论文标题
用于框受限的非凸优化程序的修改分段凸化方法
A Modification Piecewise Convexification Method for Box-Constrained Non-Convex Optimization Programs
论文作者
论文摘要
本文提出了一种分段凸化方法，以近似于与盒子约束的非凸优化问题的整个近似最佳解决方案集。在Box Division的过程中，我们首先对子盒进行了分类，仅在后续分区中仅将某些子盒划分。同时，应用基于$α$的分支机构（{\ rm $ $α$ bb}）方法，我们构建了一系列的分段凸出放松子问题，这些分段凸出子问题集体称为原始问题的分段凸化问题。然后，我们根据子盒的分类结果来定义分段凸化问题的（近似）解决方案集。随后，我们得出这些集合可用于近似具有预定义质量的全局解决方案集。最后，提出了具有新的子框选择规则的分段凸化算法，并提出了两个新的终止测试。几个实例证明这些技术有利于提高算法的性能。
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 $α$-based Branch-and-Bound ({\rm$α$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.
