The Differentiability of <i>λ</i> Functions and Its Optimization Theory

本文深入研究了模糊函数的可微性及其在优化问题中的应用。针对现有研究的不足，本文进行了以下核心贡献：1) 统一模糊函数的可微性理论：梳理了H可微、S可微、G可微等不同可微性定义的逻辑脉络，构建了统一的理论框架，为研究者选择合适的可微性定义提供了指导。2) 构建模糊优化的KKT条件：针对模糊优化问题，建立了KKT条件的统一框架，使得这些理论在实际应用中可以更灵活地解决多样化的模糊优化问题。本文进一步分析了模糊序关系对KKT条件的特殊性影响，并通过模糊截集运算的置信水平优化，提出了适用于G可微的框架。This paper conducts an in-depth study on the differentiability of fuzzy functions and their application in optimization problems. In response to the deficiencies of existing research, this paper makes the following core contributions: 1) Unifying the differentiability theory of fuzzy functions: This paper clarifies the logical connections between different differentiability definitions, such as H-differentiability, S-differentiability, and G-differentiability, and constructs a unified theoretical framework. This provides guidance for researchers to choose the appropriate differentiability definition. 2) Developing KKT conditions for fuzzy optimization: A unified framework of KKT conditions is established for fuzzy optimization problems, enabling these theories to be more flexibly applied to solve diverse fuzzy optimization problems in practice. This paper further analyzes the special influence of the fuzzy order relation on the KKT condition, and through the confidence level optimization of the fuzzy truncated set operation, proposes a framework suitable for G-differentiability.