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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 Supremum Functions 2.11 Exercises for Chapter 2 3 Remarkable Consequences of Convexity 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 Existence 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 ±à¼­ÊÖ¼Ç