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 is second-order-cone representable. Via a shooting e...

A large portion of research in science and engineering, as well as in business, concerns one similar problem: how to make things “better”? Once properly modeled (often a highly nontrivial task), this kind of question can be approached via a mathematical optimization problem. An optimal solution to a mathematical optimization problem, when interpreted properly, might correspond to new knowledge,...

where c ∈ R, g : R → R is non-linear and smooth on its domain, and Ω is a nonempty closed convex subset in R. This paper introduces sequential convex programming (SCP), a local optimization method for solving the nonconvex problem (P). We prove that under acceptable assumptions the SCP method locally converges to a KKT point of (P) and the rate of convergence is linear. Problems in the form of ...

We describe a new convex quadratic programming bound for the quadratic assignment problem (QAP). The construction of the bound uses a semideenite programming representation of a basic eigenvalue bound for QAP. The new bound dominates the well-known projected eigenvalue bound, and appears to be competitive with existing bounds in the tradeoo between bound quality and computational eeort.

The purpose of this survey article is to introduce the reader to a very elegant formulation of convex optimization problems called conic optimization and outline its many advantages. After a brief introduction to convex optimization, the notion of convex cone is introduced, which leads to the conic formulation of convex optimization problems. This formulation features a very symmetric dual prob...

Abstract Copositive optimization is a special case of convex conic programming, and it consists optimizing linear function over the cone all completely positive matrices under constraints. provides powerful relaxations NP-hard quadratic problems or combinatorial problems, but there are still many open regarding copositive matrices. In this paper, we focus on one such problem; finding (CP) facto...

In this paper, we first consider the stability analysis of a convex quadratic programming problem and its restricted Wolfe dual in which all parameters in the problem are perturbed. We demonstrate the upper semi-continuity of solution mappings for the primal problem and the restricted Wolfe dual problem and establish the Hadamard directionally differentiability of the optimal value function. By...

Horizontal placement of nodes in tree layout or layered drawings of directed graphs can be modelled as a convex quadratic program. Thus, quadratic programming provides a declarative framework for specifying such layouts which can then be solved optimally with a standard quadratic programming solver. While slower than specialized algorithms, the quadratic programming approach is fast enough for ...

Copositive optimization problems are particular conic programs: extremize linear forms over the copositive cone subject to linear constraints. Every quadratic program with linear constraints can be formulated as a copositive program, even if some of the variables are binary. So this is an NP-hard problem class. While most methods try to approximate the copositive cone from within, we propose a ...

In this paper we study several issues related to the characterization of specific classes of multivariate quadratic mappings that are nonnegative over a given domain, with nonnegativity defined by a pre-specified conic order. In particular, we consider the set (cone) of nonnegative quadratic mappings defined with respect to the positive semidefinite matrix cone, and study when it can be represe...