Convex optimization is an important branch of operations research. It
generalizes linear programming and offers powerful tools for modelling problems and discovering optimal solutions to real world problems. Mathematically it is an interesting topic because it ties together many branches: linear algebra, multivariable calculus, and numerical analysis, to name a few. Modelling a problem as a convex optimization problem can be challenging but offers many benefits. Algorithm design is critically important to ensure precision of solutions that solve with minimal computation power. From an
engineering perspective it is also incredibly useful, since many more situations can be modeled than with linear programming alone. Exponentials, distances, and many other functions arise in engineering problems all the time, and require convex optimization to optimally design.
To fully appreciate convex optimization, I believe both a solid understanding of the theory and exploring an in-depth applied problem are necessary. After studying the theory in the fall, I became interested in the facility layout problem. I read about the nature of the problem, current algorithmic approaches to solving it, and how convex optimization could be used to formulate the problem. Finally, I created a mock scenario to solve: designing a
summer camp in the best way possible. This scenario was little more than a colorful motivation to solve the facility layout problem, yet provided a sense of realism with which to create the data needed for the problem. I took a convex formulation of the problem from my textbook, changed some facets of how it was created, and expanded upon the ideas they presented. I was successful in solving my instance of the problem to optimality.
This paper will be laid out in four broad sections, building our knowledge base from the ground up and mirroring my progression through the topic. In the first, I will explain convexity of sets and functions. The material in this section is based on that presented in the textbook Convex Optimization. I will define the vocabulary to discuss optimization problems, and explain what convex optimization is and why it is a useful tool. In the second section, I will overview the facility layout problem and describe commercially
available software and the benefits and drawbacks to each. In the third section, I will present my model, the contributions I have made, and define my data. Finally, in the last section I will examine my results graphically and numerically and reflect on future research directions.