An approximate approach of global optimization for polynomial programming problems

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 xi as the addition of a discrete variable di and a small variable ϵi, a polynomial term xixj can be expanded as the sum of dixj, djϵi and ϵiϵj. A procedure is then developed to fully linearize dixj and djϵi, and to approximately linearize ϵiϵj 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.