We focus on a class of nonconvex cooperative optimization problems that involve multiple participants. We study the duality framework and provide geometric and analytic character- izations of the duality gap. The dual problem is related to a market setting in which each participant pursuits self interests at a given price of common goods. The duality … Read more
We provide a monotone non increasing sequence of upper bounds $f^H_k$ ($k\ge 1$) converging to the global minimum of a polynomial $f$ on simple sets like the unit hypercube. The novelty with respect to the converging sequence of upper bounds in [J.B. Lasserre, A new look at nonnegativity on closed sets and polynomial optimization, SIAM … Read more
Despite the fact that many vertex coloring problems are polynomially solvable on certain graph classes, most of these problems are not “under control” from a polyhedral point of view. The equivalence between optimization and separation suggests the existence of integer programming formulations for these problems whose associated polytopes admit elegant characterizations. In this work we … Read more
In this paper we discuss the complexity of a dynamic spectrum management problem within a multi-user communication system with K users and N available tones. In this problem a common utility function is optimized. In particular, so called min-rate, harmonic mean and geometric mean utility functions are considered. The complexity of the optimization problems with … Read more
We consider the problem of minimizing a continuous function f over a compact set K. We analyze a hierarchy of upper bounds proposed by Lasserre in [SIAM J. Optim. 21(3) (2011), pp. 864-ô€€€885], obtained by searching for an optimal probability density function h on K which is a sum of squares of polynomials, so that … 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 address combinatorial problems that can be formulated as minimization of a partially separable function of discrete variables (energy minimization in graphical models, weighted constraint satisfaction, pseudo-Boolean optimization, 0-1 polynomial programming). For polyhedral relaxations of such problems it is generally not true that variables integer in the relaxed solution will retain the same values in … Read more
In this paper we consider a class of non-Lipschitz and non-convex minimization problems which generalize the L2−Lp minimization problem. We propose an iterative algorithm that decides the next iteration based on the local convexity/concavity/sparsity of its current position. We show that our algorithm finds an epsilon-KKT point within O(log(1/epsilon)) iterations. The same result is also … Read more
Many interesting properties of polynomials are closely related to the geometry of their Newton polytopes. In this article we analyze the coercivity on $\mathbb{R}^n$ of multivariate polynomials $f\in \mathbb{R}[x]$ in terms of their Newton polytopes. In fact, we introduce the broad class of so-called gem regular polynomials and characterize their coercivity via conditions imposed on … Read more
We consider the problem of minimizing a general continuously differentiable function over symmetric sets under sparsity constraints. These type of problems are generally hard to solve as the sparsity constraint induces a combinatorial constraint into the problem, rendering the feasible set to be nonconvex. We begin with a study of the properties of the orthogonal … Read more