## Probabilistic covering problems

##### Abstract

This dissertation studies optimization problems that involve probabilistic covering constraints. A probabilistic constraint evaluates and requires that the probability that a set of constraints involving random coefficients with known distributions hold satisfy a minimum requirement. A covering constraint involves a linear inequality on non-negative variables with a greater or equal to sign and non-negative coefficients. A variety of applications, such as set cover problems, node/edge cover problems, crew scheduling, production planning, facility location, and machine learning, in uncertain settings involve probabilistic covering constraints.
In the first part of this dissertation we consider probabilistic covering linear programs. Using the sampling average approximation (SAA) framework, a probabilistic covering linear program can be approximated by a covering k-violation linear program (CKVLP), a deterministic covering linear program in which at most k constraints are allowed to be violated. We show that CKVLP is strongly NP-hard. Then, to improve the performance of standard mixed-integer programming (MIP) based schemes for CKVLP, we (i) introduce and analyze a coefficient strengthening scheme, (ii) adapt and analyze an existing cutting plane technique, and (iii) present a branching technique. Through computational experiments, we empirically verify that these techniques are significantly effective in improving solution times over the CPLEX MIP solver. In particular, we observe that the proposed schemes can cut down solution times from as much as six days to under four hours in some instances. We also developed valid inequalities arising from two subsets of the constraints in the original formulation. When incorporating them with a modified coefficient strengthening procedure, we are able to solve a difficult probabilistic portfolio optimization instance listed in MIPLIB 2010, which cannot be solved by existing approaches.
In the second part of this dissertation we study a class of probabilistic 0-1 covering problems, namely probabilistic k-cover problems. A probabilistic k-cover problem is a stochastic version of a set k-cover problem, which is to seek a collection of subsets with a minimal cost whose union covers each element in the set at least k times. In a stochastic setting, the coefficients of the covering constraints are modeled as Bernoulli random variables, and the probabilistic constraint imposes a minimal requirement on the probability of k-coverage. To account for absence of full distributional information, we define a general ambiguous k-cover set, which is ``distributionally-robust." Using a classical linear program (called the Boolean LP) to compute the probability of events, we develop an exact deterministic reformulation to this ambiguous k-cover problem. However, since the boolean model consists of exponential number of auxiliary variables, and hence not useful in practice, we use two linear program based bounds on the probability that at least k events occur, which can be obtained by aggregating the variables and constraints of the Boolean model, to develop tractable deterministic approximations to the ambiguous k-cover set. We derive new valid inequalities that can be used to strengthen the linear programming based lower bounds. Numerical results show that these new inequalities significantly improve the probability bounds. To use standard MIP solvers, we linearize the multi-linear terms in the approximations and develop mixed-integer linear programming formulations. We conduct computational experiments to demonstrate the quality of the deterministic reformulations in terms of cost effectiveness and solution robustness. To demonstrate the usefulness of the modeling technique developed for probabilistic k-cover problems, we formulate a number of problems that have up till now only been studied under data independence assumption and we also introduce a new applications that can be modeled using the probabilistic k-cover model.