A sequential semidefinite programming method and an application in passive reduced-order modeling

We consider the solution of nonlinear programs with nonlinear semidefiniteness constraints. The need for an efficient exploitation of the cone of positive semidefinite matrices makes the solution of such nonlinear semidefinite programs more complicated than the solution of standard nonlinear programs. In particular, a suitable symmetrization procedure needs to be chosen for the linearization of the complementarity condition. The choice of the symmetrization procedure can be shifted in a very natural way to certain linear semidefinite subproblems, and can thus be reduced to a well-studied problem. The resulting sequential semidefinite programming (SSP) method is a generalization of the well-known SQP method for standard nonlinear programs. We present a sensitivity result for nonlinear semidefinite programs, and then based on this result, we give a self-contained proof of local quadratic convergence of the SSP method. We also describe a class of nonlinear semidefinite programs that arise in passive reduced-order modeling, and we report results of some numerical experiments with the SSP method applied to problems in that class

A new line-search step based on the Weierstrass -function for minimizing a class of logarithmic barrier functions

This article is concerned with line-search procedures for a class of problems with certain nonlinear constraints. The class includes as special cases linear and convex quadratic programming problems, entropy programming problems and minimization problems over the cone of positive semidefinite matrices, [1, 2, 18, 7]. For solving a constrained optimization problem of the form minff 0 (x) j f i (x) 0 for 1 i mg (1.1) with four times continuously differentiable functions f i , we consider logarithmic barrier functions of the for

A QQP-Minimization Method for Semidefinite and Smooth Nonconvex Programs

. In several applications, semidefinite programs with nonlinear equality constraints arise. We give two such examples to emphasize the importance of this class of problems. We then propose a new solution method that also applies to smooth nonconvex programs. The method combines ideas of a predictor corrector interior-point method, of the SQP method, and of trust region methods. In particular, we believe that the new method combines the advantages---generality and robustness of trust region methods, local convergence of the SQP-method and data-independence of interior-point methods. Some convergence results are given, and some very preliminary numerical experiments suggest a high robustness of the proposed method. AMS 1991 subject classification. Primary: 90C. Key words. Predictor corrector method, SQP method, trust region method, semidefinite program. 1. Introduction This work was motivated by two applications from semidefinite programming with nonlinear equality constraints as outlin..