparse solvers consistently outperform dense counterparts for problems with low-density structures.

Real-World Applications: Lessons Learned

• Linear Programming: For problems with densities below 1%, sparse solvers can unlock dramatic performance gains. Sparse solvers complete tasks in seconds where dense solvers take minutes or hours – note not all linear programming problems are sparse.
• Cholesky Factorization: Sparse Cholesky routines scale seamlessly for large problems, while dense methods hit performance walls as problem size and complexity increase.

The Danger of Misaligned Solvers

Even experienced users can misinterpret their problem’s structure. It’s not uncommon to overlook the sparsity introduced during problem reformulation – see the example in this blog.

Key Takeaways for Optimal Solver Choice

• Assess Sparsity Early: Determine the density of your problem’s matrices and choose a solver that aligns with its structure.
• Consider Reformulations: The data your solver processes may differ from the original problem. Analyse what the solver actually sees.
• Leverage Dedicated Solvers: When available, they provide unparalleled efficiency by tailoring methods to your specific problem type.

Empower Your Optimization Workflow

Choosing the right solver isn’t just about saving time—it’s about transforming the way you approach optimization. By aligning your solver choice with the problem’s structure, you’ll achieve faster, more efficient, and scalable solutions. Sign up to receive more mathematical optimization resources.

Robust, Tested, Performant Optimization Solvers

The Optimization Modelling Suite – delivered with the nAG Library – features an extensive collection of Mathematical Optimization solvers. The solvers are accessed via an intuitive interface designed for ease of use. Key mathematical optimization areas covered include:

• Linear Programming (LP) – dense and sparse, based on simplex method and interior point method;
• Quadratic Programming (QP) – convex and nonconvex, dense and sparse;
• Second-order Cone Programming (SOCP) – covering many convex optimization problems, such as Quadratically Constrained Quadratic Programming (QCQP);
• Nonlinear Programming (NLP) – dense and sparse, based on active-set SQP methods and interior point method (IPM);
• Global Nonlinear Programming – algorithms based on multistart, branching, and metaheuristics;
• Mixed Integer Linear Programming (MILP) – for large-scale problems, based on a modern branch-and-bound approach;
• Mixed Integer Nonlinear Programming (MINLP) – for dense (possibly nonconvex) problems;
• Semidefinite Programming (SDP) – both linear matrix inequalities (LMI) and bilinear matrix inequalities (BMI);
• Derivative-free Optimization (DFO) – solvers for problems where derivatives are unavailable and approximations are inaccurate;
• Least Squares (LSQ), data fitting, calibration, regression – linear and nonlinear, constrained and unconstrained.

View nAG’s optimization solver documentation.

Continue your optimization learning with insights direct to your inbox – sign-up here.

nAG is proud to announce a new student award for 2025, designed to drive forward our commitment to diversity and inclusion in HPC and the broader tech landscape. The nAG WHPC Student Award, a collaboration between the nAG Women in Tech and WHPC Chapter Groups and NI-HPC’s WHPC Chapter, joins nAG’s prestigious Student Awards programme to recognise outstanding achievements by female students in HPC. This award highlights exceptional contributions from students at NI-HPC, representing Queen’s University Belfast and Ulster University, as we work to inspire a more inclusive and diverse future in high-performance computing.

n² Group invests selectively in technical computing companies with deep business impact in a variety of sectors, providing operational support and a collaborative approach to innovation and business transformation. The addition of X-ISS will further strengthen the Group’s already strong HPC/AI credentials, with nAG, STAC and BioTeam already adding to this space. 

X-ISS is a pioneer in Managed Services specifically designed for HPC/AI. With their in-depth understanding of hardware and software complexities within HPC and AI, they deliver highly impactful end-to-end services to clients through the integration, optimization and management of HPC/AI systems. The integration of X-ISS into n² Group aligns with the Group vision of improving the accessibility, quality and robustness of computing solutions to enable greater productivity ​​in industry.  

X-ISS will operate as an autonomous business within the n² Group, maintaining its brand, identity and ethos. n² Group’s status as an independent, member-backed organisation with no external financial stakeholders allows X-ISS to continue providing impartial advice based on the technology needs and challenges of its clients. Inter-group synergies will enable greater innovation and collaboration, advancing the Group’s position and long-term HPC/AI market impact. 

