Restricted-Recourse Bounds for Stochastic Linear Programming

Abstract

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...

Alternative Water Allocation in Kyrgyzstan: Lessons from the Lower Colorado River Basin and New South Wales

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“…Using the data given in Table 1, the model generates river inflow for different probabilistic scenarios of water supply [17].…”

“…Modeling platforms have been used to simulate water allocation decisions [9][10][11][12][13][14][15][16][17][18][19][20][21]. The impact of transition from a top-down to a user-based participatory water management approach on water use efficiency and the reallocation of saved water from the first to the second crop was analyzed for the Akbura River basin using a two-stage stochastic linear programming that takes into account the probabilistic water availability.…”

A tighter variant of Jensen’s lower bound for stochastic programs and separable approximations to recourse functions

“…The computational effort for the last three methods grows linearly as the number of random variables in the problem increases, but the numerical experiments in Birge and Wallace (1988) and Birge and Wets (1989) demonstrate that the upper bounds that are obtained by these methods can be less tight than Edmundson Madansky upper bound. Morton and Wood (1999) construct upper and lower bounds on the recourse functions by relaxing some of the constraints in the problem and penalizing the violations of the relaxed constraints. Their approach has similarities with ours, but if we work with stochastic programs that include random variables only on the right side of their constraints, then the lower bounds obtained by Morton and Wood (1999) collapse to Jensen's lower bound, whereas our lower bounds have the potential to improve Jensen's lower bound.…”