基于Scholtes松弛方法的收敛性研究

Convergence Research Based on Scholtes’ Relaxation Method

摘要: 带有切换约束的数学规划问题(MPSC)在切换控制、MPEOC、具有半连续变量的投资组合优化问题等领域有着广泛的应用。MPSC带有可行域非凸非连通的特殊结构，因而在可行点处不满足大多数约束规范，往往难于求解。松弛方法被普遍视为解决MPSC问题的有效方法。本文基于Scholtes松弛方法以及非精确稳定点的概念，证明了在较弱的$\epsilon$ -KKT点条件下能得到与KKT点条件相同的结果，即收敛到了MPSC的W-稳定点，如果增强假设条件，松弛问题的$\epsilon$ -KKT点列可收敛到MPSC的S-稳定点。

Abstract: Mathematical programming with switching constraints (MPSC) has broad applications in fields such as switching control, MPEOC, and portfolio optimization problems with semi-continuous variables. MPSC features a special structure where the feasible region is non-convex and non-connected, making it difficult to solve, as most constraint qualifications fail at feasible points. Relaxation methods are widely regarded as effective approaches for addressing MPSC problems. Based on Scholtes’ relaxation method and the concept of inexact stationary points, this paper demonstrates that under weaker $\epsilon$ -KKT point conditions, results equivalent to those under KKT point conditions can be achieved, i.e., convergence to W-stationary points of MPSC. With enhanced assumptions, sequences of $\epsilon$ -KKT points from the relaxed problem can converge to S- stationary points of MPSC.