# Convex Analysis and Global Optimization

Springer Science & Business Media, Jan 31, 1998 - Business & Economics - 339 pages
Due to the general complementary convex structure underlying most nonconvex optimization problems encountered in applications, convex analysis plays an essential role in the development of global optimization methods. This book develops a coherent and rigorous theory of deterministic global optimization from this point of view. Part I constitutes an introduction to convex analysis, with an emphasis on concepts, properties and results particularly needed for global optimization, including those pertaining to the complementary convex structure. Part II presents the foundation and application of global search principles such as partitioning and cutting, outer and inner approximation, and decomposition to general global optimization problems and to problems with a low-rank nonconvex structure as well as quadratic problems. Much new material is offered, aside from a rigorous mathematical development.
Audience: The book is written as a text for graduate students in engineering, mathematics, operations research, computer science and other disciplines dealing with optimization theory. It is also addressed to all scientists in various fields who are interested in mathematical optimization.

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### Contents

 CONVEX SETS 3 12 Convex Sets 6 13 relative Interior and Closure 8 14 Convex Hull 13 17 Representation of Convex Sets 26 18 Polars of Convex Sets 28 19 Polyhedral Convex Sets 30 110 Exercises 38
 49 Exercises 130 SUCCESSIVE PARTITIONING METHOD 133 52 Concavity Cuts 136 53 Subdivision Processes 140 54 Procedure DC 146 55 Branch and Select Algorithms 153 56 Rectangular Algorithms 159 57 Reverse Convex Programming 164

 CONVEX FUNCTIONS 41 22 Convexity Preserving Operations 46 23 Lower SemiContinuity 50 24 Lipschitz Continuity 54 25 Convex Inequalities 56 26 Approximation by Affine Functions 60 27 Subdifferential 62 28 Subdifferential Calculus 67 29 Approximate Subdifferential 69 210 Conjugate Functions 72 211 Extremal Values of Convex Functions 74 212 SaddleFunctions and Minimax 77 213 Exercises 80 DC FUNCTIONS AND DC SETS 83 32 University of DC Functions 85 33 DC Representation of Basic Composite Functions 88 35 DC Representation of Polynomials 93 36 DC Sets 95 37 Convex Minorants of a DC Function 100 38 The Minimum of a DC Function 103 39 Exercises 105 GLOBAL OPTIMIZATION 107 MOTIVATION AND OVERVIEW 109 42 Multiextrenality in the Real World 110 43 Basic Classes of Nonconvex Problems 115 44 General Nonconvex Structure 117 46 DC Inclusion Associated with a Feasible Solution 122 47 Approximate Solution 123 48 When Global Optimization Motivated? 125
 58 Exercises 174 OUTER AND INNER APPROXIMATION 177 62 Concave Minimization Under Convex Constraints 181 63 Canonical DC Programming 186 64 Polyhedral Annexation 192 65 Duality in Global Optimization 199 66 Noncanonical DC Problems 206 67 Continuous Optimization Problems 211 68 Exercises 220 Decomposition 223 72 Decomposition by Projection 227 73 Decomposition by Polyhedral Annexation 233 74 The Parametric Approach 236 75 Linear Multiplicative Programming 240 76 Convex Constraints 247 77 Reverse Convex Constraints 255 78 Network Constraints 261 79 Some PolynomialTime Solvable Problems 269 710 Exercises 274 NONCONVEX QUADRATIC PROGRAMMING 277 82 Quadratic Minimization Over Ellipsoids 281 83 Generalized Linear Programs 285 84 Quadratic Problems with Convex Constraints 287 85 Quadratic Problems with Quadratic Constraints 293 86 Exercises 316 References 319 Index 337 Copyright

### Popular passages

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