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Giorgi G., Thierfelder J. — Mathematics of Optimization: Smooth and Nonsmooth Case
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Название: Mathematics of Optimization: Smooth and Nonsmooth CaseАвторы: Giorgi G., Thierfelder J. Аннотация: The book is intended for people (graduates, researchers, but also undergraduates with a good mathematical background) involved in the study of (static) optimization problems (in finite-dimensional spaces). It contains a lot of material, from basic tools of convex analysis to optimality conditions for smooth optimization problems, for non smooth optimization problems and for vector optimization problems.
Язык: Рубрика: Математика /Статус предметного указателя: Готов указатель с номерами страниц ed2k: ed2k stats Год издания: 2004Количество страниц: 616Добавлена в каталог: 21.07.2008Операции: Положить на полку |
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174 Abadie — Mangasarian linearization theorem 249 Accessibility lemma 33 Active constraints 249 Affine combination 29 Affine hull 29 Affine set 29 Alternative (theorems of the) 59 Arcwise connected set 177 Arcwise convex function 177 Arcwise convex set 177 Asymptotic cone 531 Augmented Lagrangian 495 Bi-conjugate function 106 Binding constraints 249 Bohnenblust, Karlin and Shapley theorem 115 Bouligand tangent cone 222 C-bounded set 535 C-closed set 535 C-compact set 534 C-complete set 540 C-convex functions 526 C-convex set 527 C-convexlike functions 527 C-invex functions 526 C-pre-invex functions 526 C-pseudoconvex functions 526 C-quasibounded set 539 C-quasiconvex functions 527 C-semicompact set 534 Caratheodory's theorem 28 Clarke generalized derivative 383 Clarke subdifferential 393 Clarke tangent cone 238 Closed line segment 23 Complementarity slackness conditions 252 334 Completely differentiable functions 363 Composite functions (generalized convexity of) 161 Concave function 70 Cone 49 Cone convex 49 Cone normal 59 Cone of attainable directions 210 228 Cone of decreasing directions 438 Cone of epi-Lipschitzian directions 239 Cone of feasible directions 229 Cone of gradients 265 Cone of interior directions 231 Cone of locally constrained directions 261 Cone of quasi-interior directions 233 Cone polar or dual 53 Cone, generated by a set 49 Cone, pointed 49 Cone, polyhedral or finite 57 Conical hull 50 Conical preference orders 508 Conjugate function 88 102 Constraint qualifications 252 Constraint qualifications, Abadie c.q. 272 285 Constraint qualifications, Arrow — Hurwicz — Uzawa first c.q. 254 Constraint qualifications, Arrow — Hurwicz — Uzawa second c.q. 271 285 Constraint qualifications, Cottle — Dragonirescu c.q. 253 Constraint qualifications, generalized Kariin c.q. 339 Constraint qualifications, generalized Slater c.q. 339 Constraint qualifications, generalized strict c.q. 339 Constraint qualifications, Guignard — Gould — Tolle c.q. 264 272 Constraint qualifications, Kariin's c.q. 253 337 Constraint qualifications, Kuhn — Tucker c.q. 271 285 Constraint qualifications, Mangasarian — Fromovitz c.q. 254 285 Constraint qualifications, Modified strict c.q. 253 Constraint qualifications, Nondegeneracy c.q. 254 285 Constraint qualifications, Reverse c.q. 253 285 Constraint qualifications, Slater's c.q. 253 337 Constraint qualifications, Slater's weak c.q. 253 285 Constraint qualifications, strict c.q. 337 Constraint qualifications, strict Mangasarian — Fromovitz c.q. 307 Constraint qualifications, Zangwill c.q. 271 Constraint set 4 Constraints 6 Contingent cone 222 Continuously differentiable functions 18 Convex combination 25 Convex cone of a set 59 Convex conical hull 50 Convex function 70 Convex function proper 84 Convex function twice continuously differentiable 79 Convex function, continuity of 90 Convex function, differentiable 77 Convex function, directional derivatives of 94 Convex function, generalized 118 Convex function, generalized at a point 156 Convex function, improper 84 Convex function, strictly 80 Convex hull (of a set) 26 Convex set 23 Convex-like functions 117 D-R-quasidifferentiability 394 Differentiable functions 18 Dini directional derivative 379 Directional derivative 94 360 Distance function 87 Dual local cone approximation 400 Dual optimization problem 459 Duffin's theorem 69 Effective constraints 249 Effective domain (of a convex function) 84 Efficient points (existence theorems) 533 Efficient values 509 Epigraph 15 Extreme point (of a convex set) 47 Fan-Glicksberg and Hoffman theorem 111 Farkas — Minkowski's theorem 60 Feasible arc 209 Feasible direction 202 Feasible point 4 Feasible set 4 Fenchel — Moreau theorem 106 Fenchel's theoreme of the alternative 69 Fractional programming problem 325 Frechet differentiable functions 363 Fritz John multipliers 252 Fritz John necessary optimality conditions 251 283 Fritz John sufficient optimality conditions 291 Gale's theorems 68 Gateaux differentiability 361 Gauge function or Minkowski function 89 Generalized Abadie constraint qualification 444 Generalized directional derivative (vector case) 