凸優(yōu)化理論

前言

優(yōu)化理論與應(yīng)用是非常經(jīng)典但依然非?；钴S的研究領(lǐng)域，涉及幾乎所有的理工和管理學科以及計量社會科學學科，是系統(tǒng)工程、運籌學、計量經(jīng)濟學等學科的理論基礎(chǔ)。凸優(yōu)化是優(yōu)化理論十分重要的分支，是本書討論的重點。凸優(yōu)化是指目標函數(shù)為凸函數(shù)、約束集為凸集合的約束優(yōu)化問題。凸優(yōu)化具有重要的工程應(yīng)用背景，求解凸優(yōu)化問題的方法通常也是一般非線性規(guī)劃方法的重要基礎(chǔ)。本書是凸優(yōu)化理論與方法的重要專著和教材，主要內(nèi)容分為兩部分：凸分析和凸問題的對偶優(yōu)化理論。本書先從基本線性代數(shù)和實分析理論出發(fā)，比較詳盡地討論了凸理論和凸分析，為求解凸優(yōu)化問題建立了足夠的基礎(chǔ)。本書在引入了凸優(yōu)化的基本概念后，著重討論了對偶優(yōu)化理論。本書從比較獨特的幾何問題角度——最小共同點和最大相交點問題——引入了對偶理論框架，討論對偶性和對偶優(yōu)化解的存在性等問題。在此統(tǒng)一對偶理論框架下，本書討論了多種優(yōu)化問題如線性規(guī)劃、凸規(guī)劃、最小最大等問題的對偶性和對偶優(yōu)化理論，并討論了當目標函數(shù)非光滑時的次梯度和最優(yōu)性條件。本書的重要特點是白成體系，所需要的基礎(chǔ)知識除理工科本科線性代數(shù)和少量實分析基本概念和理論外，并不需要一般優(yōu)化理論如線性規(guī)劃、非線性規(guī)劃等作為基礎(chǔ)。所以本書既適用作研究生的教材，也可作為優(yōu)化理論與方法研究者的參考書。本書作者德梅萃·博賽克斯教授是優(yōu)化理論的國際著名學者、美國國家工程院院士，現(xiàn)任美國麻省理工學院電氣工程與計算機科學系教授，曾在斯坦福大學工程經(jīng)濟系和伊利諾伊大學電氣工程系任教，在優(yōu)化理論、控制工程、通信工程、計算機科學等領(lǐng)域有豐富的科研教學經(jīng)驗，成果豐碩。博賽克斯教授是一位多產(chǎn)作者，著有14本專著和教科書。本書是作者在優(yōu)化理論與方法的系列專著和教科書中的一本，自成體系又相互對應(yīng)。

內(nèi)容概要

本書作者德梅萃，博賽克斯教授是優(yōu)化理論的國際著名學者、美國國家工程院院士，現(xiàn)任美國麻省理工學院電氣工程與計算機科學系教授，曾在斯坦福大學工程經(jīng)濟系和伊利諾伊大學電氣工程系任教，在優(yōu)化理論、控制工程、通信工程、計算機科學等領(lǐng)域有豐富的科研教學經(jīng)驗，成果豐碩。博賽克斯教授是一位多產(chǎn)作者，著有14本專著和教科書。本書是作者在優(yōu)化理論與方法的系列專著和教科書中的一本，自成體系又相互對應(yīng)。主要內(nèi)容分為兩部分：凸分析和凸問題的對偶優(yōu)化理論。

作者簡介

作者：（美國）博賽克斯（Dimitri P.Bertsekas）

書籍目錄

1. basic concepts of convex analysis 　1.1. convex sets and functions 　1.1.1. convex functions 　1.1.2. closedness and semicontinuity 　1.1.3. operations with convex functions 　1.1.4. characterizations of differentiable convex functions 　1.2. convex and afiine hulls 　1.3. relative interior and closure 　1.3.1. calculus of relative interiors and closures 　1.3.2. continuity of convex functions 　1.3.3. closures of functions 　1.4. recession cones 　1.4.1. directions of recession of a convex function 　1.4.2. nonemptiness of intersections of closed sets 　1.4.3. closedness under linear transformations 　1.5. hyperplanes 　1.5.1. hyperplane separation 　1.5.2. proper hyperplane separation 　1.5.3. nonvertical hyperplane separation 　1.6. conjugate functions 　1.7. summary 2. basic concepts of polyhedral convexity 　2.1. extreme points 　2.2. polar cones 　2.3. polyhedral sets and functions 　2.3.1. polyhedral cones and farkas' lemma 　2.3.2. structure of polyhedral sets 　2.3.3. polyhedral functions 　2.4. polyhedral aspects of optimization 3. basic concepts of convex optimization 　3.1. constrained optimization 　3.2. existence of optimal solutions 　3.3. partial minimization of convex functions 　3.4. saddle point and minimax theory 4. geometric duality framework 　4.1. min common/max crossing duality 　4.2. some special cases 　4.2.1. connection to conjugate convex functions 　4.2.2. general optimization duality 　4.2.3. optimization with inequality constraints 　4.2.4. augmented lagrangian duality 　4.2.5. minimax problems 　4.3. strong duality theorem 　4.4. existence of dual optimal solutions 　4.5. duality and polyhedral convexity 　4.6. summary 5. duality and optimization 　5.1. nonlinear farkas' lemma 　5.2. linear programming duality 　5.3. convex programming duality 　5.3.1. strong duality theorem inequality constraints 　5.3.2. optimality conditions 　5.3.3. partially polyhedral constraints 　5.3.4. duality and existence of optimal primal solutions 　5.3.5. fenchel duality 　5.3.6. conic duality 　5.4. subgradients and optimality conditions 　5.4.1. subgradients of conjugate functions 　5.4.2. subdifferential calculus 　5.4.3. optimality conditions 　5.4.4. directional derivatives 　5.5. minimax theory 　5.5.1. minimax duality theorems 　5.5.2. saddle point theorems 　5.6. theorems of the alternative 　5.7. nonconvex problems 　5.7.1. duality gap in separable problems 　5.7.2. duality gap in minimax problems appendix a: mathematical background notes and sources supplementary chapter 6 on convex optimization algorithm

章節(jié)摘錄

插圖：Convex sets and functions are very useful in optimization models, and havea rich structure that is convenient for analysis and algorithms. Much of thisstructure can be traced to a few fundamental properties. For example, eachclosed convex set can be described in terms of the hyperplanes that supportthe set, each point on the boundary of a convex set can be approachedthrough the relative interior of the set, and each halfline belonging to aclosed convex set still belongs to the set when translated to start at anypoint in the set.Yet, despite their favorable structure, convex sets and their analysisare not free of anomalies and exceptional behavior, which cause seriousdifficulties in theory and applications.  For example, contrary to affineand compact sets, some basic operations such as linear transformation andvector sum may not preserve the closedness of closed convex sets. This inturn complicates the treatment of some fundamental optimization issues,including the existence of optimal solutions and duality.For this reason, it is important to be rigorous in the development ofconvexity theory and its applications. Our aim in this first chapter is toestablish the foundations for this development, with a special emphasis onissues that are relevant to optimization.

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