本书由4章组成，组织结构如下：在章中，我们研究了凸集和函数的基本性质，同时特别关注了一类在优化中很重要的凸函数；第2章主要研究了凸集的法线和凸函数的子梯度的基本演算规则，这是凸理论的主流；第3章涉及到凸分析的一些额外的主题，它们在很大程度上是应用性的；第4章从定性和数值的角度，全面地研究了凸分析在凸优化问题和选址问题中的应用；后，我们在本书的结尾给出了所选练习题的解答和提示。习题在每一章的末尾给出，而图表和例子则贯穿全文。参考文献中包含书籍和选定的论文，它们是与本书内容密切相关的，可能有助于读者对凸分析的进一步研究，包括研究凸分析的应用和未来的扩展。

Preface Acknowledgments List of Symbols 1 Convex Sets and Functions 1.1 Preliminaries 1.2 Convex Sets 1.3 Convex Functions 1.4 Relative Interiors of Convex Sets 1.5 The Distance Function 1.6 Exercises for Chapter 1 2 Subdifferential Calculus 2.1 Convex Separation 2.2 Normals to Convex Sets 2.3 Lipschitz Continuity of Convex Functions ... 2.4 Subgradients of Convex Functions 2.5 Basic Calculus Rules 2.6 Subgradients of Optimal Value Functions 2.7 Subgradients of Support Functions 2.8 Fenchel Conjugates 2.9 Directional Derivatives 2.10 Subgradients of Suprem Functions 2.11 Exercises for Chapter 2 3 Remarkable Consequences of Convety 3.1 Characterizations of Differentiability 3.2 Caratheodory Theorem and Farkas Lemma 3.3 Radon Theorem and HeUy Theorem 3.4 Tangents to Convex Sets 3.5 Mean Value Theorems 3.6 Horizon Cones 3.7 Minimal Time Functions and Minkowski Gauge 3.8 Subgradients of Minimal Time Functions 3.9 Nash Equilibrium 3.10 Exercises for Chapter 3 4 Applications to Optimization and Location Problems 4.1 Lower Semicontinuity and Estence of Minimizers 4.2 Optimality Conditions 4.3 Subgradient Methods in Convex Optimization 4.4 The Fermat-TorriceUi Problem 4.5 A Generalized Fermat-Torricelli Problem 4.6 A Generalized Sylvester Problem 4.7 Exercises for Chapter 4 Solutions and Hints for Selected Exercises Bibliography Authors' Biographies Index 编辑手记