We consider the global minimization of a multivariate polynomial on a semi-algebraic set \Omega defined with polynomial inequalities. We then compare two hierarchies of relaxations, namely, LP-relaxations based on products of the original constraints, in the spirit of the RLT procedure of Sherali and Adams and recent SDP (semi definite programming) relaxations introduced by the … Read more
We consider the general nonlinear optimization problem in 0-1 variables and provide an explicit equivalent positive semidefinite program in $2^n-1$ variables. The optimal values of both problems are identical. From every optimal solution of the former one easily find an optimal solution of the latter and conversely, from every solution of the latter one may … Read more
A form p on R^n (homogeneous n-variate polynomial) is called positive semidefinite (psd) if it is nonnegative on R^n. In other words, the zero vector is a global minimizer of p in this case. The famous 17th conjecture of Hilbert (later proven by Artin) is that a form p is psd if and only if … Read more
A user-friendly free Matlab package for defining Linear Matrix Inequality (LMI) problems. It acts as an interface for the Self-Dual-Minimisation package SeDuMi developed by Jos F. Sturm. The functionalities of SeDuMi Interface are the following: (1) Declare an LMI problem. Five Matlab functions allow to define completely an LMI problem which can be characterised by … Read more
SeDuMi 1.05 is an add-on for MATLAB, which lets you solve optimization problems with linear, quadratic and semidefiniteness constraints. It is possible to have complex valued data and variables in SeDuMi. Moreover, large scale optimization problems are solved efficiently, by exploiting sparsity. This paper describes how to work with this toolbox. CitationOptimization Methods and Software … Read more
The recent spectral bundle method allows to compute, within reasonable time, approximate dual solutions of large scale semidefinite quadratic 0-1 programming relaxations. We show that it also generates a sequence of primal approximations that converge to a primal optimal solution. Separating with respect to these approximations gives rise to a cutting plane algorithm that converges … Read more
We study convex conic optimization problems in which the right-hand side and the cost vectors vary linearly as a function of a scalar parameter. We present a unifying geometric framework that subsumes the concept of the optimal partition in linear programming (LP) and semidefinite programming (SDP) and extends it to conic optimization. Similar to the … Read more
We compare several semidefinite relaxations for the cut polytope obtained by applying the lift and project methods of Lov\’asz and Schrijver and of Lasserre. We show that the tightest relaxation is obtained when aplying the Lasserre construction to the node formulation of the max-cut problem. This relaxation $Q_t(G)$ can be defined as the projection on … Read more
In this paper, we introduce a transformation that converts a class of linear and nonlinear semidefinite programming (SDP) problems into nonlinear optimization problems. For those problems of interest, the transformation replaces matrix-valued constraints by vector-valued ones, hence reducing the number of constraints by an order of magnitude. The class of transformable problems includes instances of … Read more
Our concern lies in solving the following convex optimization problem: minimize cx subject to Ax=b, x \in P, where P is a closed convex set, not necessarily a cone. We bound the complexity of computing an almost-optimal solution of this problem in terms of natural geometry-based measures of the feasible region and the level-set of … Read more