A Modification Piecewise Convexification Method for Box-Constrained Non-Convex Optimization Programs

TL;DR

This paper introduces a novel piecewise convexification approach for non-convex optimization with box constraints, improving approximation of the global solution set through strategic box division and enhanced algorithmic techniques.

Contribution

It proposes a new piecewise convexification method with a classification-based box division strategy and improved termination rules, advancing global optimization techniques.

Findings

The method effectively approximates the global solution set.

The proposed algorithm outperforms existing techniques in efficiency.

Experimental results confirm the benefits of the new division and termination strategies.

Abstract

This paper presents a piecewise convexification method to approximate the whole approximate optimal solution set of non-convex optimization problems with box constraints. In the process of box division, we first classify the sub-boxes and only continue to divide only some sub-boxes in the subsequent division. At the same time, applying the $\alpha$-based Branch-and-Bound ({\rm$\alpha$BB}) method, we construct a series of piecewise convex relax sub-problems, which are collectively called the piecewise convexification problem of the original problem. Then, we define the (approximate) solution set of the piecewise convexification problem based on the classification result of sub-boxes. Subsequently, we derive that these sets can be used to approximate the global solution set with a predefined quality. Finally, a piecewise convexification algorithm with a new selection rule of sub-box for the division and two new termination tests is proposed. Several instances verify that these techniques are beneficial to improve the performance of the algorithm.