Designing approximation schemes for stochastic optimization problems, in particular for stochastic programs with recourse

Birge, John R.; Wets, Roger J.-B.

doi:10.1007/bfb0121114

Cited by 196 publications

(127 citation statements)

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“…Let /x e ~(RS;p, K) and /2 be a discrete approximation of ~ via conditional expectations (for details see [8,31]). Jensen's inequality then implies that also /2 c ~(~; p, K).…”

Section: ~(Z; P K) := {~ C ~(Z): Fz Llzll~"t~(dz)mentioning

confidence: 99%

Distribution sensitivity in stochastic programming

Römisch

Schultz

1991

Mathematical Programming

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In this paper, stochastic programming problems are viewed as parametric programs with respect to the probability distributions of the random coefficients. General results on quantitative stability in parametric optimization are used to study distribution sensitivity of stochastic programs. For recourse and chance constrained models quantitative continuity results for optimal values and optimal solution sets are proved (with respect to suitable metrics on the space of probability distributions). The results are useful to study the effect of approximations and of incomplete information in stochastic programming.

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“…Let /x e ~(RS;p, K) and /2 be a discrete approximation of ~ via conditional expectations (for details see [8,31]). Jensen's inequality then implies that also /2 c ~(~; p, K).…”

Section: ~(Z; P K) := {~ C ~(Z): Fz Llzll~"t~(dz)mentioning

confidence: 99%

Distribution sensitivity in stochastic programming

Römisch

Schultz

1991

Mathematical Programming

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“…Let us add some further bibliographical comments: Approximation techniques for solving stochastic programs with complete information on the underlying measures are presented in [4,12,37,42]. When having only incomplete information about the measures it is possible to use parametric as well as non-parametric statistical estimators instead of the true distributions.…”

Section: Introductionmentioning

confidence: 99%

Stability analysis for stochastic programs

Römisch

Schultz

1991

Ann Oper Res

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For stochastic programs with recourse and with (several joint) probabilistic constraints, respectively, we derive quantitative continuity properties of the relevant expectation functionals and constraint set mappings. This leads to qualitative and quantitative stability results for optimal values and optimal solutions with respect to perturbations of the underlying 9 probability distributions. Earlier stability results for stochastic programs with recourse and for those with probabilistic constraints are refined and extended, respectively. Emphasis is placed on equipping sets of probability measures with metrics that one can handle in specific situations. To illustrate the general stability results we present possible consequences when estimating the original probability measure via empirical ones.

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“…Piecewise-linear bounds (Birge and Wallace 1988, Wallace 1987b, Birge and Wets 1986, Birge and Wets 1989 are based on a functional approximation. These bounds can be computed by solving a sequence of restrictions of the recourse problem and thus yield upper bounds for a minimization problem (Birge and Wallace 1988).…”

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confidence: 99%

“…These bounds can be computed by solving a sequence of restrictions of the recourse problem and thus yield upper bounds for a minimization problem (Birge and Wallace 1988). For the purpose of solving stochastic LPs, it is signiÿcant that the piecewiselinear bound is complementary to the Jensen bound.…”

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confidence: 99%

Restricted-Recourse Bounds for Stochastic Linear Programming

Morton

Wood

1999

Operations Research

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We consider the problem of bounding the expected value of a linear program (LP) containing random coe cients, with applications to solving two-stage stochastic programs. An upper bound for minimizations is derived from a restriction of an equivalent, penalty-based formulation of the primal stochastic LP, and a lower bound is obtained from a restriction of a reformulation of the dual. Our "restrictedrecourse bounds" are more general and more easily computed than most other bounds because random coe cients may appear anywhere in the LP, neither independence nor boundedness of the coe cients is needed, and the bound is computed by solving a single LP or nonlinear program. Analytical examples demonstrate that the new bounds can be stronger than complementary Jensen bounds. (An upper bound is "complementary" to a lower bound, and vice versa). In computational work, we apply the bounds to a two-stage stochastic program for semiconductor manufacturing with uncertain demand and production rates.T his paper develops new techniques for bounding the expected value of a stochastic linear program, which is a linear program (LP), some or all of whose coe cients are random. The random coe cients may be discretely or continuously distributed, may be independent or contain dependencies, and may occur anywhere in the objective function, right-hand side, or constraint matrix. Calculating (or estimating) the expected value of a stochastic LP is key to solving two-stage and multi-stage stochastic programs with recourse (Dantzig 1955).It is usually impossible to compute exactly the expected value of a stochastic LP unless the random coe cients are discretely distributed and the total number of realizations of the coe cients is small, or the problem has a very special structure. As a result, solution techniques for stochastic programming with recourse usually involve approximations that are based on Monte Carlo sampling (e.g., Ermoliev 1983, Dantzig and Glynn 1990, Higle and Sen 1991 or on deterministically valid bounds. This paper involves approximations of the latter type.When initial lower and upper bounds are insu ciently tight, they can often be improved within a sequentialapproximation algorithm that iteratively partitions the support of the random variables (e.g., Kall et al. 1988). We do not discuss the details of these algorithms here, but note that our restricted-recourse bounds can be incorporated within such algorithms just as well as other bounds can, and possibly better because of improved computational e ciency and fewer technical requirements.The problem of computing the expected value of a stochastic LP also arises as a stand-alone problem. For instance, the stochastic maximum-ow problem (e.g., Evans 1976) calculates the expected value of the maximum ow through a network whose arc capacities are nonnegative random variables. Another example is the stochastic PERT problem (Fulkerson 1962), which evaluates the expected length of a longest path in a directed acyclic network with stochastic arc lengths. Both of these prob...

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Designing approximation schemes for stochastic optimization problems, in particular for stochastic programs with recourse

Abstract: Supported i n p a r t by g r a n t s of t h e N a t i o n a l Science Foundation.

Cited by 196 publications

References 32 publications

Distribution sensitivity in stochastic programming

Distribution sensitivity in stochastic programming

Stability analysis for stochastic programs

Restricted-Recourse Bounds for Stochastic Linear Programming

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