“X-ISS strengthens the n² community in the strategically important area of HPC/AI”, said Adrian Scales, Snr Director of Investments and Partnerships at n² Group. “As a respected boutique HPC service provider, X-ISS is helping clients navigate an increasingly complex landscape in terms of technologies and software integrations with AI and analytics. The acquisition strongly complements the Group’s existing HPC professional services capability, and we are delighted to have them on board.”

“This is an important milestone for X-ISS.”, said Deepak Khosla, CEO X-ISS, “The partnership with n² Group will enable us to enhance our flagship ManagedHPC solution by leveraging n²’s complementary services and product developments, allowing us to deliver even greater value to our customers. As businesses face increasing challenges with complex technologies like AI and cloud computing, we’re now better equipped to support them with the same quality, passion, and partnership that defines X-ISS. I am excited about the opportunities this can bring for current and future X-ISS customers.”

About n² Group  

At n² Group we are transforming computing and technology investment with a radical new approach. Our businesses are all established, purpose-driven market-leaders in computing products or services. We stimulate long-term sustainable growth through group-level support in strategy, business development, innovation, and operations. With no shareholders or external financial interests, we reinvest all profits back into the group or to the community, reinforcing our commitment to positive social impact through technological advancements.    

n² Group companies are at the forefront of computing and IT infrastructure, helping clients in various sectors to be more productive, innovative or reduce risk through advanced software and services. Rapidly expanding in high-performance computing, artificial intelligence, and scientific computing, our businesses maintain their unique brands and identities, but benefit from the expanded network available through the group.   

n² Group Companies   

• BioTeam: ​​Scientific computing consultancy integrating technologies, data, and cultures to accelerate science. 
• nAG: Advanced products and services in algorithms, optimization, HPC and AI. 
• STAC: Independent financial services technology research and community events. 
• VSNi: Proven statistical solutions and data expertise driving innovation and success. 
• X-ISS: Industry leading management and analytics solutions for HPC/AI systems.

Additional Benefits of Bursting from Cloud to On-Premise: 

Staged Migration and Infrastructure Flexibility: Bursting from cloud to on-premise simplifies the process of migrating workloads. Organisations can utilise existing on-premise infrastructure without disrupting users while gradually reducing its size or decommissioning. This approach also allows companies to keep sensitive data and models on-premise, ensuring compliance with regulatory requirements and safeguarding intellectual property. 

Centralised and Resilient Infrastructure: By centralising infrastructure in the cloud, organisations can leverage cloud-native tools and services for enhanced automation, resilience, logging, and monitoring, thereby reducing overall operational risk. Additionally, with data primarily stored in the cloud, the autoscaling of pre and post-processing services is no longer constrained by on-premise capabilities. 

Cost Efficiency and Data Management: Bursting from the cloud to the on-premise shifts costs from output to input, reversing the typical egress/ingress cost structure and optimising financial efficiency. Furthermore, with data stored in the cloud, businesses can take advantage of a wide array of cloud-native backup and archiving services, eliminating the need to continually expand on-premise storage. 

Alignment with Cloud-First Strategies: For companies adopting a cloud-first strategy, bursting from cloud to on-premise integrates HPC into a unified IT infrastructure, creating a more streamlined and cohesive environment. This approach not only simplifies the management of HPC resources but also aligns with broader corporate IT initiatives, making the transition to cloud-based operations less complex. 

Elimination of Additional Tooling and Downtime: Leveraging cloud resources eliminates the need for additional third-party tools to manage bursting and ensures no downtime for updates or maintenance, enhancing the overall efficiency and reliability of the HPC environment. 

Unified User Experience: From the user’s perspective, this approach creates a more homogeneous environment, satisfying the need for cloud utilisation while presenting it as a single, integrated system, minimising the disruption to their work. This makes the HPC environment more cohesive and easier to manage. 

Embracing Hybrid Cloud: Opportunities and Challenges 

But before we dive into bursting from cloud to on-prem and an example of how one company, nAG, has delivered such a model, let’s look at some of the Opportunities and Challenges.  

Opportunities of Hybrid Cloud: 

Scalability: Hybrid cloud solutions allow organisations to scale their computational resources up or down based on demand, which is crucial for industries with variable workloads. But how much do you need this extra resource? 

Cost Efficiency: Companies can optimise IT expenditure and often achieve massive savings through the best use of cloud resources vs. on-premise, taking into account existing and future demand, relative functionality, and sensitive operations. What is the best solution for your goals whilst remaining on budget? 

