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Tuy H. — Convex analysis and global optimization
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Название: Convex analysis and global optimizationАвтор: Tuy H. Аннотация: 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.
Язык: Рубрика: Математика /Статус предметного указателя: Готов указатель с номерами страниц ed2k: ed2k stats Год издания: 1998Количество страниц: 353Добавлена в каталог: 05.11.2010Операции: Положить на полку |
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138 287 138 149 166 70 Affine function 41 Affine set, manifold 3 Affinely independent 5 Approximate subdifferential 69 Barycentric coordinates 6 Basic concave programming, problem (BCP) 133 Basic outer approximation, theorem 180 Bilinear programming 290 Bisection 141 Bisection of ratio a 141 Bisection, exact 141 Branch and bound 155 Branch and cut 155 Branch and select 153 Canonical cone 139 Canonical d.c. programming, problem 118 Caratheodory core 16 Caratheodory’s theorem 14 Cl 53 Closed function 53 Closed, open 6 Closure of a convex function 53 Coercive 200 Combined algorithm 190 Complementary convex set 84 117 Complementary convex structure 83 Concave function 41 Concave majorant 100 Concave production-transportation, problem 262 Concave quadratic programming, problem 287 Concavity cut 136 Cone 7 143 Conical subdivision 143 conjugate 72 Constancy space 52 Convex envelope 48 Convex function 41 Convex hull 13 Convex hull of a function 48 Convex hull of a set 13 Convex inequality 56 Convex minorant 100 Convex set 6 Convex-concave 77 Convg 48 CS restart algorithm for BCP 148 CS restart algorithm for LRCP 169 Cutting plane method 177 D.c set 95 d.c. function 83 D.c. programming 117 D.c. representation 88 D.c. structure 83 DC feasiblity problem 123 Decomposition 223 Decomposition by polyhedral annexation 233 Decomposition by projection 227 Decoupling relaxation 301 Dimension of a convex set 10 Dimension of a convex set of a convex function 53 Direction in which / is affine 53 Direction of recession 11 Directional derivative 65 Distance function 42 64 Distinguished point 154 177 Distinguished point, member 154 Dual DC feasibility problem 193 Dual problems 202 Dualization by polyhedral, annexation 233 EDGE 33 Edge property 164 Effective domain 41 Epigraph 41 Exact bisection 141 Exact selection 155 Exhaustive filter 141 Exhaustive subdivision process 143 7 Extreme direction 24 Extreme point 24 Face 23 31 Facet 32 Factorable function 87 Filter 141 153 Full dimension 10 Gauge 10 General concave minimization, problem 181 General nonconvex quadratic, problem 293 Generalized convex multiplicative, programming problems 250 Generalized linear program 127 285 Geometric programming 126 285 global 74 Global, 123 Global, maximizer 74 Global, minimizer 74 Halfspace 6 Indefinite quadratic program 291 Indicator function 42 Infimal convolution 48 L.s.c. hull 53 Lagrange dual problem 309 Lagrangian 309 Level set (lower,upper) 50 Lineality 26 Lineality of a convex function 53 Lineality of a convex set 26 Linear matrix inequality (LMI) 129 Linear multiplicative program 240 Linear program under LMI, constraints 316 Local minimizer 109 Local optimal solution 109 Locally convexifiable 301 Locally d.c 86 Location-allocation problem 112 Low rank nonconvex problem 223 Lower linearizable 296 Lower semi-continuous (l.s.c.) 51 Minimax theorem 78 Minimum concave cost network flow problem MCNFP 262 Minkowski functional 10 Minkowski’s theorem 27 Modulus of strong convexity 71 Monotonic functions 223 Monotonic reverse convex problem 255 Multiextremal 109 Net 151 Network constraints 261 Noncanonical d.c. problems 206 Nonconvex quadratic programming 277 Nonregular problems 167 Norm 9 Norm, dual 35 Norm, Euclidean 9 Norm, Tchebychev 9 normal 22 Normal cone 22 Normal procedure DC 151 NS (normal subdivision) rule 150 Optimal visible point algorithm 209 Optimal visible point problem 209 Outer approximation 177 Parametric approach 236 Parametric d.c. feasibility problem 169 Parametric d.c. inclusion 134 205 Partition via 160 Phase (global, local) 124 polar 28 Polyhedral annexation 192 Polyhedral convex set 30 Polyhedron 30 Polytope 34 Positively homogeneous 063 Preprocessing 125 Problem PTP(2) 271 Procedure 195 Procedure DC 148 Projection of a point on a convex set 23 Prototype branch and select, algorithm 154 Quadratic constraints 293 Quadratic function 044 Quadratic minimization over, ellipsoids 281 Qualified member 154 Quasiconcave function 077 Quasiconcave minimization 116 Quasiconjugate function 199 Quasiconvex function 50 Quasiconvex maximization 116 Radial partition (subdivision) 141 Rank 30 Rank of a convex function 53 Rank of a quadratic form 279 Recession (cone, direction) 12 Rectangular subdivision 169 Refinement of a subdivision 143 Reformulation-linearization 304 Regular problem 166 Regularity asumption 167 Relative boundary 10 Relative interior 10 Relief indicator method 212 Representation theorem 26 restart 152 Reverse convex 56 Reverse convex, constraint 116 Reverse convex, inequality 84 164 Reverse convex, programming 116 164 Robust set 119 S-shaped functions 93 Saddle-function 77 Saddle-point 77 Second quasiconjugate 201 Semi-continuous (lower, upper) 51 Semi-definite program 130 Separable basic concave program 161 Separation theorem, first 19 Separation theorem, second 20 Separator 97 Shapley- Folkman’s Theorem 015 Simple OA for CDC 189 Simple outer approximation 181 Simplicial subdivision 141 Special concave program CPL(1) 269 Stochastic transportation-location, problem 264 Strictly convex 74 Strictly quasiconcave function 238 Strongly convex 71 Strongly separated 20 Subdifferentiable 62 Subdifferential 62 Subdivision (partition) via 160 Subdivision processes 140 153 Subgradient 62 Support function 29 42 Supporting hyperplane 21 Supporting hyperplane, hafspace 21 Tight convex minorant 298 Transcending stationarity 122 Transcending the incumbent 119 Trust region subproblem 282 Two phase scheme 124 Unary matrix 294 Unary program 294 Upper envelope 46 7 Vertex 033 Visibility assumption 207 Weakly convex 105 Weber problem with attraction and repulsion 130
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