基于特殊指数函数求取特征函数的光滑近似函数

Smooth Approximation of Characteristic Functions Based on Special Exponential Functions

摘要: 本文针对概率约束优化问题中特征函数的非光滑性导致的数值优化困难，提出了一种基于特殊指数函数构造的新型光滑近似函数。概率约束优化问题在航空航天、铁路调度、通信网络及能源管理等领域具有广泛应用，但特征函数的非连续性与非光滑性使得直接进行梯度计算或数值优化极其困难。为克服这一挑战，本文首先分析了传统Sigmoid型光滑近似函数在收敛速度和参数调节灵活性方面的局限性，进而通过倒数变换与指数函数构造了新的光滑近似函数。理论分析验证了该函数满足值域规范、极限一致、分段单调性及无穷阶可微等核心性质。数值实验表明，相较于常规近似函数，本文提出的方法具有更快的收敛速度和更敏锐的参数调节机制，能够更好地适应不同约束特征和随机分布下的概率约束优化场景，为该类问题的求解提供了新的有效工具。

Abstract: To address the numerical optimization difficulties caused by the non-smoothness of characteristic functions in probabilistically constrained optimization problems, this paper proposes a novel smooth approximation function constructed based on special exponential functions. Probabilistically constrained optimization problems have extensive applications in aerospace, railway scheduling, communication networks, and energy management. However, the discontinuity and non-smoothness of characteristic functions make direct gradient computation or numerical optimization extremely challenging. To overcome this difficulty, this paper first analyzes the limitations of traditional Sigmoid-type smooth approximation functions in terms of convergence speed and parameter adjustment flexibility, and then constructs a new smooth approximation function through reciprocal transformation and exponential functions. Theoretical analysis verifies that the proposed function satisfies core properties including range normalization, limit consistency, piecewise monotonicity, and infinite-order differentiability. Numerical experiments demonstrate that compared with conventional approximation functions, the proposed method exhibits faster convergence speed and more sensitive parameter adjustment mechanisms, enabling better adaptation to probabilistically constrained optimization scenarios with different constraint characteristics and random distributions, thus providing a new effective tool for solving such problems.