An Introduction to Mathematical Optimization

For example, let's say you are the owner of a bakery. You need to decide which types of bread to make and how many of each for tomorrow, based on the ingredients you have on hand. In such a situation, how do you decide the number of each bread to make? You would want to make the decision that maximizes your profit, right? Mathematical optimization is one of the technologies that supports this kind of decision-making.

Mathematical optimization is a field of mathematical science concerned with mathematical optimization problems—which involve minimizing (or maximizing) an objective function (such as cost or profit) under limited conditions—and their solution methods. Mathematical optimization problems are important issues that arise in various fields such as engineering, social sciences, and statistics. They have been studied for a long time, but recently, with the development and advancement of the information society, the mathematical optimization problems that need to be solved are becoming larger and more complex. Therefore, developing solutions for these problems has become an urgent task.

Mathematical optimization consists of two complementary technologies: modeling, which formulates real-world problems into mathematical optimization problems, and algorithms for solving these modeled problems. In the Narushima Laboratory, we conduct research on the large-scale and complex mathematical optimization problems mentioned above from both aspects: modeling and algorithm development.

● Research on large-scale optimization problems and algorithms to solve them. Due to the trend of the advanced information society in recent years, the amount of data to be handled is increasing. As a result, so-called large-scale optimization problems have been on the rise. However, large-scale optimization problems cannot always be solved by existing methods. Therefore, we are conducting research on modeling and algorithms for solving large-scale optimization problems.

● Research on robust optimization problems. For example, consider an optimization problem where you want to maximize sales profit by determining the production volume of multiple products. In this case, the unit profit of a product is modeled as a constant value, but in reality, it typically changes due to external factors (such as fluctuations in transportation costs due to soaring gasoline prices). We are conducting research on methods that provide robust optimal solutions for such optimization problems that include uncertain situations.

● Research on equilibrium problems. Normally, when there is a single decision-maker, the problem is a mathematical optimization problem. However, when there are multiple decision-makers who influence each other, creating so-called competition, the resulting problem is called an equilibrium problem, which is more complex than a standard optimization problem. Therefore, we are conducting research on the modeling of equilibrium problems and algorithms for solving them.