Mixed-Integer Programming Subject to Uncertain Data

Abstract

The here presented thesis deals with optimization problems where the underlying problem data are subject to uncertainty. Sources of data uncertainty in practical problems are manifold, and so are the ways to model uncertainty in a mathematical programming context. The position taken in this thesis is that the underlying problem is a linear or mixedinteger program where some part of the problem data, e.g., the constraint matrix, is described by a set of possible matrices instead of a single one. There are two opposite viewpoints on this: The optimist assumes that he can influence the uncertainty and, thus, can choose a constraint matrix along with values for the variables of the underlying problem. The pessimist, however, assumes that he has to take a decision without having this possibility to choose and, therefore, assumes the worst case. The former viewpoint is expressed by a so called generalized mixed-integer program, the latter by a so called robust mixed-integer program.

In the first part of this thesis, robust problems with uncertainty in the cost vector are investigated. Here, the emphasis lies on considering simply structured uncertainties that allow the reduction of a problem with uncertainty to a series of problems of the same type but without uncertainty. It is known from the literature that this is possible for robust 0-1 programs and the robust minimum-cost flow problem if the uncertainty is a (higher dimensional) interval where the upper bound corner is cut off by a single cardinality constraint; this constraint permits control over the amount of robustness in the problem. In this thesis, it is demonstrated that this is still possible for uncertainties where the upper bound is cut off by arbitrarily many knapsack constraints with non-negative coefficients, which permits more detailed control. For the robust minimum-cost flow problem, a subgradient optimization approach is proposed; this is more practical than the binary search method proposed in literature.

The second part of this thesis is concerned with more general uncertainties, mainly polyhedral ones, and robust and generalized mixed-integer programs. Reformulations of these problems as mixed-integer programs are discussed, and some useful tools known from linear programming, like duality and Farkas' lemma, are reviewed for linear programs with uncertainty. With help of these, it is shown that lattice-free cuts for robust mixed-integer programs are generated by generalized linear programs while lattice-free cuts for generalized mixed-integer programs are generated by robust linear programs. Strengthening procedures, known from literature for the non-uncertain case, and, finally, problems with uncertainties described by convex conic sets are investigated.

The performance of the lattice-free cuts for robust mixed-integer programs is assessed in terms of the amount of gap closed and the time spent for cut generation by a computational study.