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Kuhn D. — Generalized Bounds For Convex Multistage Stochastic Programs
Kuhn D. — Generalized Bounds For Convex Multistage Stochastic Programs

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Название: Generalized Bounds For Convex Multistage Stochastic Programs

Автор: Kuhn D.

Аннотация:

This book investigates convex multistage stochastic programs whose objective and constraint functions exhibit a generalized nonconvex dependence on the random parameters. Although the classical Jensen and Edmundson-Madansky type bounds or their extensions are generally not available for such problems, tight bounds can systematically be constructed under mild regularity conditions. A distinct primal-dual symmetry property is revealed when the proposed bounding method is applied to linear stochastic programs. Exemplary applications are studied to assess the performance of the theoretical concepts in situations of practical relevance. It is shown how market power, lognormal stochastic processes, and risk-aversion can be properly handled in a stochastic programming framework. Numerical experiments show that the relative gap between the bounds can typically be reduced to a few percent at reasonable problem dimensions.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2005

Количество страниц: 192

Добавлена в каталог: 16.02.2010

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$\sigma$-Algebra of events      7
Adjusted recourse problem      88 95 101
Anticipative policy      see “Policy”
Attouch      81
Augmented probability space      30
Auxiliary recourse problem      73
Auxiliary stochastic program      73 74
Barycentric approximation scheme      5 53
Barycentric approximation, lower      62
Barycentric approximation, upper      63
Barycentric coordinates      54
Barycentric probability measures      71
Barycentric transition probabilities      71
Barycentric weights      54
Barycentric weights, (generalized)      55
Barycentric weights, (generalized) history-dependent      70
Barycentric weights, history-dependent      69
Biconcave function      40
Biconjugate function      148
Biconvex function      46
Birge      3 6 27 46 65
Border      16
Bounding measure      53
Bounding set      78 111
Bounds on the recourse functions      74—77 88—89 99—100 102 110—111
Branch and bound algorithm      114
Branching factor      53
Capacity constraints      116
Caratheodory map      17
Caroe      115
Closed function      148
Closure of a function      147
Complementary slackness      56
Component coupling constraints      115
Concave conjugate function      148
Concave function      35
Concave function on nonconvex domain      36
Concave function, vector-valued      36
Conditional correction terms for nonconvex constraints      95
Conditional correction terms for nonconvex objective      87
Conditional correction terms for nonconvex objective and constraints      101
Conditional expectation, regular      9
Conditional expectation, regular w.r.t. barycentric measures      71
Conditional probability, regular      9
Conditional probability, regular w.r.t. barycentric measures      71
Confidence ellipse      117
Conjugate duality      147; see “Duality”
Conjugate function      148
Constraint function      11
Constraint function, defined on the augmented probability space      30
Constraint multifunction      12
Constraint multifunction, non-anticipative      12
Constraint qualification      33
Constraint qualification, Slater      33—34
Convex function      35
Convex function on nonconvex domain      35—188
Convex function, vector-valued      35
Convex hull      148
Correction terms for linear stochastic programs      110
Correction terms for nonconvex constraints      94
Correction terms for nonconvex objective      84
Correction terms for nonconvex objective and constraints      101
Cross-simplex      54
Curse of dimensionality      53
d.c. function      105
Danzig      102
Decision history      10
Decision rule      10
Decision rule, anticipative      10
Decision rule, non-anticipative      10
Deng      120
Dentcheva      115
Dokov      3
Dual problem conjugate duality      150
Dual problem Lagrangian duality with equality constraints      161
Dual problem Lagrangian duality without equality constraints      157
Duality gap      152
Duality, conjugate      147
Duality, Lagrangian      155
Duality, strong      152
Duality, weak      152
Dula      3
Dupacova      3 56
Dynamic constraints      114
Dynamic version      32
Edirisinghe      3 144
Edmundson      3
Edmundson — Madansky inequality      59
Effective profit function      14
Effective profit function, defined on the augmented probability space      31
End effects      127
Energy balance equation      116
Energy conversion factor      118
Epi-convergence      81
Event      7
EVPI      see “Expected value of perfect information”
Exact penalty functions      164
Expectation functional      15 32
Expected value of perfect information      7 26
