We develop a new modeling and exact solution method for stochastic programming problems that include a joint probabilistic constraint in which the multirow random technology matrix is discretely distributed. We binarize the probability distribution of the random variables in such a way that we can extract a threshold partially defined Boolean function (pdBf) representing the … Read more
A common structure in convex mixed-integer nonlinear programs is separable nonlinear functions. In the presence of such structures, we propose three improvements to the outer approximation algorithms. The first improvement is a simple extended formulation, the second is a refined outer approximation, and the third is a heuristic inner approximation of the feasible region. These … Read more
In this paper we study valid inequalities for a fundamental set that involves a continuous vector variable x in [0,1]^n, its associated quadratic form x x’ and its binary indicators. This structure appears when deriving strong relaxations for mixed integer quadratic programs (MIQPs). We treat valid inequalities for this set as lifted from QPB, which … Read more
We consider N-fold 4-block decomposable integer programs, which simultaneously generalize N-fold integer programs and two-stage stochastic integer programs with N scenarios. In previous work [R. Hemmecke, M. Koeppe, R. Weismantel, A polynomial-time algorithm for optimizing over N-fold 4-block decomposable integer programs, Proc. IPCO 2010, Lecture Notes in Computer Science, vol. 6080, Springer, 2010, pp. 219–229], … Read more
Detecting and solving aircraft conflicts, which occur when aircraft sharing the same airspace are too close to each other according to their predicted trajectories, is a crucial problem in Air Traffic Management. We focus on mixed-integer optimization models based on speed regulation. We first solve the problem to global optimality by means of an exact … Read more
We study the convex hull of the intersection of a convex set E and a linear disjunction. This intersection is at the core of solution techniques for Mixed Integer Conic Optimization. We prove that if there exists a cone K (resp., a cylinder C) that has the same intersection with the boundary of the disjunction … Read more
We propose a branch-and-bound algorithm for minimizing a not necessarily convex quadratic function over integer variables. The algorithm is based on lower bounds computed as continuous minima of the objective function over appropriate ellipsoids. In the nonconvex case, we use ellipsoids enclosing the feasible region of the problem. In spite of the nonconvexity, these minima … Read more
In this paper we consider sparse approximation problems, that is, general $l_0$ minimization problems with the $l_0$-“norm” of a vector being a part of constraints or objective function. In particular, we first study the first-order optimality conditions for these problems. We then propose penalty decomposition (PD) methods for solving them in which a sequence of … Read more
A novel approach for obtaining globally optimal solutions to design of networks with nonlinear resistances and potential driven flows is proposed. The approach is applicable to networks where the potential loss on an edge in the network is governed by a convex and strictly monotonically increasing function of flow rate. We introduce a relaxation of … Read more
We consider a chance-constrained optimization problem (CCOP), where the random variables follow finite discrete distributions. The problem is in general nonconvex and can be reformulated as a mixed-integer program. By exploiting the special structure of the probabilistic constraint, we propose an alternating direction method for finding suboptimal solutions of CCOP. At each iteration, this method … Read more