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Auslender A., Teboulle M. — Asymptotic Cones and Functions in Optimization and Variational Inequalities
Auslender A., Teboulle M. — Asymptotic Cones and Functions in Optimization and Variational Inequalities



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Название: Asymptotic Cones and Functions in Optimization and Variational Inequalities

Авторы: Auslender A., Teboulle M.

Аннотация:

The book will serve as useful reference and self-contained text for researchers and graduate students in the fields of modern optimization theory and nonlinear analysis.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2002

Количество страниц: 249

Добавлена в каталог: 11.06.2008

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
Affine      2 89
Affine hull      2
Affine subspace      2 206
Asymptote      44
Asymptotic      36
Asymptotic approximation kernel      75 166
Asymptotic cone of function      55
Asymptotic cones      25—31
Asymptotic cones, dual characterization      31
Asymptotic cones, operations with      30
Asymptotic constraint qualification      134
Asymptotic direction      55 177
Asymptotic functions      47—60
Asymptotic functions as support functions      55
Asymptotic functions of conjugate      55
Asymptotic functions of maximal monotone maps      197 203 214
Asymptotic functions of spectral function      71
Asymptotic functions, calculus with      60
Asymptotic functions, convex cases.      50
Asymptotic functions, examples      51
Asymptotic functions, smoothing with      72
Asymptotic linear sets      36 93
Asymptotic optimality      229
Asymptotic polyhedral set      37
Asymptotically directional constant function      85
Asymptotically level stable function      94 99 162
Asymptotically linear function      94
Asymptotically well-behaved functions      124—132
Asymptotically well-behaved maximal monotone maps      219
Ball      3
Biconjugate      14 147
Biconjugate of marginal function      147 160
Bidual      148 149
Bipolar cone      6
Boundary      3
Boundary ray      44
Boundary relative      3
Bronsted — Rockafellar theorem      123
Caratheodory's theorem      3
Closed      2
Closed convex concave      see convex-concave functionals
Closure      3 33
Closure criteria      33 46
Closure of sum of convex sets      39
Coercivity      81 160
Coercivity of maps      214
Coercivity, characterization      83
Coercivity, conditions      84
Coercivity, weak      85 129
Cofinite function      64
Compactness for convex problems      84
Compactness of optimal solution set      82
Compactness of primal dual solution sets      160
Complementarity      224
Complementarity problem      213
Cone      5
Cone barrier      18
Cone convex      5
Cone, finitely generated      8
Cone, generated by set      7
Cone, ice cream      68
Cone, normal      6
Cone, pointed      6 42
Cone, polar      6
Cone, polyhedral      7
Cone, tangent      7
Conical hull      7
Conjugate function      13 198
Conjugate function of marginal      147
Constancy space      55 160
Constancy space of infimal convolution      90
Constancy space, dual representation of      57
Constraint qualification      120 160
Continuous sets      33
Continuous sets, characterization of      45
Continuous sets, convex      44
Convex combinations      2
Convex feasibility problem      116
Convex functions      9—13
Convex hull of functions      13
Convex sets      1—8
Convex sets, operations with      2
Convex, hull      2
Convex-concave functionals      170
Convex-concave functionals, closed      173 194
Convex-valued maps      206
Directional derivative      15
Directional local boundedness      208
Domains and ranges      20 see
Dual objective      158
dual operations      108
Duality      13
Duality and asymptotic functions      166—170
Duality and stationary sequences      178—181
Duality for generalized equations      221
Duality for nonconvex problems      148
Duality for semidefinite optimization      169
Duality, abstract      145—153
Duality, conjugate      145 222
Duality, gap      150
Duality, perturbational      145
Duality, strong      151 160
Duality, weak      148 154 159
Effective domains      9
Ekeland's principle      85 122
Epigraph      9
Epigraph, operations with      9
Error bounds      125
Error bounds for convex systems      133
Error bounds, global      125
Error bounds, Lipschitz      140
Error bounds, local      125
Error bounds, sharp      134
Euclidean vector space      1
Extension map      207 210
Extreme      8
Extreme point      8
Extreme ray      8
Fenchel      14
Fenchel duality      154—157
Fenchel duality theorem      155
Fenchel inequality      14
Fenchel — Moreau theorem      14
Fenchel, duality      154
Gap function      213
Gauge function      60
Generalized equations      212 216
Generalized equations, solving      218
Graph of maps      184
Half-spaces      2
Helly's theorem      116
hessian      15
Hoffman's error bound      138
homogeneous functions      18
Hyperplane      2
Indicator function      11
Inf-projection      11