Data Security: Sensitive data can be kept on-premise, reducing the risk of exposure, while the cloud can be leveraged for less critical tasks. How critical is the security of your data, and what steps are you taking to ensure its protection? 

Performance Optimisation: Tasks requiring low latency and high performance can be executed on-premise while the cloud handles other computations, ensuring overall efficiency. How does your current infrastructure handle high-performance tasks, and where can improvements be made? 

Challenges of Hybrid Cloud: 

Integration Complexity: Ensuring seamless integration between cloud and on-premise systems can be technically challenging, requiring custom interfaces and workflows. 

Data Management: Managing data across hybrid environments can be difficult, especially with large datasets. Consistency, latency, and bandwidth issues need to be addressed. 

Cost Management: Hybrid solutions can be cost-effective but require careful monitoring and management to avoid unexpected expenses. 

nAG’s Hybrid Cloud, Bursting to On-prem Solution: A Case Study 

Successfully implementing a comprehensive HPC solution to fully realise these advantages, and to solve those challenges requires careful planning and expertise. nAG has been at the forefront of developing innovative HPC solutions, including the groundbreaking concept of bursting from cloud to on-premise.  

This approach to HPC solutions emphasises performance engineering and optimisation. By working closely with clients, nAG ensures that HPC systems are tailored to performance and cost requirements.  

Step 1: Assessment and Planning 

The project began with an in-depth assessment and planning phase. This step involved close consultation with the client to fully understand their requirements and the specific workloads they needed to manage. Workload analysis was critical to determining the computational needs, data movement patterns, and potential bottlenecks. Based on this analysis, a detailed proposal was developed outlining the architecture, timelines, and expected outcomes. 

Step 2: Design Architecture 

The next phase focused on designing the architecture. The network configuration was carefully planned to support the data transfer requirements between cloud and on-premise systems, ensuring low latency and high throughput. Topology considerations included redundancy and failover capabilities to maintain system availability. The software stack included SLURM for workload management and other essential HPC tools and libraries. 

Step 3: Procurement and Setup 

With the architecture in place, we moved on to procurement and setup. The network setup required custom routing to ensure seamless connectivity between cloud and on-premise environments. A parallel file system was configured for storage for efficient data access and distribution. A custom data movement solution was implemented to ensure that data is always available when needed, minimising latency and optimising performance. Importantly, all cloud infrastructure was deployed using an Infrastructure-as-Code (IaC) approach, ensuring consistency, repeatability, and scalability. 

Step 4: Software Installation and Configuration 

The software installation and configuration phase was critical, particularly in the context of bursting from cloud to on-prem.  

The SLURM configuration was the most pivotal and innovative aspect of this phase. SLURM needed to be configured and customised to facilitate communication between cloud and on-premise nodes, enabling bidirectional data and task management. This ensured that jobs could be dynamically distributed across cloud and on-premise resources based on real-time demand. 

Building the images for the on-premise nodes was another critical step. These custom images were designed to communicate effectively with the SLURM controller in the cloud, enabling seamless integration and automated on-premise node provisioning. This automation was crucial for scalability and efficiency, allowing the system to quickly adapt to changing computational needs. 

Highlighted Techniques: 

SLURM customisation: Enabled two-way communication between cloud and on-premise nodes, a key innovation for the bursting system. 

Custom image building: Ensured on-premises nodes could integrate seamlessly with cloud-based SLURM, allowing for automated provisioning. 

Step 5: Integration and Testing 

Finally, integration and testing was conducted to ensure the entire system functioned as expected. This included implementing corporate firewall rules to protect both cloud and on-premise environments. Authentication services were configured to work across the hybrid system, ensuring secure access and data integrity. Rigorous testing was performed to validate the bursting functionality, focusing on performance, reliability, and security. 

Summary of the Case Study 

This case study highlights the meticulous planning, innovative engineering, and technical expertise required to develop a hybrid cloud solution that bursts from cloud to on-premise. By focusing on key elements such as SLURM customisation, custom image building, and automated provisioning, the project successfully delivered a solution that meets the high demands of many industries. 

The result is a flexible, scalable, and secure HPC environment that empowers organisations to fully leverage the benefits of cloud computing while maintaining control over critical data and workloads. 

