Accelerating the LSTRS Algorithm

In a recent paper [Rojas, Santos, Sorensen: ACM ToMS 34 (2008), Article 11] an efficient method for solvingthe Large-Scale Trust-Region Subproblem was suggested which is based on recasting it in terms of a parameter dependent eigenvalue problem and adjusting the parameter iteratively. The essential work at each iteration is the solution of an eigenvalue problem for the smallest eigenvalue of the Hessian matrix (or two smallest eigenvalues in the potential hard case) and associated eigenvector(s). Replacing the implicitly restarted Lanczos method in the original paper with the Nonlinear Arnoldi method makes it possible to recycle most of the work from previous iterations which can substantially accelerate LSTRS

Copositivity and constrained fractional quadratic programs

Abstract We provide Completely Positive and Copositive Optimization formulations for the Constrained Fractional Quadratic Problem (CFQP) and Standard Fractional Quadratic Problem (StFQP). Based on these formulations, Semidefinite Programming (SDP) relaxations are derived for finding good lower bounds to these fractional programs, which can be used in a global optimization branch-and-bound approach. Applications of the CFQP and StFQP, related with the correction of infeasible linear systems and eigenvalue complementarity problems are also discussed