Santanu S. Dey Mixed-integer quadratic programming (MIQP) is the problem of optimizing a quadratic function over points in a polyhedral set where some of the components are restricted to be integral. In this paper, we prove that the decision version of mixed-integer quadratic programming is in NP, thereby showing that it is NP-complete. This is established by showing … Read more
Sparse cutting-planes are often the ones used in mixed-integer programing (MIP) solvers, since they help in solving the linear programs encountered during branch-\&-bound more efficiently. However, how well can we approximate the integer hull by just using sparse cutting-planes? In order to understand this question better, given a polyope $P$ (e.g. the integer hull of … Read more
We consider the integer L-shaped method for two-stage stochastic integer programs. To improve the performance of the algorithm, we present and combine two strategies. First, to avoid time-consuming exact evaluations of the second-stage cost function, we propose a simple modification that alternates between linear and mixed-integer subproblems. Then, to better approximate the shape of the … Read more
Abstract: Given a set of n random events in a probability space, represented by n Bernoulli variables (not necessarily independent,) we consider the probability that at least k out of n events occur. When partial distribution information, i.e., individual probabilities and all joint probabilities of up to m (m< n) events, are provided, only an ... Read more
In this work, we introduce and study the forbidden-vertices problem. Given a polytope P and a subset X of its vertices, we study the complexity of linear optimization over the subset of vertices of P that are not contained in X. This problem is closely related to finding the k-best basic solutions to a linear … Read more
Let $C$ be a full-dimensional pointed closed convex cone in $R^m$ obtained by taking the conic hull of a strictly convex set. Given $A \in Q^{m \times n_1}$, $B \in Q^{m \times n_2}$ and $b \in Q^m$, a simple conic mixed-integer set (SCMIS) is a set of the form $\{(x,y)\in Z^{n_1} \times R^{n_2}\,|\,\ Ax +By … Read more
The $pq$-relaxation for the pooling problem can be constructed by applying McCormick envelopes for each of the bilinear terms appearing in the so-called $pq$-formulation of the pooling problem. This relaxation can be strengthened by using piecewise-linear functions that over- and under-estimate each bilinear term. The resulting relaxation can be written as a mixed integer linear … Read more
We study a variant of the classical transportation problem in which suppliers with limited capacities have a choice of which demands (markets) to satisfy. We refer to this problem as the transportation problem with market choice (TPMC). While the classical transportation problem is known to be strongly polynomial-time solvable, we show that its market choice … Read more
Let B be the set of n-dimensional binary vectors and let V be a subset of m of its elements. We give an extended formulation of the convex hull of B\V which is polynomial in n and m. In developing this result, we give a two-sided extension of a result in Laurent and Sassano (1992) … Read more
We consider a class of linear programs involving a set of covering constraints of which at most k are allowed to be violated. We show that this covering linear program with violation is strongly NP-hard. In order to improve the performance of mixed-integer programming (MIP) based schemes for these problems, we introduce and analyze a … Read more