Abstract
We consider the objective function of a simple recourse problem with fixed technology matrix and integer second-stage variables. Separability due to the simple recourse structure allows to study a one-dimensional version instead.
Based on an explicit formula for the objective function, we derive a complete description of the class of probability density functions such that the objective function is convex. This result is also stated in terms of random variables.
Next, we present a class of convex approximations of the objective function, which are obtained by perturbing the distributions of the right-hand side parameters. We derive a uniform bound on the absolute error of the approximation. Finally, we give a representation of convex simple integer recourse problems as continuous simple recourse problems, so that they can be solved by existing special purpose algorithms.
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Research of the third author has been made possible by a fellowship of the Royal Netherlands Academy of Arts and Sciences.
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Klein Haneveld, W., Stougie, L. & van der Vlerk, M. Simple integer recourse models: convexity and convex approximations. Math. Program. 108, 435–473 (2006). https://doi.org/10.1007/s10107-006-0718-4
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DOI: https://doi.org/10.1007/s10107-006-0718-4