Key Considerations for Implementing Comprehensive HPC Solutions 

So, what were the key focuses of this project? How did it all happen? We would recommend these steps to ensure optimal performance and prepare organisations for future technological advancements: 

Evaluation: Before adopting HPC, thorough benchmarking or proof-of-concept studies is crucial. Setting up test environments to run specific workloads provides valuable insights into performance metrics and cost implications, helping organisations make informed decisions about hybrid solutions. 

Professional Implementation: Optimising HPC systems, whether on-premise, hybrid, or cloud-based, requires strategic and knowledgeable guidance. Expert services ensure systems are configured to achieve peak performance, facilitate smooth transitions to cloud environments, and design elastic HPC setups. 

Managed Services: Maintaining an HPC system involves comprehensive monitoring, regular updates, and reviews. Proactive management ensures systems run smoothly and efficiently, minimising downtime and allowing organisations to focus on innovation. 

Strategic Enhancement: Continuous assessment and optimisation of HPC environments are essential for sustained success. Leveraging historical data and feedback loops provides actionable insights, driving meaningful system improvements and aligning HPC infrastructure with evolving organisational objectives. 

What Does This All Mean for HPC Users? The Power of HPC in Industry 

HPC has revolutionised various industries by enabling advanced research and development. For example, HPC-driven simulations can significantly reduce the time and cost associated with crash testing and vehicle design in the automotive industry, allowing for thousands of virtual crash scenarios. Similarly, in the oil & gas sector, HPC accelerates seismic data analysis, enhancing exploration and production efficiency. In aerospace, HPC supports the design and testing of new aircraft, while nuclear energy relies on HPC for reactor design and safety analysis. 

The Evolution and Impact of HPC Technologies 

The rapid evolution of HPC technologies, driven by advancements in processor design, parallel computing, and machine learning, has significantly enhanced the capabilities of HPC systems. As Moore’s Law approaches its limits, the focus is shifting from hardware to software optimisation, making performance engineering crucial for maximising HPC efficiency. The integration of AI and ML with HPC is also opening new frontiers, enabling predictive maintenance, optimised supply chains, and advanced decision-making across various sectors. 

Embracing Hybrid Cloud: An Invitation to Innovate 

The hybrid cloud model represents a significant opportunity for industries to enhance their computational capabilities. Combining the best aspects of cloud and on-premise HPC allows organisations to achieve unprecedented efficiency, scalability, and performance. This approach enables businesses to be more agile, responding quickly to changing demands and market conditions. 

Summary and Conclusion 

Hybrid cloud solutions are transforming industries by merging the scalability of cloud computing with the control and performance of on-premise systems.  

The innovative concept of bursting from cloud to on-premise represents a significant advancement in HPC. By enabling organisations to keep sensitive tasks on-premise and utilise cloud resources for less critical workloads, this strategy addresses critical needs like compliance, data sovereignty, and performance optimisation. This capability is particularly beneficial for sectors such as aerospace and nuclear energy, where regulatory requirements and performance demands are stringent. 

Successfully implementing a comprehensive HPC solution requires careful planning and expertise. Thorough benchmarking helps organisations understand the feasibility and benefits of hybrid solutions. Strategic professional services ensure systems are configured for peak performance, while proactive managed services maintain system reliability and efficiency. Continuous strategic enhancement is essential to adapt to evolving technologies and business goals, ensuring HPC systems remain cost-effective and efficient. 

Mastering these elements is crucial for businesses aiming to achieve their strategic objectives with exceptional efficiency. Seamlessly integrating cloud and on-premise HPC environments to burst either way, can not only meet current computational demands but also prepare businesses for future challenges and opportunities. 

In conclusion, embracing hybrid cloud and the innovative strategy of bursting from cloud to on-premise is more than a technological upgrade; it is a strategic move towards sustained innovation, agility, and competitive advantage in the fast-evolving HPC landscape. 

n² Group invests selectively in advanced and scientific computing companies with deep operational impact in a variety of sectors, providing operational support and central services to support innovation and strategic advancement of the group businesses.

STAC insights empower the finance sector to achieve faster, smarter, and more efficient solutions. Their integration into the n² Group solidifies the commitment of n² to improving performance and robustness of computing solutions for the financial services community. STAC will operate as an independent business within n², maintaining its brand, identity and ethos.

n² Group’s status as an independent, member backed organisation with no external financial stakeholders has enabled nAG to provide trusted solutions to FSI for decades. As such, STAC will maintain its impartial, neutral status while benefiting from the strong technical and research community that nAG, VSNi and further acquired business can provide. 