Infimal convolution      14 90
Inner product      1
Interior      3
Isotone      78 166
Jensen's inequality      11
Krein — Milman theorem      8 37
Lagrangian duality      157—162
Lagrangian duality for VI problems      224
Legendre transform      see conjugate
Level bounded function      83
Level sets      9
Level sets, operations with      9
Line segment principle      3
Lineality space      55
Linear mapping      3
Lipschitz functions      12 53
Lower semicontinuity      10
Lower semicontinuity of maps      22
lsc functions      see lower semicontinuity
Marginal function      100 146 150 151 159
Marginal function for minimax problems      170
Matrix funtions      68
Maximal inonotonicity      183—186
Maximal monotone maps, compactness      211
Maximal monotone maps, convexity of      196
Maximal monotone maps, directional local boundedness      208
Maximal monotone maps, domains and ranges      195
Maximal monotone maps, locally bounded      211
Maximal monotone maps, range of sum      198
Maximal monotone maps, sum of      203
Metrically regular      139 140
Minimax theorems      175
Minimax theory      170 193
Minimax theory for convex-concave problems      173
Minimizing sequence      see sequences
Minkowski — Weyl theorem      8
Minkowski's theorem      8
Minty's theorem      186
Monotone maps      184
Monotone maps, continuous      185
Monotone maps, Lipschitz      189
Monotone maps, local boundedness      206
Monotone maps, maximal      185
Monotone maps, nonexpansive      185 189
Monotone maps, resolvent      189 193
Monotone maps, single valued      184 189
Monotone maps, star      199
Monotone maps, strict      184
Monotone maps, strongly coercive      200
Multivalued maps      20
Nonexpansivity      185 193
Norm      1
Normal cone      192
Normal cone, formulas for      17
Normal cones      see cone
Normal cones, operations with      112
Normal map      213
Optimality conditions      119
Optimality conditions, approximate      121
Optimality conditions, Fermat principle      120
Optimality conditions, KKT theorem      120
Optimality conditions, primal-dual      153
Orthant      6
Orthogonal complement      193
Orthogonal subspace      6
Palais — Smale condition      85 123
Perturbation function      see marginal function
Piecewise linear quadratic      95
Pointed      see cone
polar      see cone
Polar, operations with      7
Polyhedral      7
Polynomial convex function      86
Positive hull      7 58
Positively homogeneous      18 58
Primal-dual problems      147 148
Projection      89 126 192
Proper functions      9
Proximal map      192
Recession directions      84
Relative interior      see interior
Saddle      171
Saddle functions      171
Saddle points      171 177 193
Semibounded      42
Semibounded function      63
Semibounded set      42
Semibounded weakly      42
Semidefinite optimization      66
Semidefinite optimization, smoothing of      75
Separation      5
Separation of convex sets      5
Separation, polyhedral      7
Separation, proper      5
Separation, strong      5
Sequences      125
Sequences, asymptotically residual      140
Sequences, minimizing      125 142
Sequences, stationary      125 128 140 219
Set-valued maps      20—23
simplex      2
Slater's condition      120 160 170 228
Smoothing      see asymptotic functions
Smoothing, examples of      75
Spectral functions      68
Stability      100
Stability, convex case      112
Stationary sequence      see sequences
Strict convexity      11
Strong duality      151 153
Strong duality for minimax problems      175
Strong duality, dual attainment      151
Strong duality, primal attainment      153
Subdifferential      15 151 191 197 199 202 209
Subdifferential calculus      108 110
Subgradient      15 184 194 199
Sublinear function      18 78
Support functions      17—20
Supporting hyperplane      5
Symmetric functions      68
Symmetric matrices      67
Symmetric matrices, cone of psd      68
Symmetric matrices, eigenvalues of      68
Symmetric matrices, minimum eigenvalue      151
Tangent cone      see cone
Upper semicontinuity      22
Upper semicontinuity of functions      82
Upper semicontinuity of maps      22
Upper semicontinuity of maximal monotone maps      211
Variational inequalities, dual problems      225
Variational inequalities, existence results for      213
Variational inequalities, Lagrangian for      218
Variational inequalities, primal-dual      223
Variational inequalities, problems      212
Variational inequalities, solutions set      216
Weak coercivity      see coercivity
Weak coercivity of maps      215
Weak coercivity, characterization of      87
Weak duality      see duality
Weak duality for minimax      171
Weakly analytic      163
Weakly analytic, convex programs      163
Weakly analytic, functions      163
Weierstrass theorem      82
Well-behaved functions      see asymptotically
Well-behaved functions, dual characteriztion      130
Zero duality gap      150 151
Zero duality gap for special convex problems      16
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