Rebel Brown

Constrained optimization has long been a powerful approach to solve an array of problems in applied mathematics and to drive better business decisions and efficiency.

Using a variety of mathematical approaches such as Lagrange multipliers, substitution methods, and quadratic programming, constrained optimization is a perfect solution whenever there is a requirement to allocate or prioritize scarce or important resources within a dynamic and complex environment.

In simple terms, constrained optimization is the mathematical processes and calculations you use to decide how to do more with less, or how to use less to do more.

As we all know, this is a high priority in any organization.

## Constrained Optimization and Business Decisions

Constrained optimization enables you to represent business problems mathematically. Once these problems are in mathematical constructs, you can use them for analytical, decision-support computations as part of larger business applications and processes.

Consider some of the ways we use constrained optimization in our daily lives:

• When we make large buying decisions, they are often constrained by the requirement that we have to be able to afford what we choose to buy.
• Families decide how to prioritize study, work, play, travel, and more based on the constraints of their available time and income/budget.
• We often make decisions on the route weâ€™ll drive based on an objective (our destination), constraints (time, traffic number of lights) applied to our variables (the optional routes.)

In like manner, the majority of economic business decisions require applying constraints (cost, volume, time) to a set of variables (trucks, SKUs, people) with an objective of minimizing (cost) or maximizing (profit) outcomes. Such economic-driven constrained optimization problems are manifold in an organization.

A few examples:

• Companies often make manufacturing and packaging decisions so that they can maximize their profit margins, within the constraints of finite manufacturing or shelf space.
• Retail stores minimize costs as they are related the constraints of shelf space, availability, time to ship and more.
• Employers seek to minimize payroll costs while maximizing customer satisfaction during peak seasons by optimizing holiday work schedules based on the above objectives.

As you can see, there is an enormous number of decisions we make every day that are drive by constrained optimization, whether we recognize it or not.

Which is why QCI chose to focus Qatalyst, the first ready-to-run quantum software, on this powerful and much needed computational challenge.

## Optimization and Quantum Computing

If constrained optimization is such a powerful way to improve business decisions, why arenâ€™t businesses using it more widely?

Well, many are. But many are also falling behind in optimizing their business goals due to issues with their current constrained optimization approaches. In a nutshell, hereâ€™s why:

• The volume of business data from which to make better decisions is outpacing our computational capabilities today. Classical computers werenâ€™t designed to handle the volume of data we are seeing today. And the exorbitant time it takes to analyze large datasets proves it.
• Classical constrained optimization software isnâ€™t designed to effectively compute very large variable problems with specific discrete constraints (e.g., one truck vs 1.23 trucks) and objectives.
• In part, because of the computation load required to process any large datasets in a reasonable timeframe, classical computations only deliver one result, rather than a range of possible outcomes that may be more optimal.

Thatâ€™s where quantum computing comes in. Quantum computing techniques empower constrained optimization to a new level of accuracy and performance. That’s because:

• Quantum computers allow you to represent the variables, constraints and objectives within a multi-dimensional state. That means, you can use quantum techniques to arrange your data, requirements and goals in a way that represents the real world, including inter-relationships and dependencies. As the data is processed, it changes states to represent the probabilistic cause and effect relationships, and the real-world scenarios that result. Â So you get to view the probable outcomes of the various business decisions you could make, and select the one that best meets your demands.
• Quantum computers process complex computations to return a diversity of answers, not just one. Every answer that meets the optimized state you need is delivered to you. You get exposure to more viable options than with classical processors and can select the one that best matches your specific situation right now. This is a much better way to make decisions vs the classical software approaches that only provide a single answer as your only option.
• Quantum computers are far more accurate in the answers that they deliver to you. The results are a better representation of your real-world complexities and nuances. Unlike classical computations, there’s no need for abstractions, assumptions, or data samples compressed to be small enough to process on a classical computer.

Quantum computing offers better insights to make better decisions. That’s why there’s so much excitement about it.

## The Bottom Line

Navigating dynamic market changes requires harnessing the power of constrained optimization to improve your competitive advantage and bottom line results. The bottom line is that constrained optimization is about improving your bottom line.

Constrained optimization, supercharged with quantum-ready techniques, can and will deliver the deeper real-world insights every organization needs to get ahead and stay ahead. Those who embrace this technology early will be those who achieve market advantage.

In our next article, we’ll dive more deeply into how to transform business decisions into constrained optimization computations.