references - springer978-1-4757-2809-5/1.pdf · references 321 [24] p.c. chen, p. hansen and b....
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Index
Index
affine function 41 affine set; manifold 3 affinely independent 5 approximate subdifferential 69 e-approximate optimal
solution 166 barycentric coordinates 6 basic concave programming
problem (BCP) 133 basic outer approximation
theorem 180 bilinear programming 290 bisection 141; exact 141
of ratio a 141 branch and cut 155 branch and bound 155 branch and select 153 cone 7,143 canonical cone 139 canonical d.c. programming
problem 118 Caratheodory core 16 Caratheodory's Theorem 14 elosed function 53 elosure of a convex function 53 elf. 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 contave quadratic programming
problem 287 concavity cut 136 conical subdivision 143
conjugate 72 coruştancy space 52 convex envelope 48 convex function 41 convex hulI 13; of a set 13
of a function 48 convex inequality 56 convex minorant 100 convg 48 convex set 6 convex-concave 77
337
CS restart algorithm for BCP 148 for LRCP 169
cutting plane method 177 DC feasiblity problem 123 d.c. function 83 d.c. programming 117 d.c. representation 88 d.c set 95 d.c. structure 83 decomposition 223; by projection 227
by polyhedral annexation 233 decoupling relaxation 301 dimension of a convex set 10
of a convex function 53 directional derivative 65 direction of recession 11 direction in which f is affine 53 distance function 42, 64 distinguished point 154, 177
member 154 dual DC feasibility problem 193 dual problema 202 dualization by polyhedral
annexation 233 edge 33 edge property 164 effective domain 41 epigraph 41 exact bisection 141 exact selection 155 exhaustive fl.lter 141
338 CONVEX ANALYSIS AND GLOBAL QpTIMIZATION
exhaustive subdivision process 143 'Y-extension 138; 287 extreme direction 24 extreme point 24 face 23; 31 facet 32 faclorable 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 global 74; - maximizer 74
- minimizer 74 - c-optimal solution 123
geometric programming 126; 285 halfspace 6; closed, open 6 indefinite quadratic program 291 indicator function 42 infimal convolution 48 Lagrange dual problem 309 Lagrangian 309 level set (lower, upper ) 50 lineality 26
- of a convex set 26 - of a convex function 53
linear matrix inequality (LMI) 129 linear multiplicative program 240 linear program under LMI
constraints 316 locally convexifiable 301 locally d.c. 86 local optimal solution 109 . local minimizer 109 location-allocation problem 112 lower linearizable 296 low rank nonconvex problem 223
lower semi-continuous (l.s.c.) 51 l.s.c. hulI 53 minimax theorem 78 minimum concave cost network
flow problem MCNFP 262 Minkowski functional 10 Minkowski's Theorem 27 modulus of strong convexity 71 monotonie functions 223 multiextremal 109 monotonie reverse convex problem 255 net 151 network constraints 261 noncanonical d.c. problems 206 nonconvex quadratic programming 277 nonregular problems 167 norm 9; Tchebychev 9
Euclidean 9; dual 35 normal 22; normal cone 22 NS (normal subdivision) rule 150 normal procedure DC 151 outer approximation 177 optimal visible point problem 209 optimal visible point algorithm 209 parametric approach 236 parametric d.c .. feasibility problem 169 parametric d.c. inclusion 134,205 partition via (v, j) 160 phase (global, local) 124 polar 28 polyhedral convex set 30 polyhedral annexation 192 polyhedron 30 polytope 34 positively homogeneous 063 preprocessing 125 procedure DC 148; DC* 195 problem PT P(2) 271 projection of a point
on a convex set 23 prototype branch and select
algorithm 154
Index
quadratic function 044 quadratic minimization over
ellipsoids 281 qualified member 154 quadratic problem with
quadratic constraints 293 quasiconcave function 077 quasiconcave minimization 116 quasiconvex function 50 quasiconvex maximization 116 quasiconjugate function 199 radial partition (subdivision) 141 rank 30
of a convex function 53 of a quadratic form 279
recession (cone, direction) 12 rect angular subdivision 169 reformulation-linearization 304 refinement of a subdivision 143 regular problem 166 regularity asumption 167 relative interior 10 relative boundary 10 relief indicator method 212 representation theorem 26 restart 152 reverse convex 56
constraint 116 - inequality 84; 164 - programming 116; 164
robust set 119 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; second 20) separator 97 Shapley-Folkman's Theorem 015 simple outer approximation 181 simple OA for CDC 189
339
simplicial subdivision 141 special concave program CPL(I) 269 S-shaped functions 93 stochastic transportation-Iocation
problem 264 strictly convex 74 strictly quasiconcave function 238 strongly convex 71 strongly separated 20 subdifferential 62 subdifferentiable 62 subdivision processes 140; 153 w-subdivision 149 subdivision (partition) via (v,j) 160 subgradient 62 g-subgradient 70 support function 29 42 supporting hyperplane 21
- hafspace 21 tight convex minorant 298 transcending the incumbent 119 transcending stationarity 122 trust region subproblem 282 two phase scheme 124 unary matrix 294 unary program 294 upper envelope 46 ')'-valid concavity cut 138 vertex 033 visibility assumption 207 weakly convex 105 Weber problem with attraction
and repulsion 130