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
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. Citation Optimization Methods and … 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
This article considers continuous trajectories of the vector fields induced by two different primal-dual potential-reduction algorithms for solving linear programming problems. For both algorithms, it is shown that the associated continuous trajectories include the central path and the duality gap converges to zero along all these trajectories. For the algorithm of Kojima, Mizuno, and Yoshise, … Read more
This paper provides a simple approach for solving a semidefinite program, SDP\@. As is common with many other approaches, we apply a primal-dual method that uses the perturbed optimality equations for SDP, $F_\mu(X,y,Z)=0$, where $X,Z$ are $n \times n$ symmetric matrices and $y \in \Re^n$. However, we look at this as an overdetermined system of … Read more
This software package is a MATLAB implementation of infeasible path-following algorithms for solving conic programming problems whose constraint cone is a product of semidefinite cones, second-order cones, and/or nonnegative orthants. It employs a predictor-corrector primal-dual path-following method, with either the HKM or the NT search direction. The basic code is written in Matlab, but key … Read more