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About the Convergence of Parametric Sum of Squares Relaxations

Philipp Renner (philipp.renner***at***business.uzh.ch)

Abstract: In this paper we look at a parametric polynomial optimization problem. We assume, that the parameter space, and, for each parameter, the feasible set is compact. We show, that under these assumptions, the optimal solution of the sum of squares relaxations, using Schmüdgen's Positivstellensatz, converge uniformly over the parameter space to the optimal value functions. The proof's key idea is to use the bound obtained in a paper by Schweighofer (2004), and generalize it to the parametric case. By adding some additional constraints, we obtain as a consequence explicit bounds on the degree of the relaxation, i.e. there are no unknown constants contained in it.

Keywords: parametric optimization, schmüdgen's positivstellensatz, convergence rate

Category 1: Global Optimization (Theory )

Category 2: Nonlinear Optimization (Constrained Nonlinear Optimization )

Citation: Chair for quantitative business administration Moussonstrasse 15 8044 Zürich July 2012

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Entry Submitted: 07/04/2012
Entry Accepted: 07/05/2012
Entry Last Modified: 01/31/2013

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