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Tuy H. — Convex analysis and global optimization
Tuy H. — Convex analysis and global optimization



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Название: Convex analysis and global optimization

Автор: Tuy H.

Аннотация:

Due to the general complementary convex structure underlying most nonconvex optimization problems encountered in applications, convex analysis plays an essential role in the development of global optimization methods. This book develops a coherent and rigorous theory of deterministic global optimization from this point of view. Part I constitutes an introduction to convex analysis, with an emphasis on concepts, properties and results particularly needed for global optimization, including those pertaining to the complementary convex structure. Part II presents the foundation and application of global search principles such as partitioning and cutting, outer and inner approximation, and decomposition to general global optimization problems and to problems with a low-rank nonconvex structure as well as quadratic problems. Much new material is offered, aside from a rigorous mathematical development.
Audience: The book is written as a text for graduate students in engineering, mathematics, operations research, computer science and other disciplines dealing with optimization theory. It is also addressed to all scientists in various fields who are interested in mathematical optimization.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1998

Количество страниц: 353

Добавлена в каталог: 05.11.2010

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$\Gamma$-extension      138 287
$\gamma$-valid concavity cut      138
$\omega$-subdivision      149
$\varepsilon$-approximate, optimal solution      166
$\varepsilon$-subgradient      70
Affine function      41
Affine set, manifold      3
Affinely independent      5
Approximate subdifferential      69
Barycentric coordinates      6
Basic concave programming, problem (BCP)      133
Basic outer approximation, theorem      180
Bilinear programming      290
Bisection      141
Bisection of ratio a      141
Bisection, exact      141
Branch and bound      155
Branch and cut      155
Branch and select      153
Canonical cone      139
Canonical d.c. programming, problem      118
Caratheodory core      16
Caratheodory’s theorem      14
Cl $\textit{f}.$      53
Closed function      53
Closed, open      6
Closure of a convex function      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
Concave quadratic programming, problem      287
Concavity cut      136
Cone      7 143
Conical subdivision      143
conjugate      72
Constancy space      52
Convex envelope      48
Convex function      41
Convex hull      13
Convex hull of a function      48
Convex hull of a set      13
Convex inequality      56
Convex minorant      100
Convex set      6
Convex-concave      77
Convg      48
CS restart algorithm for BCP      148
CS restart algorithm for LRCP      169
Cutting plane method      177
D.c set      95
d.c. function      83
D.c. programming      117
D.c. representation      88
D.c. structure      83
DC feasiblity problem      123
Decomposition      223
Decomposition by polyhedral annexation      233
Decomposition by projection      227
Decoupling relaxation      301
Dimension of a convex set      10
Dimension of a convex set of a convex function      53
Direction in which / is affine      53
Direction of recession      11
Directional derivative      65
Distance function      42 64
Distinguished point      154 177
Distinguished point, member      154
Dual DC feasibility problem      193
Dual problems      202
Dualization by polyhedral, annexation      233
EDGE      33
Edge property      164
Effective domain      41
Epigraph      41
Exact bisection      141
Exact selection      155
Exhaustive filter      141
Exhaustive subdivision process      143 7
Extreme direction      24
Extreme point      24
Face      23 31
Facet      32
Factorable 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
Geometric programming      126 285
global      74
Global, $\varepsilon$-optimal solution      123
Global, maximizer      74
Global, minimizer      74
Halfspace      6
Indefinite quadratic program      291
Indicator function      42
Infimal convolution      48
L.s.c. hull      53
Lagrange dual problem      309
Lagrangian      309
Level set (lower,upper)      50
Lineality      26
Lineality of a convex function      53
Lineality of a convex set      26
Linear matrix inequality (LMI)      129
Linear multiplicative program      240
Linear program under LMI, constraints      316
Local minimizer      109
Local optimal solution      109
Locally convexifiable      301
Locally d.c      86
Location-allocation problem      112
Low rank nonconvex problem      223
Lower linearizable      296
Lower semi-continuous (l.s.c.)      51
Minimax theorem      78
Minimum concave cost network flow problem MCNFP      262
Minkowski functional      10
Minkowski’s theorem      27
Modulus of strong convexity      71
Monotonic functions      223
Monotonic reverse convex problem      255
Multiextremal      109
Net      151
Network constraints      261
Noncanonical d.c. problems      206
Nonconvex quadratic programming      277
Nonregular problems      167
Norm      9
Norm, dual      35
Norm, Euclidean      9
Norm, Tchebychev      9
normal      22
Normal cone      22
Normal procedure DC      151
NS (normal subdivision) rule      150
Optimal visible point algorithm      209
Optimal visible point problem      209
Outer approximation      177
Parametric approach      236
Parametric d.c. feasibility problem      169
Parametric d.c. inclusion      134 205
Partition via $(\textit{v}, \textit{j})$      160
Phase (global, local)      124
polar      28
Polyhedral annexation      192
Polyhedral convex set      30
Polyhedron      30
Polytope      34
Positively homogeneous      063
Preprocessing      125
Problem PTP(2)      271
Procedure $DC^{*}$      195
Procedure DC      148
Projection of a point on a convex set      23
Prototype branch and select, algorithm      154
Quadratic constraints      293
Quadratic function      044
Quadratic minimization over, ellipsoids      281
Qualified member      154
Quasiconcave function      077
Quasiconcave minimization      116
Quasiconjugate function      199
Quasiconvex function      50
Quasiconvex maximization      116
Radial partition (subdivision)      141
Rank      30
Rank of a convex function      53
Rank of a quadratic form      279
Recession (cone, direction)      12
Rectangular subdivision      169
Refinement of a subdivision      143
Reformulation-linearization      304
Regular problem      166
Regularity asumption      167
Relative boundary      10
Relative interior      10
Relief indicator method      212
Representation theorem      26
restart      152
Reverse convex      56
Reverse convex, constraint      116
Reverse convex, inequality      84 164
Reverse convex, programming      116 164
Robust set      119
S-shaped functions      93
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
Separation theorem, second      20
Separator      97
Shapley- Folkman’s Theorem      015
Simple OA for CDC      189
Simple outer approximation      181
Simplicial subdivision      141
Special concave program CPL(1)      269
Stochastic transportation-location, problem      264
Strictly convex      74
Strictly quasiconcave function      238
Strongly convex      71
Strongly separated      20
Subdifferentiable      62
Subdifferential      62
Subdivision (partition) via $(\textit{v}, \textit{j})$      160
Subdivision processes      140 153
Subgradient      62
Support function      29 42
Supporting hyperplane      21
Supporting hyperplane, hafspace      21
Tight convex minorant      298
Transcending stationarity      122
Transcending the incumbent      119
Trust region subproblem      282
Two phase scheme      124
Unary matrix      294
Unary program      294
Upper envelope      46 7
Vertex      033
Visibility assumption      207
Weakly convex      105
Weber problem with attraction and repulsion      130
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