Shabbir Ahmed In this paper, we examine a mixed integer linear programming (MIP) reformulation for mixed integer bilinear problems where each bilinear term involves the product of a nonnegative integer variable and a nonnegative continuous variable. This reformulation is obtained by first replacing a general integer variable with its binary expansion and then using McCormick envelopes to … Read more
In this paper we develop convex relaxations of chance constrained optimization problems in order to obtain lower bounds on the optimal value. Unlike existing statistical lower bounding techniques, our approach is designed to provide deterministic lower bounds. We show that a version of the proposed scheme leads to a tractable convex relaxation when the chance … Read more
We provide a probabilistic comparison of split and type 1 triangle cuts for mixed-integer programs with two rows and two integer variables. Under a simple probabilistic model of the problem parameters, we show that a simple split cut, i.e. a Gomory cut, is more likely to be better than a type 1 triangle cut in … Read more
We study the modeling of non-convex piecewise linear functions as Mixed Integer Programming (MIP) problems. We review several new and existing MIP formulations for continuous piecewise linear functions with special attention paid to multivariate non-separable functions. We compare these formulations with respect to their theoretical properties and their relative computational performance. In addition, we study … Read more
Given a finite ground set N and a value vector a in R^N, we consider optimization problems involving maximization of a submodular set utility function of the form h(S)= f (sum_{i in S} a_i), S subseteq N, where f is a strictly concave, increasing, differentiable function. This function appears frequently in combinatorial optimization problems when … Read more
We study approximations of chance constrained problems. In particular, we consider the Sample Average Approximation (SAA) approach and discuss convergence properties of the resulting problem. A method for constructing bounds for the optimal value of the considered problem is discussed and we suggest how one should tune the underlying parameters to obtain a good approximation … Read more
We consider optimization problems with some binary variables, where the objective function is linear in these variables. The stability region of a given solution of such a problem is the polyhedral set of objective coefficients for which the solution is optimal. A priori knowledge of this set provides valuable information for sensitivity analysis and re-optimization … Read more
We propose a sample average approximation (SAA) method for stochastic programming problems involving an expected value constraint. Such problems arise, for example, in portfolio selection with constraints on conditional value-at-risk (CVaR). Our contributions include an analysis of the convergence rate and a statistical validation scheme for the proposed SAA method. Computational results using a portfolio … Read more
We study approximations of optimization problems with probabilistic constraints in which the original distribution of the underlying random vector is replaced with an empirical distribution obtained from a random sample. We show that such a sample approximation problem with risk level larger than the required risk level will yield a lower bound to the true … Read more
Linear programs with joint probabilistic constraints (PCLP) are difficult to solve because the feasible region is not convex. We consider a special case of PCLP in which only the right-hand side is random and this random vector has a finite distribution. We give a mixed-integer programming formulation for this special case and study the relaxation … Read more