Computations with Disjunctive Cuts for Two-Stage Stochastic Mixed Integer Programs

Two-stage stochastic mixed-integer programming (SMIP) problems with recourse are generally difficult to solve. This paper presents a first computational study of a disjunctive cutting plane method for stochastic mixed 0-1 programs that uses lift-and-project cuts based on the extensive form of the two-stage SMIP problem. An extension of the method based on where the data … Read more

Solving Max-Cut to Optimality by Intersecting Semidefinite and Polyhedral Relaxations

In this paper we present a method for finding exact solutions of Max-Cut, the problem of finding a cut of maximum weight in a weighted graph. We use a Branch-and-Bound setting, that applies a dynamic version of the bundle method as bounding procedure. This approach uses Lagrangian duality to obtain a “nearly optimal” solution of … Read more

The kernel average for two convex functions and its application to the extension and representation of monotone operators

We provide and analyze a based average for two convex functions, based on a kernel function. It covers several known averages such as the arithmetic average, epigraphical average, and the proximal average. When applied to the Fitzpatrick function and the conjugate of Fitzpatrick function associated with a monotone operator, our average produces an autoconjugate (also … Read more

The Speed of Shor’s R-Algorithm

Shor’s r-algorithm is an iterative method for unconstrained optimization, designed for minimizing nonsmooth functions, for which its reported success has been considerable. Although some limited convergence results are known, nothing seems to be known about the algorithm’s rate of convergence, even in the smooth case. We study how the method behaves on convex quadratics, proving … Read more

Semidefinite Programming versus the Reformulation-Linearization Technique for Nonconvex Quadratically Constrained Quadratic Programming

We consider relaxations for nonconvex quadratically constrained quadratic programming (QCQP) based on semidefinite programming (SDP) and the reformulation-linearization technique (RLT). From a theoretical standpoint we show that the addition of a semidefiniteness condition removes a substantial portion of the feasible region corresponding to product terms in the RLT relaxation. On test problems we show that … Read more

Two Algorithms for the Minimum Enclosing Ball Problem

Given $\cA := \{a^1,\ldots,a^m\} \subset \R^n$ and $\eps > 0$, we propose and analyze two algorithms for the problem of computing a $(1 + \eps)$-approximation to the radius of the minimum enclosing ball of $\cA$. The first algorithm is closely related to the Frank-Wolfe algorithm with a proper initialization applied to the dual formulation of … Read more

An integer programming approach for linear programs with probabilistic constraints

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