8,339 research outputs found
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.
Problem-driven scenario generation: an analytical approach for stochastic programs with tail risk measure
Scenario generation is the construction of a discrete random vector to
represent parameters of uncertain values in a stochastic program. Most
approaches to scenario generation are distribution-driven, that is, they
attempt to construct a random vector which captures well in a probabilistic
sense the uncertainty. On the other hand, a problem-driven approach may be able
to exploit the structure of a problem to provide a more concise representation
of the uncertainty.
In this paper we propose an analytic approach to problem-driven scenario
generation. This approach applies to stochastic programs where a tail risk
measure, such as conditional value-at-risk, is applied to a loss function.
Since tail risk measures only depend on the upper tail of a distribution,
standard methods of scenario generation, which typically spread their scenarios
evenly across the support of the random vector, struggle to adequately
represent tail risk. Our scenario generation approach works by targeting the
construction of scenarios in areas of the distribution corresponding to the
tails of the loss distributions. We provide conditions under which our approach
is consistent with sampling, and as proof-of-concept demonstrate how our
approach could be applied to two classes of problem, namely network design and
portfolio selection. Numerical tests on the portfolio selection problem
demonstrate that our approach yields better and more stable solutions compared
to standard Monte Carlo sampling
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.
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