Optimization Under Uncertainty

Most optimization problems in real life do not have accurate estimates of the problem parameters at the optimization phase. Stochastic optimization models have been studied widely in the literature to address this problem. The expected value optimization is reasonable in a repeated decision making framework. However, it does not sufficiently guard against the worst case future in more risk averse applications. The broad purpose of this thesis is to study optimization approaches under uncertainty that overcome this shortcoming of a traditional stochastic optimization model. We consider new models of uncertainty namely, the “demand-robust” model and the “chance constrained” model and introduce these in the framework of general covering problems. We consider uncertainty in the right hand side of the constraints which is referred to as the demand uncertainty. In the two-stage model of “demand-robustness”, we are interested in finding a solution such that the worst case cost over all realizations of uncertainty is minimized. We prove a general structural lemma about special types of first stage solutions and provide approximation algorithms for covering problems such as Steiner tree, min-cut, minimum multi-cut, vertex cover and facility location. The structural lemma essentially exploits the following idea: In a two-stage solution, if the first stage help is at least as costly as the second stage solution for some realization of the uncertain parameters (referred to as a scenario), then a solution for that scenario can be constructed completely in the first stage while only losing a factor two in the total cost. We further extend this idea to develop a ‘guess-and-prune’ algorithm where we ‘guess’ the worst case second stage cost which allows us to ‘prune’ away a set of scenarios for which a complete solution in the second stage has cost at most the worst case cost. For specific covering problems such as minimum cut and shortest path, we show that an approximate first stage solution can be constructed for the remaining scenarios using ideas from the structural lemma as well as the combinatorial structure of the problem. The robust optimization approach guards against the worst case future but tends to be overly conservative if there are some outlier scenarios. To overcome this, we consider a chance constrained model where we are given a reliability level p and the idea is to select a “p fraction” of the scenarios and find a robust solution on the selected scenarios. The remaining (1-p) fraction of the scenarios are considered as outliers and can be ignored. We consider both one-stage and two-stage chance constrained covering problems with demand-uncertainty. While it is easy to obtain bi-criteria approximations for the chance-constrained problems that violate the chance-constraint by a small factor, we consider the problem of satisfying the chance-constraint strictly. We show that the covering problems in both onestage and two-stage chance-constrained models where uncertainty is specified as