一类光滑函数在概率约束优化问题中的应用

Application of a Class of Smooth Functions to Probabilistically Constrained Optimization Problems

摘要: 概率约束优化问题(PCOP)在新能源调度、金融风险管理和网络优化等领域有广泛应用，但其非凸非光滑的特性给数值求解带来困难。本文基于光滑近似思想，提出了特征函数的一类光滑近似函数，讨论了该类函数中的两个具体函数的性质，包括极限逼近、单调性、凸凹性、无穷阶可微性等。基于这两个函数，编制了序列凸近似(SCA)算法的程序，用于求解概率约束优化问题。实验结果表明：两个实例在计算效率上表现优异(迭代次数极少)，但收敛稳定性依赖于光滑参数的选取。这一发现为光滑函数的构造提供了新的见解：在数值算法中，纯粹的高阶“光滑性”可能并非越高越好。

Abstract: Probabilistically constrained optimization problems (PCOP) have wide applications in areas such as renewable energy dispatch, financial risk management, and network optimization; however, their nonconvex and nonsmooth nature brings difficulties to numerical solution. Based on the idea of smooth approximation, this paper proposes a class of smooth approximation functions for the characteristic function, and discusses the properties of two specific functions in this class, including limit approximation behavior, monotonicity, convexity/concavity, and infinite differentiability. Using these two functions, a program implementing the sequential convex approximation (SCA) algorithm is developed to solve probabilistically constrained optimization problems. Experimental results show that two test instances perform excellently in computational efficiency (with very few iterations), yet the convergence stability depends on the selection of the smoothing parameter. This finding provides new insight into the construction of smoothing functions: in numerical algorithms, purely higher-order “smoothness” may not necessarily be better.