Convex approximations for a class of mixed-integer recourse models

We consider mixed-integer recourse (MIR) models with a single recourse constraint.We relate the secondstage value function of such problems to the expected simple integer recourse (SIR) shortage function. This allows to construct convex approximations for MIR problems by the same approach used for SIR models.

Convex approximations for a class of mixed-integer recourse models

We consider mixed-integer recourse (MIR) models with a single recourse constraint. We relate the second-stage value function of such problems to the expected simple integer recourse (SIR) shortage function. This allows to construct convex approximations for MIR problems by the same approach used for SIR models

Approximation in stochastic integer programming

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. Only recently the complexity of stochastic programming problems has been addressed, indeed confirming that these problems are harder than most combinatorial optimization problems.

Convex approximations for complete integer recourse models

We consider convex approximations of the expected value function of a two-stage integer recourse problem. The convex approximations are obtained by perturbing the distribution of the random right-hand side vector. It is shown that the approximation is optimal for the class of problems with totally unimodular recourse matrices. For problems not in this class, the result is a convex lower bound that is strictly better than the one obtained from the LP relaxation.

On multiple simple recourse models

We consider multiple simple recourse (MSR) models, both continuous and integer versions, which generalize the corresponding simple recourse (SR) models by allowing for a refined penalty cost structure for individual shortages and surpluses. It will be shown that (convex approximations of) such MSR models can be represented as explicitly specified continuous SR models, and thus can be solved efficiently by existing algorithms.