微分与凸分析

R. Chambers
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引用次数: 0

摘要

提出并讨论了论证所必需的数学工具。重点是从凸分析和变分分析文献中借鉴的概念。本章首先介绍对应、上半连续性和下半连续性的概念。然后介绍了实值函数的超微分和次微分对应,研究了它们的基本性质及其在全局最优解刻画中的作用。介绍了凸集,并将其与函数凹性(convexity)联系起来。研究了泛函的凹性(凸性)、超可微性(次可微性)和单向导数的存在性之间的关系。讨论了凸共轭理论和本质共轭对偶结果。讨论的主题包括Berge极大定理、超微分(次微分)对应的循环单调性、凹(凸)共轭和双共轭、Fenchel不等式、fenchell - rockafellar共轭对偶定理、支持函数、超线性函数、次线性函数、最小卷积和最大卷积理论以及Fenchel对偶定理。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Differentials and Convex Analysis
Mathematical tools necessary to the argument are presented and discussed. The focus is on concepts borrowed from the convex analysis and variational analysis literatures. The chapter starts by introducing the notions of a correspondence, upper hemi-continuity, and lower hemi-continuity. Superdifferential and subdifferential correspondences for real-valued functions are then introduced, and their essential properties and their role in characterizing global optima are surveyed. Convex sets are introduced and related to functional concavity (convexity). The relationship between functional concavity (convexity), superdifferentiability (subdifferentiability), and the existence of (one-sided) directional derivatives is examined. The theory of convex conjugates and essential conjugate duality results are discussed. Topics treated include Berge's Maximum Theorem, cyclical monotonicity of superdifferential (subdifferential) correspondences, concave (convex) conjugates and biconjugates, Fenchel's Inequality, the Fenchel-Rockafellar Conjugate Duality Theorem, support functions, superlinear functions, sublinear functions, the theory of infimal convolutions and supremal convolutions, and Fenchel's Duality Theorem.
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