Second-Order Variational Analysis in Optimization, Variational Stability, and Control: Theory, Algorithms, Applications
暫譯: 優化中的二階變分分析、變分穩定性與控制：理論、演算法、應用
Mordukhovich, Boris S.
相關主題
商品描述
This fundamental work is a sequel to monographs by the same author: Variational Analysis and Applications (2018) and the two Grundlehren volumes Variational Analysis and Generalized Differentiation: I Basic Theory, II Applications (2006). This present book is the first entirely devoted to second-order variational analysis with numerical algorithms and applications to practical models. It covers a wide range of topics including theoretical, numerical, and implementations that will interest researchers in analysis, applied mathematics, mathematical economics, engineering, and optimization. Inclusion of a variety of exercises and commentaries in each chapter allows the book to be used effectively in a course on this subject. This area has been well recognized as an important and rapidly developing area of nonlinear analysis and optimization with numerous applications. Consisting of 9 interrelated chapters, the book is self-contained with the inclusion of some preliminaries in Chapter 1.
Results presented are useful tools for characterizations of fundamental notions of variational stability of solutions for diverse classes of problems in optimization and optimal control, the study of variational convexity of extended-real-valued functions and their specifications and variational sufficiency in optimization. Explicit calculations and important applications of second-order subdifferentials associated with the achieved characterizations of variational stability and related concepts, to the design and justification of second-order numerical algorithms for solving various classes of optimization problems, nonsmooth equations, and subgradient systems, are included. Generalized Newtonian algorithms are presented that show local and global convergence with linear, superlinear, and quadratic convergence rates. Algorithms are implemented to address interesting practical problems from the fields of machine learning, statistics, imaging, and other areas.
Results presented are useful tools for characterizations of fundamental notions of variational stability of solutions for diverse classes of problems in optimization and optimal control, the study of variational convexity of extended-real-valued functions and their specifications and variational sufficiency in optimization. Explicit calculations and important applications of second-order subdifferentials associated with the achieved characterizations of variational stability and related concepts, to the design and justification of second-order numerical algorithms for solving various classes of optimization problems, nonsmooth equations, and subgradient systems, are included. Generalized Newtonian algorithms are presented that show local and global convergence with linear, superlinear, and quadratic convergence rates. Algorithms are implemented to address interesting practical problems from the fields of machine learning, statistics, imaging, and other areas.
商品描述(中文翻譯)
這本基礎著作是同一作者的專著的續篇：變分分析與應用（2018年）以及兩卷的Grundlehren系列書籍變分分析與廣義微分：I 基本理論，II 應用（2006年）。本書首次完全專注於二階變分分析，並結合數值演算法及其在實際模型中的應用。內容涵蓋廣泛的主題，包括理論、數值方法及實作，將吸引分析、應用數學、數學經濟學、工程學及優化領域的研究人員。每章節中包含各種練習題和評論，使本書能有效用於相關課程的教學。這一領域已被廣泛認可為非線性分析和優化的重要且快速發展的領域，並具有眾多應用。全書由9個相互關聯的章節組成，並在第一章中包含了一些初步內容，使其自成體系。
所呈現的結果是用於表徵各類優化和最優控制問題解的變分穩定性基本概念的有用工具，研究擴展實值函數的變分凸性及其規範，以及在優化中的變分充分性。明確的計算和與變分穩定性及相關概念的表徵相關的二階次微分的重要應用，將用於設計和證明解決各類優化問題、非光滑方程和次梯度系統的二階數值演算法。書中介紹了廣義牛頓演算法，顯示出局部和全局收斂性，並具有線性、超線性和二次收斂速率。這些演算法被實作以解決來自機器學習、統計、影像處理及其他領域的有趣實際問題。
作者簡介
Boris S. Mordukhovich is Distinguished Professor of Mathematics at Wayne State University. He has more than 500 publications including several monographs. Among his best known achievements are the introduction and development of powerful constructions of generalized differentiation and their applications to broad classes of problems in variational analysis, optimization, equilibrium, control, economics, engineering, and other fields. Mordukhovich is a SIAM Fellow, an AMS Fellow, and a recipient of many international awards and honors including Doctor Honoris Causa degrees from six universities over the world. He is a Highly Cited Researcher in Mathematics. His research has been supported by continued grants from the National Science Foundations and the Air Force Office of Scientific Research.
作者簡介(中文翻譯)
Boris S. Mordukhovich 是韋恩州立大學的數學特聘教授。他擁有超過500篇的出版物，包括幾本專著。他最著名的成就之一是引入和發展強大的廣義微分構造，並將其應用於變分分析、優化、均衡、控制、經濟學、工程及其他領域的廣泛問題。Mordukhovich 是美國工業與應用數學學會（SIAM）和美國數學學會（AMS）的會士，並獲得多項國際獎項和榮譽，包括來自全球六所大學的榮譽博士學位。他是數學領域的高被引研究者。他的研究得到了國家科學基金會和空軍科學研究辦公室的持續資助。
