Filled Function Methods For General Constrained Global Optimization Problems

It is should be noted that most of the existing filled function methods focus only on solving unconstrained global optimization problems, whereas them have difficulty in solving constrained global optimization problems. Z.Y.Wu et al has proposed an new filled function method for constrained global optimization problem, but the method proposed in [7] focus only on solving global optimization problem with inequality constraint. The filled function method for global optimization problem with both inequality and equality constraints is an even tougher area to tackle. However, different filled function methods have different theoretic and algorithm. Consequently, it is worth using filled function method to solve continuously constrained programming problems, specially, to solve global optimization problem with both equality and inequality constraints.In this paper we proposed an new filled function method referring to [6] and [7] for continuously general constrained programming problems, specially, for continuous programming problems with equality constraint. This paper is organized as follows: In Chapter 1, the development and the key idea of filled function method are introduced simply. In order to introduce filled function method for continuous optimization problem systematically, in Chapter 2, the filled function method to solve continuously unconstrained programming problem proposed in [6] is introduced briefly, and a illustrative numerical example is given. In Chapter 3, the filled function method proposed in [7] for solving continuous programming problem with inequality constraint is introduced primitively, and a numerical example is given to demonstrate the efficiency and credibility of the method. In Chapter 4, we present an new filled function method to solve the general constrained global optimization problem, namely, continuously global optimization problem with both equality and inequality constraints, to obtain an approximate global minimizer. Chapter 4 is organized as follows: firstly, an auxiliary function for general constrained global optimization problem is introduced, and then some basic properties of the proposed auxiliary function are discussed. The auxiliary function is used to find an approximate feasible point of problem (P). Secondly, a filled function for general constrained global optimization problem is introduced by combining the idea of filled function in unconstrained global optimization and the idea of penalty function in constrained optimization, and some basic properties of the proposed filled function are discussed. Especially, it is shown that an improved approximate feasible point can be obtained by solving locally an unconstrained or a box constrained optimization problem constructed via the present filled function. Finally, an global optimization method using the present auxiliary function and the filled function is proposed to obtain an approximate global minimizer of problem (P). Besides, two numerical results are reported to demonstrate the efficiency of the present global optimization method.