Extended arithmetic      36
Feasible set mapping      12 32
Feasible set mapping, nested      12 32
Filtration      7
Filtration, induced      8
Fleten      115
Frauendorrnr      3 5 29 53 54 66
Fubini's theorem (generalized)      20
Gassmann      3
Generalized barycenters      61 63
Generalized barycenters, history-dependent      70
Generalized barycentric weights      see “Barycentric weights”
Generalized feasible set      13
Generalized feasible set, defined on the augmented probability space      32
Generalized Fubini theorem      20
Groewe-Kuska      115
Guessow      115
Hartman      106
Here-and-now      25
Huang      65
Hypograph      147
Induced constraints      12
Induced filtration      see “Filtration”
Inverse demand elasticity      119
Inverse demand function      119
Jensen      2 3
Jensen's inequality      57
Kall      3 65
Klein Haneveld      22
Lagrange multiplier      161
Lagrangian function      150 157 160
Lagrangian relaxation      114
Level set      13
Linear stochastic program      see “Stochastic program”
Louveaux      27 46
Madansky      3 26
MILP      see “Mixed integer linear program”
Mixed integer linear program      114
Moment problem      53
Morton      3
Multifunction      see “Set-valued mapping”
Multistage stochastic program      see “Stochastic program”
Natural domain of the expectation functional      15 32
Natural domain of the recourse function      15 32
Nested feasible set mapping      see “Feasible set mapping”
Non-anticipative constraint multifunction      see “Constraint multifunction”
Non-anticipative policy      see “Policy”
Non-anticipativity constraints      114
Normal form of a linear stochastic program      104
Normal integrand      12 19
Normality      152
Objective function coefficients      103
Observation      8
Ostermaier      115
Outcome      7
Outcome history      10 30
Partition of a conditional probability      68
Partition of a probability measure      64
Partition of a set      64
Penalty formulation      163—164
Pereira      114
Pflug      4 78
Pinto      114
Policy      10
Policy, anticipative      10
Policy, non-anticipative      10
Primal problem conjugate duality      150
Primal problem Lagrangian duality with equality constraints      161
Primal problem Lagrangian duality without equality constraints      157
Profit function      14
Profit function, defined on the augmented probability space      30
Proper concave function      147
Proximal bundle method      114
Pseudo-probabilities      60 63
Pseudo-probabilities, history-dependent      70
Rachev      4
Raiffa      26
Random variable      8
Random vector      8
Recourse function      15 32
Recourse matrix      103
refinement      65 71
Refinement parameter      64 68
Regular conditional expectation      see “Conditional expectation”
Regular cross-simplex      see “Cross-simplex”
Regular refinement strategy      72
Regularity conditions      17 22 24 37 45 46 85 91 100 107
Regularizable constraint function      89
Regularizable profit function      84
Revenue balance equation      125
Rhs vector      see “Right hand side vector”
Right hand side vector      103
Risk aversion      123
Rockafellar      12 16 40 47 147
Roemisch      4 114 115
Sample space      7
Scenario generation      1 51
Scenario problem      25
Scenario tree      52
Scenario tree node      52
Scenario tree path probability      52
Schlaifer      26
Schultz      114 115
SDDP      see “Stochastic dual dynamic programming”
SDP      see “Stochastic dynamic programming”
Set-valued mapping bounded      16
Set-valued mapping continuous      16
Set-valued mapping lower semicontinuous      16
Set-valued mapping upper semicontinuous      16
Shadow price      122
Shapiro      106
Slater point      34
Slater's constraint qualification      see “Constraint qualification”
Stability      152
State space      8
Static version      14 31
Stochastic dual dynamic programming      114
Stochastic dynamic programming      113
Stochastic process      8
Stochastic process, adapted      8
Stochastic process, block-diagonal autoregressive      36
Stochastic process, deterministic      8
Stochastic process, nonlinear autoregressive      16
Stochastic process, previsible      8
Stochastic program      1
Stochastic program, convex      29
Stochastic program, dynamic version      14 32
Stochastic program, linear      102
Stochastic program, non-linear      14
Stochastic program, static version      14 31
Strategy      10
Strategy, anticipative      10
Strategy, non-anticipative      10
Subdifferentiability      47—49
Subgradient method      114
Sup-projection      40 169—174
Technology matrix      103
Transition probability      52 68
Truncation      116
Upper semicontinuity      147
Usc hull      147
Utility function      123
Value of the stochastic solution      7 27
VSS      see “Value of the stochastic solution”
Wait-and-see      26
Wallace      115
Weak convergence      66
Wets      3 6 12 16 43 65 81 173
Ziemba      3 115
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