572 Generalized Fritz John conditions 447 Generalized Gould — Tolle constraint qualification 444 Generalized gradient 393 Generalized Jacobian 576 577 Generalized Kuhn — Tucker conditions 448 Generalized Lagrangian 494 Generalized Mangasarian — Fromovitz constraint qualification 444 Generalized marginal function 487 Generalized r-th mean 173 Generalized subdifferential (vector case) 573 Geometric programming problem 326 Gordan's Theorem 67 Gould — Tolle problem 276 Gould — Tolle regularity 262 279 Gradient vector 18 360 Guignard — Gould — Tolle constraint qualification 264 Hadamard differentiability 364 Helly's theorem 46 Hessian matrix 19 Hypertangent cone 238 Hypograph 15 Implicit function theorem 20 Indicator function 88 Infimal convolution 108 Inner product (of two vectors) 13 Interval (of real numbers) 12 Invex function 180 Jacobian matrix 18 Jensen's inequality 73 K-directional derivative 401 K-subdifferential 405 K-subgradient 405 Krein — Milman's theorem 48 Kuhn — Tucker necessary optimality conditions 257 278 Kuhn — Tucker sufficient optimality conditions 288 292 Ky Fan's theorem 63 Lagrange dual problem 475 Lagrange duality 469 Lagrange — Fritz John conditions 216 Lagrangian function 212 Lagrangian multipliers 214 Leading principal minors 206 Limit inferior 16 Limit superior 16 Linear programming problem 324 460 Linearization theorem 249 Linearizing cone 211 261 438 Lipschitz condition 17 Local cone approximation 221 396 Local minimum 4 Logarithmic convex function 166 Lower Dini directional derivative 379 Lower level set 15 Lower semicontinuous function 5 Mangasarian's theorems of the alternative 68 Marginal function 481 Matrix 13 Matrix, identity 14 Matrix, nonnegative 14 Matrix, positive 14 Matrix, semipositive 14 Matrix, skew symmetric 13 Matrix, symmetric 13 Merely quasiconvex (merely pseudoconvex) function 170 Merely subdefinite matrix 170 Michel — Penot directional derivative 391 Min-max problem 326 Minimax theorem 472 Minkowski functional 566 Modularity property 55 Monotonicity (generalized) 149 Motzkin's theorem 68 Nonlinear fractional function (generalized convexity of) 164 Norm (of a vector) 13 Normal problem 491 North West principal minors 206 Objective function 4 Open line segment 23 Optimal point 4 Optimal solution 4 Optimality conditions (vector nonsmooth case) 580 Optimality conditions (vector smooth case) 546 Perturbation function 487 Perturbed optimization problems 486 Polarity property 55 Polyhedron or polyhedral set 48 Polytope 48 Positive subdefinite matrix 170 Positively homogeneous function 93 Pre-invex function 184 Primal optimization problem 459 Principal minors 206 Projection cone 58 Proper efficiency 512 Proper separation hyperplane 40 Proper separation theorem 44 Prototangent cone 246 Pseudo-invex function 181 Pseudoconvex function 136 Pseudoconvex function, non differentiable 137 Pseudoconvex function, strictly 139 Pseudoconvex function, twice continuously differentiable 137 Pseudomonotone or pseudolinear function 138 Pseudotangent cone 227 Quadratic form 14 206 217 Quadratic form, definite 14 206 Quadratic form, indefinite 14 Quadratic form, semidefinite 14 206 Quadratic form, subdefinite 170 Quadratic function 168 Quadratic programming problem 324 Quasi-invex function 181 Quasi-minimum vector 562 Quasiconvex function 118 Quasiconvex function, continuous 122 Quasiconvex function, differentiable 120 Quasiconvex function, semistrictly 134 Quasiconvex function, strictly 132 Quasiconvex function, twice continuously differentiable 123 Quasidifferentiability in the sense of Demyanov and Rubinov see “D-R-quasidifferentiability” Quasidifferentiable function 380 Quasidifferential 394 Quasimonotone or quasilinear function 130 r-convex function 175 Radial minimum point 148 Radial tangent cone 230 Recession cone 247 Regular locally convex function 381 Regularity conditions 321 Rockafetlar upper subderivative 388 Saddle point (for a function f) 207 Saddle point (of a Fritz John — Lagrange function) 336 Saddle point (of a Lagrangian function) 333 Scalarization techniques 554 Second-order constraint qualification 303 Second-order optimality condition 303 Semiconvex function 136 Sensitivity Theorem 219 316 Separation hyperplane 40 Separation theorem 43 Sets of natural numbers 11 Sets of rational numbers 11 Sets of real numbers 11 Sets, algebraic sum of 11 Sets, boundary of 12 Sets, cardinality of 12 Sets, cartesian product of 11 Sets, closure of 12 Sets, complement of 11 Sets, difference of 11 Sets, empty 11 Sets, exterior of 12 Sets, interior of 12 Sets, relative interior of 12 30 simplex 48 Slack variable 7 248 Slater's theorem of the alternative 68 Stable problem 491 Star-shaped set 269 Stiemke's theorem 68 Strict complementarity slackness conditions 316 Strict differentiability 365 Strict separation hyperplane 40 Strictly subdefinite matrix 170 Strong duality 460 Strong duality (vector case) 554 Strong separation function 116 Strong separation hyperplane 41 Strong separation theorem 43 Strongly active constraint 307 Subadditive function 93
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