內容簡介
閱讀本書僅需要線性代數和數學分析的基本知識。通過學習本書，可以了解凸分析和最佳化領域的主要結果，掌握有關理論的本質內容，提高分析和解決最最佳化問題的能力。因此，所有涉足最最佳化與系統分析領域的理論研究人員和實際工作者均可從學習或閱讀本書中獲得益處。此外，本書也可用作高年級大學生或研究生學習凸分析方法和最最佳化理論的教材或輔助材料。
圖書目錄
1. Basic Convexity Concepts
1.1. Linear Algebra and Real Analysis
1.1.1. Vectors and Matrices
1.1.2. Topological Properties
1.1.3. Square Matrices
1.1.4. Derivatives
1.2. Convex Sets and Functions
1.3. Convex and Affine Hulls
1.4. Relative Interior, Closure, and Continuity
1.5. Recession Cones
1.5.1. Nonemptiness of Intersections of Closed Sets
1.5.2. Closedness Under Linear Transformations
1.6. Notes, Sources, and Exercises
2. Convexity and Optimization
2.1. Global and Local Minima
2.2. The Projection Theorem
2.3. Directions of Recession and Existence of Optimal Solutions
2.3.1. Existence of Solutions of Convex Programs
2.3.2. Unbounded Optimal Solution Sets
2.3.3. Partial Minimization of Convex Functions
2.4. Hyperplanes
2.5. An Elementary Form of Duality
2.5.1. Nonvertical Hyperplanes
2.5.2. Min Common/Max Crossing Duality
2.6. Saddle Point and Minimax Theory
2.6.1. Min Common/Max Crossing Framework for Minimax
2.6.2. Minimax Theorems
2.6.3. Saddle Point Theorems
2.7. Notes, Sources, and Exercises
3. Polyhedral Convexity
3.1. Polar Cones
3.2. Polyhedral Cones and Polyhedral Sets
3.2.1. Farkas' Lemma and Minkowski-Weyl Theorem
3.2.2. Polyhedral Sets
3.2.3. Polyhedral Functions
3.3. Extreme Points
3.3.1. Extreme Points of Polyhedral Sets
3.4. Polyhedral Aspects of Optimization
3.4.1. Linear Programming
3.4.2. Integer Programming
3.5. Polyhedral Aspects of Duality
3.5.1. Polyhedral Proper Separation
3.5.2. Min Common/Max Crossing Duality
3.5.3. Minimax Theory Under Polyhedral Assumptions
3.5.4. A Nonlinear Version of Farkas' Lemma
3.5.5. Convex Programming
3.6. Notes, Sources, and Exercises
4. Subgradients and Constrained Optimization
4.1. Directional Derivatives
4.2. Subgradients and Subdifferentials
4.3. e-Subgradients
4.4. Subgradients of Extended Real-Valued Functions
4.5. Directional Derivative of the Max Function
4.6. Conical Approximations
4.7. Optimality Conditions
4.8. Notes, Sources, and Exercises
5. Lagrange Multipliers
5.1. Introduction to Lagrange Multipliers
5.2. Enhanced Fritz John Optimality Conditions
5.3. Informative Lagrange Multipliers
5.3.1. Sensitivity
5.3.2. Alternative Lagrange Multipliers
5.4. Pseudonormality and Constraint Qualifications
5.5. Exact Penalty Functions
5.6. Using the Extended Representation
5.7. Extensions Under Convexity Assumptions
5.8. Notes, Sonrces, and Exercises
6. Lagrangian Duality
6.1. Geometric Multipliers
6.2. Duality Theory
6.3. Linear and Quadratic Programming Duality
6.4. Existence of Geometric Multipliers
6.4.1. Convex Cost Linear Constraints
6.4.2. Convex Cost Convex Constraints
6.5. Strong Duality and the Primal Function
6.5.1. Duality Gap and the Primal Function
6.5.2. Conditions for No Duality Gap
6.5.3. Subgradients of the Primal Function
6.5.4. Sensitivity Analysis
6.6. Fritz John Conditions when there is no Optimal Solution
6.6.1. Enhanced Fritz John Conditions
6.6.2. Informative Geometric Multipliers
6.7. Notes, Sources, and Exercises
7. Conjugate Duality
7.1. Conjugate Functions
7.2. Fenchel Duality Theorems
7.2.1. Connection of Fenchel Duality and Minimax Theory
7.2.2. Conic Duality
7.3. Exact Penalty Functions
7.4. Notes, Sources, and Exercises
8. Dual Computational Methods
8.1. Dual Derivatives and Subgradients
8.2. Subgradient Methods
8.2.1. Analysis of Subgradient Methods
8.2.2. Subgradient Methods with Randomization
8.3. Cutting Plane Methods
8.4. Ascent Methods
8.5. Notes, Sources, and Exercises
References
Index
前言
本書針對最最佳化問題介紹凸分析方法。第1章介紹凸集、凸函式、上境圖、凸包、仿射包、相對內點、回收錐等凸分析的基本概念及其相關性質；第2章討論凸性在最最佳化問題中的基本作用，介紹最優解集的存在性定理、投影定理、凸集分離定理、極小公共點與極大交叉點對偶問題以及一般性的極小極大定理和鞍點定理；第3章討論凸集為多面體的情況，介紹線性Farkas引理、凸多面體的Minkowski Weyl表示定理、線性規劃的基本定理、凸多面體的極小極大定理以及非線性Farkas引理；第4章介紹方嚮導數、次梯度、次微分、切錐、法錐等基本概念及其相關性質，給出Danskin定理和抽象可行集描述的約束最佳化問題最優性條件；第5章討論由抽..
作者簡介
Dimitri P.Bertsekas，美國國家工程院院士，麻省理工學院（MIT）McAfee教授。本作者Dimitri P. Bertsekas是美國麻省理工學院電氣工程和計算機科學系的資深教授，他是“動態規劃與隨機控制”、“約束最佳化與Lagrange乘子方法”、“非線性規劃”、“連續和離散模型的網路最佳化”、“離散時間隨機最優控制”、“並行和分布計算中的數值方法”等十餘部教科書的主要作者，這些教科書的大部分被用作麻省理工學院的研究生或本科生教材，本書就是其中之一。
