An approximate approach of global optimization for polynomial programming problems

Abstract

Many methods for solving polynomial programming problems can only find locally optimal solutions. This paper proposes a method for finding the approximately globally optimal solutions of polynomial programs. Representing a bounded continuous variable x(i) as the addition of a discrete variable d(i) and a small variable epsilon(i), a polynomial term x(i)x(j) can be expanded as the sum of d(i)x(j), d(j) epsilon(i) and epsilon(i) epsilon(j). A procedure is then developed to fully linearize d(i)x(j) and d(j) epsilon(i), and to approximately linearize epsilon(i) epsilon(j) with an error below a pre-specified tolerance. This linearization procedure can also be extended to higher order polynomial programs. Several polynomial programming examples in the literature are tested to demonstrate that the proposed method can systematically solve these examples to find the global optimum within a pre-specified error. (C) 1998 Elsevier Science B.V.