J. M. Borwein, Qi.Zhu — Techniques of Variational Analysis (CMS Books in Mathematics) :: Электронная библиотека попечительского совета мехмата МГУ

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J. M. Borwein, Qi.Zhu — Techniques of Variational Analysis (CMS Books in Mathematics)

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Название: Techniques of Variational Analysis (CMS Books in Mathematics)

Авторы: J. M. Borwein, Qi.Zhu

Аннотация:

Variational arguments are classical techniques whose use can be traced back to the early development of the calculus of variations and further. Rooted in the physical principle of least action, they have wide applications in diverse fields. This book provides a concise account of the essential tools of infinite-dimensional first-order variational analysis. These tools are illustrated by applications in many different parts of analysis, optimization and approximation, dynamical systems, mathematical economics and elsewhere. Much of the material in the book grows out of talks and short lecture series given by the authors in the past several years. Thus, chapters in this book can easily be arranged to form material for a graduate level topics course. A sizeable collection of suitable exercises is provided for this purpose. In addition, this book is also a useful reference for researchers who use variational techniques - or just think they might like to.

Язык:

Рубрика: Математика/

Статус предметного указателя: Готов указатель без номеров страниц

ed2k: ed2k stats

Год издания: 2005

Количество страниц: 366

Добавлена в каталог: 30.04.2020

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Предметный указатель

      2
$(X, \|\cdot\|)$       5
      3
      299
      78
      2
      317
      317
      3
      2
      321
      2
$B_{sa}$       317
      220
      220
      149
      3
      23
      166
      66
$E_{N \times M}$       299
$f \circ g$       222
      52
      39
$f\ \framebox{\makebox[\totalheight]{\itshape }}\ g$       3
      134
$f^{**}$       134
$f^{-1}$       4
$f^{\circ}$       188
      177
$f_{+}$       98
      299
      299
$G_{\delta}$       243
$K(x^*, \varepsilon)$       10
      292
$K^{oo}$       136
$K^{o}$       83
      21
      190
      41
      196
      299
      22 302
      141
$P_{\gamma}(a,b)$       11
$P_{\pi}(x,y)$       321
$S(f, S, \alpha)$       267
      299
      301
      213
      47
      292
      292
      292
      209
      190
      209
      291
$x\ \odot \ x$       318
$x\ \prec \ y$       21
      2
      48
$x^{+}$       69
$x^{-}$       69
$x^{\downarrow}$       21
      299
      12
      15
$\#(S)$       23
$\chi_S$       45
$\delta_{ij}$       317
$\Gamma(a,b)$       275
$\gamma_C$       111
$\iota(S; \cdot)$       3
$\iota_S$       3
$\lambda(A)$       302
$\langle\cdot, \cdot\rangle$       2
$\mathcal{A}(N)$       25
$\mathcal{B}$       318
$\mathcal{K}$       223
$\mathcal{N}$       317
$\overline(S)$       3
$\overline{conv}\ S$       3
$\overline{d}(S; \cdot)$       3
$\partial$       117
$\partial^F$       44
$\partial^{\infty}$       196
$\partial^{\infty}_C$       191
$\partial_C$       189
$\partial_F$       39
$\partial_L$       196
$\partial_P$       46
$\partial_{VF}$       40
$\phi^*$       292
$\phi_*$       292
$\pi$       292
$\Pi^*$       292
$\rho(\cdot, \cdot)$       30
$\sigma(A)$       302
$\sigma_C$       111
$\tau_n$       62
$\textbf{C}$       317
$\wedge[f_1,...,f_N](S)$       47
${\rm I\!R} \cup \{+\infty\}$       3
${\rm I\!R}$       3
${\rm I\!R}^N$       19
${\rm I\!R}^N_{+}$       21
${\rm I\!R}^N_{\geq}$       302
:doubly stochastic pattern      159
Analytic center      139
approximate chain rule      87 90 93 107
approximate chain rule, strong      87
approximate chain rule, weak      88
approximate critical point      10
approximate extremal principle      104 107 254
approximate Fermat principle      19 20 27
approximate local sum rule      54 71 74 81 87 102 107 197 225 245 253 254
approximate local sum rule, Lipschitz      245
approximate local sum rule, strong      54 57 60 74
approximate local sum rule, strong, failure of      74
approximate local sum rule, weak      57 62
approximate mean value theorem      81 82 83 167 193
approximate multidirectional mean value inequality      95 253 254
approximate multiplier rule      62
approximate multiplier rule, weak      64
approximate nonlocal sum rule      48 51 55 95 105 128 253 254 260
approximate projection      232
Argmin      166
Asplund space      243
Asplund space and Fr $\' e$ chet smooth space      245
Asplund space, separable      245
Asplund space, subdifferential characterization      254
Asplund space, sum rule characterization      252
asymptotically regular      145
Atlas      291
attainment in Fenchel problems      136
attractive set      89
Baire category theorem      33 203
Banach fixed point theorem      15
bd S      3
Biconjugate      134
Bilinear form      321
Bipolar      136
Birkhoff theorem      25 323
Bishop — Phelps cone      10 59 134
Bishop — Phelps theorem      10
Borwein — Preiss variational principle      30 34
Borwein — Preiss variational principle in finite dimensional spaces      19
Borwein — Preiss variational principle, general form      32

