Chapter 1: Introduction to Mathematical Optimization

1.1 What is mathematical optimization?
1.2 Optimization Problem
1.3 Representative optimization problems
1.4 Structure of this book
1.5 Summary

Chapter 2 Linear Programming

2.1 Formulation of the linear programming problem
2.2 Group Law
2.3 Relaxation Problem and Dual Theorem
2.4 Summary

Chapter 3 Nonlinear Programming

3.1 Formulation of the Nonlinear Programming Problem
3.2 Unconstrained optimization problems
3.3 Constrained optimization problems
3.4 Summary

Chapter 4 Integer Programming and Combinatorial Optimization

4.1 Formulation of the integer programming problem
4.2 Evaluation of algorithm performance and problem difficulty
4.3 Combinatorial optimization problems that are solved efficiently
4.4 Quarterly Limitation and Abstinence Plane Methods
4.5 Approximation Algorithm
4.6 Local Search Algorithm
4.7 Metaheuristics
4.8 Summary

Mathematical optimization is one of the ways to rationally solve real-world problems.
To quickly understand mathematical optimization, you need to learn modeling methods for optimizing problems and examine optimization problems to which efficient algorithms are applied.
This book covers modeling optimization problems and basic optimization algorithms to solidify the fundamentals of mathematical optimization thinking.
It also includes concrete examples and practice problems that are easy to recall to aid understanding.


[Structure of this book]

Chapter 1: Introduction to Mathematical Optimization

Mathematical optimization is an optimization problem that seeks a solution that minimizes (or maximizes) the objective function value under given constraints, and is a means of solving real-world decision-making and problems.
Chapter 1 provides an overview of mathematical optimization with examples.

Chapter 2: Linear Programming

Linear programming problems are the most fundamental optimization problems, and effective algorithms have been developed to solve them on large scales with realistic computational means.
We will learn about the formalization of linear programming problems, the simplex method, a representative algorithm for linear programming problems, and the dual problem and relaxation problem, which are the most important concepts in mathematical optimization.

Chapter 3: Nonlinear Planning

Because nonlinear programming problems have such a wide range of applications, it is difficult to develop a general algorithm that can efficiently solve diverse nonlinear programming problems.
After formalizing the nonlinear programming problem and explaining the characteristics of nonlinear programming problems that can be solved efficiently, we explain representative algorithms for unconstrained and constrained optimization problems.

Chapter 4: Integer Programming and Combinatorial Optimization Problems

In linear programming problems, integer programming problems, in which variables have only integer values, are one of the general optimization problems that can be used to formulate real-world problems in a wide range of fields, including industry and academia.
Learn about the formalization of integer programming problems and the fundamental thinking behind computational complexity theory, which evaluates the difficulty of combinatorial optimization problems.
In addition, after explaining efficient algorithms for some special integer programming problems and representative algorithms for integer programming problems such as the branch-and-bound method and the cut plane method, we will explain approximate algorithms that guarantee approximate performance and find feasible solutions for arbitrary problems as examples, as well as local search algorithms and metaheuristics that can find high-quality feasible solutions for many problem cases.

[Target Readers]

- Students and researchers interested in optimization theory and practitioners engaged in mathematical optimization and related work.
- Readers who want to study the application of optimization algorithms in artificial intelligence or other various industries, even if they are not majoring in mathematics.