Stochastic Integer Programming

Abstract

Approximation algorithms are the prevalent solution methods in the field of stochastic programming. Problems in this field are very hard to solve. Indeed, most of the research in this field has concentrated on designing solution methods that approximate the optimal solutions. However, efficiency in the complexity theoretical sense is usually not taken into account. Quality statements mostly remain restricted to convergence to an optimal solution without accompanying implications on the running time of the algorithms for attaining more and more accurate solutions.However, over the last twenty years also some studies on performance analysis of approximation algorithms for stochastic programming have appeared. In this direction we find both probabilistic analysis and worst-case analysis. There have been studies on performance ratios and on absolute divergence from optimality.Recently the complexity of stochastic programming problems has been addressed, indeed confirming that these problems are harder than most combinatorial optimization problems. Polynomial time approximation algorithms and their performance guarantees for stochastic linear and integer programming problems have seen increasing research attention only very recently.Approximation in the traditional stochastic programming sense will not be discussed in this chapter. The reader interested in this issue is referred to surveys on stochastic programming, like the Handbook on Stochastic Programming [38] or the text books [3,21,35]. We concentrate on the studies of approximation algorithms which are more similar in nature to those for combinatorial optimization.

Convex Approximations for Totally Unimodular Integer Recourse Models: A Uniform Error Bound

“…The main reason that integer recourse models are considerably more difficult to solve than continuous recourse models is that the integer recourse function Q is generally non-convex [8]. A possible approach to deal with this difficulty is to construct convex approximations of the recourse function Q by modifying the recourse data (MRD) [13], which comprises the parameters and structure of the model, and the distributions of the random variables involved.…”

Pseudo-Valid Cutting Planes for Two-Stage Mixed-Integer Stochastic Programs with Right-Hand-Side Uncertainty

“…All these solution methods aim to find the exact optimal solution for MISPs, but generally have difficulties scaling up to solve large problems. This is not surprising, because contrary to their continuous counterparts, these MISPs are nonconvex in general (Rinnooy Kan and Stougie 1988). This means that efficient techniques from convex optimization cannot be used to solve these problems.…”

Approximate and exact convexification approaches for solving two-stage mixed-integer recourse models

“…This situation is undesirable from a computational point of view, and thus we exclude it by assuming complete recourse, see Definition 1.1.Definition 1.1. The recourse is complete if and only if for every s ∈ R m , there exists a y ∈ Y such that Wy ≥ s. Then, v(ω, x) < +∞ for every ω ∈ Ω and x ∈ R n .Assume that the recourse is complete and sufficiently expensive, and the random data (h(ω), T(ω)) satisfy the weak covariance condition.(i) Q is a finite-valued, convex, continuous, and subdifferentiable function onwhere λ ω is a vector of optimal dual multipliers of the second-stage problem (1.6)The convexity of Q in Theorem 1.2 enables the use of techniques from convex optimization to efficiently solve continuous recourse models, see also Section 1.3.1.If integer restrictions are imposed on the recourse actions y, however, then convexity of Q is lost, see, e.g., [54], resulting in significant computational challenges.From the perspective of computational complexity, however, the difficulties posed by integer recourse actions are dominated by those caused by the curse of is a linear program and thus can be solved efficiently. Moreover, (MP) is a relaxation of the original problem (1.7), and thus, if an optimal solution ( x, θ) of (MP) is feasible in (1.7), i.e., if θ ≥ Q( x), then ( x, θ) is also optimal in (1.7).…”

“…Finally, if ω follows a finite discrete distribution, then finite convergence can be established under mild conditions [43]. Indeed, we only need a finite number of optimality cuts to completely describe Q, because Q is a convex polyhedral function.The L-shaped method cannot be used to solve general two-stage MIR models, because convexity of Q is lost if integer restrictions are imposed on the recourse actions [54]. Typically, Benders' decomposition algorithms that solve general MIR models combine ideas from the L-shaped method and deterministic mixed-integer In this thesis, we propose novel solution methods for two-stage MIR models that are inspired by Benders' decomposition.…”

“…If integer restrictions are imposed on the recourse actions y, however, then convexity of Q is lost, see, e.g., [54], resulting in significant computational challenges.…”