Error bounds for parametric polynomial systems with applications to higher-order stability analysis and convergence rates

Guoyin Li University of New South Wales Boris Mordukhovich University of Wayne, USA T.T.A. Nghia Oakland University, USA T.S. Pham University of Dalat

Numerical Analysis and Scientific Computing Optimization and Control mathscidoc:1904.25015

Mathematical Programming, 168, 313–346, 2018
The paper addresses parametric inequality systems described by polynomial functions in finite dimensions, where state-dependent infinite parameter sets are given by finitely many polynomial inequalities and equalities. Such systems can be viewed, in particular, as solution sets to problems of generalized semi-infinite programming with polynomial data. Exploiting the imposed polynomial structure together with powerful tools of variational analysis and semialgebraic geometry, we establish a far-going extension of the Łojasiewicz gradient inequality to the general nonsmooth class of supremum marginal functions as well as higher-order (Hölder type) local error bounds results with explicitly calculated exponents. The obtained results are applied to higher-order quantitative stability analysis for various classes of optimization problems including generalized semi-infinite programming with polynomial data, optimization of real polynomials under polynomial matrix inequality constraints, and polynomial second-order cone programming. Other applications provide explicit convergence rate estimates for the cyclic projection algorithm to find common points of convex sets described by matrix polynomial inequalities and for the asymptotic convergence of trajectories of subgradient dynamical systems in semialgebraic settings.
Polynomial optimization, Semialgebraic functions, Error bounds, Variational analysis, Higher-order stability analysis, Convergence rate of algorithms
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@inproceedings{guoyin2018error,
  title={Error bounds for parametric polynomial systems with applications to higher-order stability analysis and convergence rates},
  author={Guoyin Li, Boris Mordukhovich, T.T.A. Nghia, and T.S. Pham},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20190430215629933808310},
  booktitle={Mathematical Programming},
  volume={168},
  pages={313–346},
  year={2018},
}
Guoyin Li, Boris Mordukhovich, T.T.A. Nghia, and T.S. Pham. Error bounds for parametric polynomial systems with applications to higher-order stability analysis and convergence rates. 2018. Vol. 168. In Mathematical Programming. pp.313–346. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20190430215629933808310.
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