Jem Davies, Chair of n² Group, said “The acquisition of STAC is a great milestone in our committed long-term strategy of expanding and strengthening the group.”

Adrian Tate, CEO of n² Group added “I am delighted to welcome STAC, an unparalleled resource for the finance industry, to the n² Group. Countless businesses trust STAC for its independent technology assessments using community-developed benchmark standards and use STAC’s events as the primary place to discuss strategic technology. n² Group will preserve STAC’s unique position while strengthening and accelerating the value that it delivers to its community and other n² Group clients.”

Peter Lankford, Founder of STAC: “n² Group’s technical and algorithmic expertise, together with its deep research roots and community orientation make it uniquely qualified to help STAC grow further while protecting its independence. I look forward to helping STAC and n² Group make the most of the synergies in the months ahead.”

Jack Gidding, CEO of STAC: “The acquisition by n² Group comes at an exciting time of innovation in technologies important to low-latency trading, big compute, big data, and increasingly sophisticated AI/ML and cloud services. Joining forces with n² Group will enable STAC to expand the information, tools, and services that our subscribers can leverage in strategic areas of their businesses.”

About n² Group

At n² Group we are transforming computing and technology investment with a radical new approach. Our businesses are all established, purpose-driven market-leaders in computing products or services. We stimulate long-term sustainable growth through group-level support in strategy, business development, innovation and operations. With no shareholders or external financial interests, we reinvest all profits back into the group or to the community, reinforcing our commitment to positive social impact through technological advancements.   

n² Group companies are at the forefront of computing and IT infrastructure, helping clients in various sectors to be more productive, innovative or reduce risk through advanced software and services. Rapidly expanding in high-performance computing, artificial intelligence, and scientific computing, our businesses maintain their unique brands and identities, but benefit from the expanded network available through the group.  

n² Group Companies

• nAG: Advanced products and services in algorithms, optimization, high-performance computing and AI.
• STAC: Independent financial services technology research and community events.
• VSNi: Proven statistical solutions and data expertise driving innovation and success.

What is Mixed Integer Linear Programming?

Mixed Integer Linear Programming (MILP) is a mathematical method used to solve optimization problems where some of the variables are required to be integer values. It is particularly useful in scenarios where decisions are discrete, such as scheduling, resource allocation, and, as in our case, managing operations of BESS. nAG introduced a new high-performance MILP solver at Mark 29.3 of the nAG Library, and we’ve used this in our latest optimization example.

Mathematical Modelling of BESS

The mathematical model for optimizing a BESS involves several components:

• Objective Function: The goal is to minimize the total operation cost of the utility including generators and batteries.
• Variables: These include battery and generator schedules, battery specification and imported power.
• Constraints: The model includes load balance, power limit, up and down time limit and power rating limit.

Example Scenario

Consider a simple scenario where a BESS is used to store electricity generated or imported at a lower cost, and supply to the utilities when cost is high. The optimization model needs to decide the best times to charge or discharge the battery to maximize profits over a given time horizon.

Implementing the Model in Python

Using Python, with the new MILP solver in the nAG Library, you can implement and solve the BESS optimization model. The process involves:

1. Defining the problem parameters (e.g., time intervals, electricity prices, battery specifications).
2. Setting up the objective function and constraints in a form that the MILP solver can understand.
3. Using the solver to find the optimal charging and discharging schedule.

Benefits and Challenges

Implementing BESS with MILP offers several benefits, including improved efficiency and profitability in energy storage and the ability to integrate seamlessly with renewable energy sources. However, challenges such as modelling accuracy, computational complexity, and the dynamic nature of energy markets also need to be addressed. Using Mixed Integer Linear Programming provides a clear pathway to enhance energy storage management, making it more cost-effective and aligned with energy demands. As technology advances, the integration of such models will become increasingly important in our shift towards sustainable energy solutions.

View the Modelling Process. Try the Solver

At Mark 29.3 the nAG Library features a new Mixed Integer Linear Programming (MILP) solver. Try the solver with a no-obligation 30-day trial or arrange a call with our Optimization team to discuss your challenge. Follow the links to learn more.