Convex Analysis and Global Optimization

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Springer Science & Business Media, Jan 31, 1998 - Business & Economics - 339 pages
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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

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Page 328 - Quasiconjugates of functions, duality relationship between quasiconvex minimization under a reverse convex constraint and quasiconvex maximization under a reverse convex constraint and applications, Journal of Mathematical Analysis and Applications 159 (1991) 199-322.
Page 329 - An Outer Approximation Method for Globally Minimizing a Concave Function over a Compact Convex Set', Acta Mathematica Vietnamica, 8, 21-40, 1983.
Page 333 - SW (1982), A Design Centering Algorithm for Nonconvex Regions of Acceptability, IEEE Transactions on Computer- Aided- Design of Integrated Circuits and Systems CAD-1, 13-24.
Page 329 - A finite algorithm for solving linear programs with an additional reverse convex constraint, Lecture Notes in Economics and Mathematical Systems, 225 ,291-302 (1985).
Page 333 - A Global Optimization Algorithm (GOP) for Certain Classes of Nonconvex NLPs — II. Application of Theory and Test Problems. Computers and Chemical Engineering, 14(12), 14191434. Visweswaran, V., and Floudas, CA (1992). Unconstrained and Constrained Global Optimization of Polynomial Functions in One Variable.

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