We consider linear programs with uncertain parameters, lying in some prescribed uncertainty set, where part of the variables must be determined before the realization of the uncertain parameters (‘‘non-adjustable variables’’), while the other part are variables that can be chosen after the realization (‘‘adjustable variables’’). We extend the Robust Optimization methodology ([1, 3-6, 9, 13, 14]) to this situation by introducing the Adjustable Robust Counterpart (ARC) associated with an LP of the above structure. Often the ARC is significantly less conservative than the usual Robust Counterpart (RC), however, in most cases the ARC is computationally intractable (NP-hard). This difficulty is addressed by restricting the adjustable variables to be affine functions of the uncertain data. The ensuing Affinely Adjustable Robust Counterpart (AARC) problem is then shown to be, in certain important cases, equivalent to a tractable optimization problem (typically an LP or a Semidefinite problem), and in other cases, having a tight approximation which is tractable. The AARC approach is illustrated by applying it to a multi-stage inventory management problem.

https://www2.isye.gatech.edu/~nemirovs/MP_Elana_2004.pdf

Adjustable robust solutions of uncertain linear programs

This paper gives an algorithm for L-shaped linear programs which arise naturally in optimal control problems with state constraints and stochastic linear programs (which can be represented in this form with an infinite number of linear constraints). The first section describes a cutting hyperplane algorithm which is shown to be equivalent to a partial decomposition algorithm of the dual program. The two last sections are devoted to applications of the cutting hyperplane algorithm to a linear optimal control problem and stochastic programming problems.

https://www.math.ucdavis.edu/~rjbw/mypage/Stochastic_Optimization_files/VnsW69_Lshape.pdf

L-shaped linear programs with applications to optimal control and stochastic programming.

Multistage stochastic linear programs model problems in financial planning, dynamic traffic assignment, economic policy analysis, and many other applications. Equivalent representations of such problems as deterministic linear programs are, however, excessively large. This paper develops decomposition and partitioning methods for solving these problems and reports on computational results on a set of practical test problems.

Decomposition and Partitioning Methods for Multistage Stochastic Linear Programs

A problem of minimizing a sum of many convex piecewise-linear functions is considered. In view of applications to two-stage linear programming, where objectives are marginal values of lower level problems, it is assumed that domains of objectives may be proper polyhedral subsets of the space of decision variables and are defined by piecewise-linear induced feasibility constraints. We propose a new decomposition method that may start from an arbitrary point and simultaneously processes objective and feasibility cuts for each component. The master program is augmented with a quadratic regularizing term and comprises an a priori bounded number of cuts. The method goes through nonbasic points, in general, and is finitely convergent without any nondegeneracy assumptions. Next, we present a special technique for solving the regularized master problem that uses an active set strategy and QR factorization and exploits the structure of the master. Finally, some numerical evidence is given.

A regularized decomposition method for minimizing a sum of polyhedral functions

This paper examines the effects on the optimal ordering policy and its cost of allowing for interactions among retail outlets. Specifically, transshipments are allowed as recourse actions occurring after demands are realized but before they must be satisfied. The resultant savings in holding and shortage costs are balanced against the costs of transshipment. A base stock ordering policy is optimal for this model. If the final period base stock order-up-to point is nonnegative, then it will be the base stock order-up-to point for all periods; unfortunately, it can be found analytically only for two special cases, where either the cost parameters are equal at every outlet or there are only two outlets. These two special cases are used to validate a heuristic solution technique employing Monte Carlo integration, which is then compared to an easily calculated base policy in order to gauge the contribution of this model. The additional savings from using this heuristic policy are significant, particularly for problems with many retail outlets and low transshipment costs.

Optimal and Approximate Policies in Multiperiod, Multilocation Inventory Models with Transshipments

: A possible method for compensating for uncertainty in linear- programming problems is to replace the random elements by expected values or by pessimistic estimates of these values, or to recast the problem into a two-stage program so that, in the second stage, one can compensate for inaccuracies in the first stage. The purpose of this analysis is to examine the last of these methods in detail. More precisely, it investigates the conditions under which the first-stage decisions are optimal. In addition, formulas for using various existing computational algorithms to obtain an optimal solution are given.

George B Dantzig

Papers

On the solution of two-stage linear programs under uncertainty. notes on linear programming and extensions. part 55