Given a semi algebraic set S of R^n we provide a numerical approximation procedure that yields upper and lower bounds on mu(S), for measures mu that satisfy some given moment conditions. The bounds are obtained as solutions of positive semidefinite programs that can be solved via standard software packages like the LMI MATLAB toolbox. Citation … Read more
In this paper we present a primal-dual inexact infeasible interior-point algorithm for semidefinite programming problems (SDP). This algorithm allows the use of search directions that are calculated from the defining linear system with only moderate accuracy, and our analysis does not require feasibility to be maintained even if the initial iterate happened to be a … Read more
In this paper we introduce a conic optimization formulation for inequality-constrained convex programming, and propose a self-dual embedding model for solving the resulting conic optimization problem. The primal and dual cones in this formulation are characterized by the original constraint functions and their corresponding conjugate functions respectively. Hence they are completely symmetric. This allows for … Read more
This article describes a generalization of the PBM method by Ben-Tal and Zibulevsky to convex semidefinite programming problems. The algorithm used is a generalized version of the Augmented Lagrangian method. We present details of this algorithm as implemented in a new code PENNON. The code can also solve second-order conic programming (SOCP) problems, as well … Read more
We formulate a block-iterative algorithmic scheme for the solution of systems of linear inequalities and/or equations and analyze its convergence. This study provides as special cases proofs of convergence of (i) the recently proposed Component Averaging (CAV) method of Censor, Gordon and Gordon ({\it Parallel Computing}, 27:777–808, 2001), (ii) the recently proposed Block-Iterative CAV (BICAV) … Read more
We study approximation bounds for the SDP relaxation of quadratically constrained quadratic optimization: min f^0(x) subject to f^k(x)
Primal-dual interior-point methods (IPMs) have shown their power in solving large classes of optimization problems. However, at present there is still a gap between the practical behavior of these algorithms and their theoretical worst-case complexity results, with respect to the strategies of updating the duality gap parameter in the algorithm. The so-called small-update IPMs enjoy … Read more
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 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