New Second-Order Bounds on the Expectation of Saddle Functions with Applications to Stochastic Linear Programming

Edirisinghe, N. C. P.

doi:10.1287/opre.44.6.909

Cited by 25 publications

(39 citation statements)

References 23 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Furthermore, we envision the primal and dual bounds being applied like other bounds within sequentialapproximation procedures. The gaps in the bounds of Table 1 are reasonably consistent with the initial gapsprior to reÿning the bounds by applying them in a conditional fashion-found in the computational work of Edirisinghe and You (1996), Edirisinghe andZiemba (1996), andFrauendorfer (1992).…”

Section: Computational Resultssupporting

confidence: 78%

“…The approach is applicable to LPs with possibly dependent stochastic right-hand sides and objectives. Edirisinghe andZiemba (1994a, 1994b) extend the bounds to the convex-concave case by applying them over more general, possibly unbounded, polyhedra. Edirisinghe (1996) and Edirisinghe and You (1996) develop distributional bounds on simplices and show how to tighten these bounds by using second-order information and by using simplices with least possible volume.…”

mentioning

confidence: 99%

“…It is possible to circumvent these di culties (Edirisinghe 1996, Edirisinghe and You 1996, Frauendorfer 1992) by generating manageable numbers of scenarios via Monte Carlo sampling and by applying the bounds to the resulting empirical model rather than the original model. For large sample sizes, this approach is justiÿed by the theory of epi-convergence (King and Wets 1991).…”

mentioning

confidence: 99%

“…We note that several of the extensions of the Jensen and E-M bounds, as well as some bounds based on functional approximations (e.g., Birge andDulÃ a 1991, Birge andWets 1986) may be derived in an alternative manner-in particular, as solutions of generalized moment problems (Birge and Wets 1987;Kall 1988;Frauendorfer 1992;Edirisinghe andZiemba 1994a, 1994b). This interpretation of bounds for stochastic programs begins with the work of DupaÄ covÃ a in a game-theoretic setting ( Ä ZÃ aÄ ckovÃ a 1966) and also has applications in models with incomplete information regarding the probability distribution (DupaÄ covÃ a 1994).…”

mentioning

confidence: 99%

See 3 more Smart Citations

Restricted-Recourse Bounds for Stochastic Linear Programming

Morton

Wood

1999

Operations Research

View full text Add to dashboard Cite

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...

show abstract

Section: Computational Resultssupporting

confidence: 78%

mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

See 2 more Smart Citations

Restricted-Recourse Bounds for Stochastic Linear Programming

Morton

Wood

1999

Operations Research

View full text Add to dashboard Cite

show abstract

“…!ver DECOM?. The reason why we used QDECOM in our tests is the following: DECOMP requires me !npiur_ data in tne s~andard S-ivlPS datafomal [I] which does not allow far thc a i k c sums (3). Wc prefer howcvcr to carry out our tests with test batteries inwlving (3) because these kind of problems proved to be much harder from the numerical point of view.…”

Section: Test Runsmentioning

confidence: 99%

On the role of bounds in stochastic linear programming

Kall

Mayer

2000

Optimization

View full text Add to dashboard Cite

Convex relaxations for global optimization under uncertainty described by continuous random variables

Shao

Scott

2018

AIChE Journal

View full text Add to dashboard Cite

This article considers nonconvex global optimization problems subject to uncertainties described by continuous random variables. Such problems arise in chemical process design, renewable energy systems, stochastic model predictive control, etc. Here, we restrict our attention to problems with expected-value objectives and no recourse decisions. In principle, such problems can be solved globally using spatial branch-and-bound (B&B). However, B&B requires the ability to bound the optimal objective value on subintervals of the search space, and existing techniques are not generally applicable because expected-value objectives often cannot be written in closed-form. To address this, this article presents a new method for computing convex and concave relaxations of nonconvex expected-value functions, which can be used to obtain rigorous bounds for use in B&B. Furthermore, these relaxations obey a secondorder pointwise convergence property, which is sufficient for finite termination of B&B under standard assumptions. Empirical results are shown for three simple examples.

show abstract

New Second-Order Bounds on the Expectation of Saddle Functions with Applications to Stochastic Linear Programming

Cited by 25 publications

References 23 publications

Restricted-Recourse Bounds for Stochastic Linear Programming

Restricted-Recourse Bounds for Stochastic Linear Programming

On the role of bounds in stochastic linear programming

Convex relaxations for global optimization under uncertainty described by continuous random variables

Contact Info

Product

Resources

About