Some Conditions for Optimal Deterministic Solutions to Stochastic Dynamic Linear Programs

Many problems that require decisions made over time can be formulated as dynamic linear programs. Complications arise in solving these programs when one allows stochastic elements to alter the state to state transitions. Finding the stochastic linear programming solutions may be very difficult since their formulation often greatly increases the problem size. This paper shows that, under certain conditions, a simple deterministic solution technique obtains the same optimal controls as more complicated stochastic methods

Designing Approximation Schemes for Stochastic Optimization Problems, in Particular for Stochastic Programs with Recourse

Various approximation schemes for stochastic optimization problems involving either approximates of the probability measures and/or approximates of the objective functional, are investigated. We discuss their potential implementation as part of general procedures for solving stochastic programs with recourse

Approximations and Error Bounds in Stochastic Programming

We review and complete the approximation results for stochastic programs with recourse. Since this note is to serve as a preamble to the development of software for stochastic programming problems, we also address the question of how to easily find a (starting) solution

A Standard Input Format for Computer Codes Which Solve Stochastic Programs with Recourse and a Library of Utilities to Simplify its Use

We explain our suggestions for standardizing input formats for computer codes which solve stochastic programs with recourse. The main reason to set some conventions is to allow programs implementing different methods of solution to be used interchangeably. The general philosophy behind our design is a) to remain fairly faithful to the de facto standard for the statement of LP problems established by IBM for use with MPSX and subsequently adopted by the authors of MINOS, b) to provide sufficient flexibility so that a variety of problems may be expressed in the standard format, c) to allow problems originally formulated as deterministic LP to be converted to stochastic problems with a minimum of effort, d) to permit new options to be added as the need arises, and e) to provide some routines to facilitate the task of reading files specified in the standard format

Single‐commodity stochastic network design under demand and topological uncertainties with insufficient data

Stochastic network design is fundamental to transportation and logistic problems in practice, yet faces new modeling and computational challenges resulted from heterogeneous sources of uncertainties and their unknown distributions given limited data. In this article, we design arcs in a network to optimize the cost of single‐commodity flows under random demand and arc disruptions. We minimize the network design cost plus cost associated with network performance under uncertainty evaluated by two schemes. The first scheme restricts demand and arc capacities in budgeted uncertainty sets and minimizes the worst‐case cost of supply generation and network flows for any possible realizations. The second scheme generates a finite set of samples from statistical information (e.g., moments) of data and minimizes the expected cost of supplies and flows, for which we bound the worst‐case cost using budgeted uncertainty sets. We develop cutting‐plane algorithms for solving the mixed‐integer nonlinear programming reformulations of the problem under the two schemes. We compare the computational efficacy of different approaches and analyze the results by testing diverse instances of random and real‐world networks. © 2017 Wiley Periodicals, Inc. Naval Research Logistics 64: 154–173, 2017Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/137236/1/nav21739_am.pdfhttps://deepblue.lib.umich.edu/bitstream/2027.42/137236/2/nav21739.pd

A Standard Input Format for Multiperiod Stochastic Linear Programs

Data conventions for the automatic input of multiperiod stochastic linear programs are described. The input format is based on the MPSX standard and is designed to promote the efficient conversion of originally deterministic problems by introducing stochastic variants in separate files. A flexible "header" syntax generates a useful variety of stochastic dependencies. An extension using the NETGEN format is proposed for stochastic network programs

Models and model value in stochastic programming

Finding optimal decisions often involves the consideration of certain random or unknown parameters. A standard approach is to replace the random parameters by the expectations and to solve a deterministic mathematical program. A second approach is to consider possible future scenarios and the decision that would be best under each of these scenarios. The question then becomes how to choose among these alternatives. Both approaches may produce solutions that are far from optimal in the stochastic programming model that explicitly includes the random parameters. In this paper, we illustrate this advantage of a stochastic program model through two examples that are representative of the range of problems considered in stochastic programming. The paper focuses on the relative value of the stochastic program solution over a deterministic problem solution.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/44253/1/10479_2005_Article_BF02031741.pd

A polynomial oracle-time algorithm for convex integer minimization

In this paper we consider the solution of certain convex integer minimization problems via greedy augmentation procedures. We show that a greedy augmentation procedure that employs only directions from certain Graver bases needs only polynomially many augmentation steps to solve the given problem. We extend these results to convex $N$-fold integer minimization problems and to convex 2-stage stochastic integer minimization problems. Finally, we present some applications of convex $N$-fold integer minimization problems for which our approach provides polynomial time solution algorithms.Comment: 19 pages, 1 figur