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Giorgi G., Thierfelder J. — Mathematics of Optimization: Smooth and Nonsmooth Case
Giorgi G., Thierfelder J. — Mathematics of Optimization: Smooth and Nonsmooth Case

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Название: Mathematics of Optimization: Smooth and Nonsmooth Case

Авторы: 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.

The development of the subjects are self-contained and the bibliographical references are usually treated in different books (only a few books on optimization theory deal also with vector problems), so the book can be a starting point for further readings in a more specialized literature.

Assuming only a good (even if not advanced) knowledge of mathematical analysis and linear algebra, this book presents various aspects of the mathematical theory in optimization problems. The treatment is performed in finite-dimensional spaces and with no regard to algorithmic questions. After two chapters concerning, respectively, introductory subjects and basic tools and concepts of convex analysis, the book treats extensively mathematical programming problems in the smmoth case, in the nonsmooth case and finally vector optimization problems.

• Self-contained
• Clear style and results are either proved or stated precisely with adequate references
• The authors have several years experience in this field
• Several subjects (some of them non usual in books of this kind) in one single book, including nonsmooth optimization and vector optimization problems
• Useful long references list at the end of each chapter


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2004

Количество страниц: 616

Добавлена в каталог: 21.07.2008

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$\mathrm{r}^{+}$-convex function      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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