Optimization Methods

Abstract

This chapter describes how optimization problems can be solved and which different types of optimization methods exist for discrete optimization problems. The goal of optimization methods is to find an optimal or near-optimal solution with low computational effort. The effort of an optimization method can be measured as the time (computation time) and space (computer memory) that is consumed by the method. For many optimization methods, and especially for modern heuristics, there is a trade-off between solution quality and effort, as with increasing effort solution quality increases.

We can distinguish between two different types of optimization methods: Exact optimization methods that guarantee finding an optimal solution and heuristic optimization methods where we have no guarantee that an optimal solution is found. Usually, an exact optimization method is the method of choice if it can solve an optimization problem with effort that grows polynomially with the problem size. The situation is different if problems are NP-hard as then exact optimization methods need exponential effort. Then, even medium-sized problem instances often become intractable and cannot be solved any more using exact methods. To overcome these problems, we can use heuristic optimization methods. Usually, such optimization methods are problem-specific as they exploit properties of the problem. Furthermore, they often show good performance for many NP-complete problems and problems of practical relevance.