A Study on Efficient Algorithms for some Numerical Optimization Problems

Abstract

This thesis is mainly divided into two parts: parameter estimation problem in linear differential equations and a minimization algorithm which is applicable to some industrial problem. In general, mathematical optimization problems are to find optimal ele- ments of a set which minimize (or maximize) the value of a given objective function. It is well known problem and arises in a various field of applications such as science, engineering, business and so on. It has a long history and there are still very much a work in progress. Optimization problems usually depend on the properties of objective func- tions involved. If functions are simple, e.g., linear, the the problem is easy to solve and moreover mathematical theories completed. If it is complex, however, it is hard to solve it theoretically and/or numerically. In this thesis, we suggest algorithms which is related to two specific opti- mization problems. These problems are both include nonlinear objective func- tions. The first part is to find a optimal parameter function of a differential equation and the second part is to find optimal solution of a facility location problem. Each parts contains theories about the solution, such as the exis- tence and the uniqueness of the optimal solution, and numerical examples are included.