We analyze a version of the Mizuno-Todd-Ye predictor-corrector interior point algorithm for the P_*(\kappa)-matrix linear complementarity problem (LCP). We assume the existence of a strictly positive feasible solution. Our version of the Mizuno-Todd-Ye predictor-corrector algorithm is a generalization of Potra’s (2002) conclusions on the LCP with P_*(\kappa)-matrices. To derive a formulation of the complexity for … Read more
Using a moment interpretation of recent results on sum-of-squares decompositions of non-negative polynomial matrices, we propose a hierarchy of convex linear matrix inequality (LMI) relaxations to solve non-convex polynomial matrix inequality (PMI) optimization problems, including bilinear matrix inequality (BMI) problems. This hierarchy of LMI relaxations generates a monotone sequence of lower bounds that converges to … Read more
We present a primal-dual interior-point algorithm for second-order conic optimization problems based on a specific class of kernel functions. This class has been investigated earlier for the case of linear optimization problems. In this paper we derive the complexity bounds $O(\sqrt{N})(\log N)\log\frac{N}{\epsilon})$ for large- and $O(\sqrt{N})\log\frac{N}{\epsilon}$ for small- update methods, respectively. Here $N$ denotes the … Read more
In this paper, we first develop “economical” representations for positive semidefinitness of well-structured sparse symmetric matrix. Using the representations, we then reformulate well-structured large-scale semidefinite problems into smooth convex-concave saddle point problems, which can be solved by a Prox-method with efficiency ${\cal O}(\epsilon^{-1})$ developed in \cite{Nem}. Some numerical implementations for large-scale Lovasz capacity and MAXCUT … Read more
Unconstrained and inequality constrained sparse polynomial optimization problems (POPs) are considered. A correlative sparsity pattern graph is defined to find a certain sparse structure in the objective and constraint polynomials of a POP. Based on this graph, sets of supports for sums of squares (SOS) polynomials that lead to efficient SOS and semidefinite programming (SDP) … Read more
We extend the Q method to the symmetric cone programming. An infeasible interior point algorithm and a Newton-type algorithm are given. We give convergence results of the interior point algorithm and prove that the Newton-type algorithm is good for CitationAdvOl-Report#2004/18 McMaster University, Advanced Optimization Laboratory Hamilton, Ontario, Canada October 2004ArticleDownload View PDF
The second-order cone programming problem is reformulated into several new systems of nonlinear equations. Assume the perturbation of the data is in a certain neighborhood of zero. Then starting from a solution to the old problem, the semismooth Newton’s iterates converge Q-quadratically to a solution of the perturbed problem. The algorithm is globalized. Numerical examples … Read more
Based on the Q method for SDP, we develop the Q method for SOCP. A modified Q method is also introduced. Properties of the algorithms are discussed. Convergence proofs are given. Finally, we present numerical results. CitationAdvOl-Report#2004/15 McMaster University, Advanced Optimization LaboratoryArticleDownload View PDF
There is a natural norm associated with a starting point of the homogeneous self-dual (HSD) embedding model for conic convex optimization. In this norm two measures of the HSD model’s behavior are precisely controlled independent of the problem instance: (i) the sizes of epsilon-optimal solutions, and (ii) the maximum distance of epsilon-optimal solutions to the … Read more
This paper studies a new primal-dual interior/exterior-point method for linear programming. We begin with the usual perturbed primal-dual optimality equations $F_\mu(x,y,z)=0$. Under nondegeneracy assumptions, this nonlinear system is well-posed, i.e. it has a nonsingular Jacobian at optimality and is not necessarily ill-conditioned as the iterates approach optimality. We use a simple preprocessing step to eliminate … Read more