Borwein — Preiss variational principle, subdifferential form      42
Boundary      3
Bregman distance      149
Bump function      33
bump function, range      36
Canonical projection      292
Caristi — Kirk fixed point theorem      17
Chain rule      87 91 93
chain rule for permutation invariant function      307 310
chain rule for spectral function      307 311
chain rule limiting      198
chain rule on manifolds      296
chain rule smooth      94 297
chain rule, approximate      87 88 90 94 107
chain rule, convex subdifferential      129
Chebyshev set      217
Clarke directional derivative      188
Clarke normal cone      190
Clarke normal cone representation      195
Clarke representation      195
Clarke singular subdifferential      191
Clarke subdifferential      189
Clarke subdifferential maximal      206
Clarke subdifferential of distance function      215
Clarke subdifferential representation      193 195
Clarke tangent cone      190 209
Closed convex hull      3
Closest point      215 239
closest point density      217
Closure      3
Coderivative      220
coderivative chain rule limiting      227
coderivative chain rule strong      226
coderivative sum rule limiting      226
coderivative sum rule strong      224
coderivative sum rule weak      221
coderivative, Fr $\' e$ chet      220
coderivative, limiting      220
Comparison theorem      50
Concave conjugate      139
Concave function      111
cone bipolar      136
cone monotonicity      83
cone polar      83 136
Conjugation      134
Constrained optimization problem      41 62 65 108 109 190 211 212 298
constrained optimization problem and inequality      69
constrained optimization problem equilibrium constraint      77
constrained optimization problem, multifunction constraint      76
constrained optimization problem, variational inequality constraint      76
Constraint linear      136 139
Constraint qualification      64 66 120 161
constraint qualification calmness      68 72
constraint qualification Mangasarian — Fromovitz      64 67 72
Contingent cone      209
Contraction      15
contraction directional      16
conv S      3
Convex feasibility problem      140
Convex function      111
convex function regularity      192
convex function, difference of      150
convex function, differentiability of      121
convex function, recognizing      124
Convex hull      3
convex multidirectional mean value theorem      130
Convex program      135
convex quasi      83
convex separation      209
convex series(cs) closed      113
convex series(cs) compact      113
Convex set      111
convex subdifferential      117
convex subdifferential calculus      129 135 136
convex subdifferential failure of calculus      134
convex subdifferential sum rule      129
convex subgradient      777 134
convex, normal cone      117
Convexity      80 167
Core      114
core versus interior      116
core(S)      114
Cotangent bundle      292
Cotangent space      292
Cotangent vectors      292
cotangent vectors limit of      293
critical sets      89
cusco      174 190
cusco minimal      174 215
DAD problem      159
DC function      124 214
decoupled infimum      47
decoupling lemma      127 178
decrease principle      97 98
demi-closed      187 184
det(a)      302
Determinant      302
Deville — Godefroy — Zizler variational principle      33
diag      303
diag*      303
diam S      3
Diameter      3 267
Differentiability of convex functions      121
differentiability of distance function      150
differentiability, Fr $\' e$ chet      39 243
differentiability, G $\^ a$ teaux      121 258 330
Directional contraction      16
Directional derivative      117
directional derivative and subgradients      119
directional derivative of convex function      118
directional derivative sublinear      118
directional derivative, Clarke      188
Distance function      3 38 214 230
dom f      3
Domain      3
domain of subdifferential      117
domain of subdifferential not convex      123
domF      4
Doubly stochastic matrix      23 159
doubly stochastic matrix and majorization      26
Drop      12 95
drop and flower petal      13
drop theorem      13
Dual space      2
duality entropy maximization      157
duality inequalities      150
Duality map      178
Duality pairing      2
Duality weak      135 136
e-lim inf_{i \to \infty}\ f_i$      170
e-lim sup $_{i \to \infty}\ f_i$       171
e-lim $_{i \to \infty}\ f_i$       171
Eigenvalue function      38 312
eigenvalue largest      77
eigenvalue mapping      302 320
Ekeland variational principle      6 14
Ekeland variational principle alternative forms      8
Ekeland variational principle and Bishop — Phelps theorem      14
Ekeland variational principle and completeness      14
Ekeland variational principle geometric picture      5
Enlargement      3
Entropy maximization      157 159
entropy maximization duality      157 161
entropy maximization in infinite dimensional space      161
Entropy maximum      137 139
entropy, Boltzmann — Shannon      149 158
epi f      3
Epi-convergence      170
epi-limit      171
epi-limit characterization      171

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