Bounded Global Optimization for Polynomial Programming using Binary Reformulation and Linearization

TL;DR

This paper introduces an approximate global optimization method for polynomial programming with bounded variables, transforming the problem into mixed binary-linear programs to efficiently estimate bounds.

Contribution

It presents a reformulation and linearization technique that converts polynomial problems into mixed binary-linear programs, enabling bounds approximation based on user-defined error tolerances.

Findings

Bounds become tighter as error tolerances decrease

The method converges to the true solution with increasing reformulation size

Provides a systematic way to control approximation accuracy

Abstract

This paper describes an approximate method for global optimization of polynomial programming problems with bounded variables. The method uses a reformulation and linearization technique to transform the original polynomial optimization problem into a pair of mixed binary-linear programs. The solutions to these two integer-linear reformulations provide upper and lower bounds on the global solution to the original polynomial program. The tightness of these bounds, the error in approximating each polynomial expression, and the number of constraints that must be added in the process of reformulation all depend on the error tolerance specified by the user for each variable in the original polynomial program. As these error tolerances approach zero the size of the reformulated programs increases and the calculated interval bounds converge to the true global solution.