With the fast-growing demand in the electricity market of the last decades, attention has been focused on alternative and flexible sources of energy such as hydro valleys. Managing the hydroelectricity produced by the plants in hydro valleys is called the hydro unit commitment problem. This problem consists in finding the optimal power production schedule of … Read more
We study the polyhedral convex hull of a mixed-integer set S defined by a collection of multilinear equations over the 0-1-cube. Such sets appear frequently in the factorable reformulation of mixed-integer nonlinear optimization problems. In particular, the set S represents the feasible region of a linearized unconstrained binary polynomial optimization problem. We define an equivalent … Read more
We propose two exact approaches for non-convex quadratic integer minimization subject to linear constraints where lower bounds are computed by considering ellipsoidal relaxations of the feasible set. In the first approach, we intersect the ellipsoids with the feasible linear subspace. In the second approach we penalize exactly the linear constraints. We investigate the connection between … Read more
We present polynomial-time algorithms for constrained optimization problems overwhere the intersection graph of the constraint set has bounded tree-width. In the case of binary variables we obtain exact, polynomial-size linear programming formulations for the problem. In the mixed-integer case with bounded variables we obtain polynomial-size linear programming representations that attain guaranteed optimality and feasibility bounds. … Read more
We study the convex hull of a set arising as a relaxation of difficult convex mixed integer quadratic programs (MIQP). We characterize the extreme points of our set and the extreme points of its continuous relaxation. We derive four quadratic cutting surfaces that improve the strength of the continuous relaxation. Each of the cutting surfaces … Read more
In this paper we consider the use of extended formulations in LP-based algorithms for mixed integer conic quadratic programming (MICQP). Extended formulations have been used by Vielma, Ahmed and Nemhauser (2008) and Hijazi, Bonami and Ouorou (2013) to construct algorithms for MICQP that can provide a significant computational advantage. The first approach is based on … Read more
We present a solution algorithm for problems from steady-state gas transport optimization. Due to nonlinear and nonconvex physics and engineering models as well as discrete controllability of active network devices, these problems lead to hard nonconvex mixed-integer nonlinear optimization models. The proposed method is based on mixed-integer linear techniques using piecewise linear relaxations of the … Read more
In this paper we consider an aggregation technique introduced by Yildiran, 2009 to study the convex hull of regions defined by two quadratic or by a conic quadratic and a quadratic inequality. Yildiran shows how to characterize the convex hull of open sets defined by two strict quadratic inequalities using Linear Matrix Inequalities (LMI). We … Read more
We present a specialized branch-and-bound (b&b) algorithm for the Euclidean Steiner tree problem (ESTP) in R^n and apply it to a convex mixed-integer nonlinear programming (MINLP) formulation of the problem, presented by Fampa and Maculan. The algorithm contains procedures to avoid difficulties observed when applying a b&b algorithm for general MINLP problems to solve the … Read more
This paper illustrates the fundamental connection between nonconvex quadratic optimization and copositive optimization—a connection that allows the reformulation of nonconvex quadratic problems as convex ones in a unified way. We intend the paper for readers new to the area, and hence the exposition is largely self-contained. We focus on examples having just a few variables … Read more