The convex hull relaxation (CHR) method (Albornoz 1998, Ahlatçıoğlu 2007, Ahlatçıoğlu and Guignard 2010) provides lower bounds and feasible solutions (thus upper bounds) on convex 0-1 nonlinear programming problems with linear constraints. In the quadratic case, these bounds may often be improved by a preprocessing step that adds to the quadratic objective function terms which … Read more
The Kantorovich function $(x^TAx)( x^T A^{-1} x)$, where $A$ is a positive definite matrix, is not convex in general. From a matrix or convex analysis point of view, it is interesting to address the question: When is this function convex? In this paper, we prove that the 2-dimensional Kantorovich function is convex if and only … Read more
Bound tightening is an important component of algorithms for solving nonconvex Mixed Integer Nonlinear Programs. A {\em probing} algorithm is a bound-tightening procedure that explores the consequences of restricting a variable to a subinterval with the goal of tightening its bounds. We propose a variant of probing where exploration is based on iteratively applying a … Read more
We describe the search for explicit general linear methods in Nordsieck form for which the stability function has only two nonzero roots. This search is based on state-of-the-art optimization software. Examples of methods found in this way are given for order p = 5, p = 6, and p = 7. ArticleDownload View PDF
The wide applicability of chance-constrained programming, together with advances in convex optimization and probability theory, has created a surge of interest in finding efficient methods for processing chance constraints in recent years. One of the successes is the development of so-called safe tractable approximations of chance-constrained programs, where a chance constraint is replaced by a … Read more
In this paper, we establish new bounds for restricted isometry constants (RIC) in low-rank matrix recovery. Let $\A$ be a linear transformation from $\R^{m \times n}$ into $\R^p$, and $r$ the rank of recovered matrix $X\in \R^{m \times n}$. Our main result is that if the condition on RIC satisfies $\delta_{2r+k}+2(\frac{r}{k})^{1/2}\delta_{\max\{r+\frac{3}{2}k,2k\}}
In this paper, we propose an efficient approach for solving a class of convex optimization problems in Hilbert spaces. Our feasible region is a (possibly infinite-dimensional) simple convex set, i.e. we assume that projections on this set are computationally easy to compute. The problem we consider is the minimization of a convex function over this … Read more
The classical column generation is based on optimal solutions of the restricted master problems. This strategy frequently results in an unstable behaviour and may require an unnecessarily large number of iterations. To overcome this weakness, variations of the classical approach use interior points of the dual feasible set, instead of optimal solutions. In this paper, … Read more
Energy consumption of communication systems is becoming a fundamental issue and, among all the sectors, wireless access networks are largely responsible for the in- crease in consumption. In addition to the access segment, wireless technologies are also gaining popularity for the back- haul infrastructure of cellular systems mainly due to their cost and easy deployment. … Read more
This is the second part of a work devoted to the theory of epigraphical cones and their applications. A convex cone $K$ in the Euclidean space $\mathbb{R}^{n+1}$ is an epigraphical cone if it can be represented as epigraph of a nonnegative sublinear function $f: \mathbb{R}^n\to \mathbb{R}$. We explore the link between the geometric properties of … Read more