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

• 成果类型: Article
• 署名作者: Li, G.; Mordukhovich, B. S.; Nghia, T. T. A.; Pham, T. S.
• 署名单位: University of New South Wales Sydney; Wayne State University; Oakland University
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-016-1014-6
• 发表日期: 2018
• 页码: 313-346
• 关键词: tilt stability; projection algorithm; matrix inequalities; semiinfinite; optimization; nonsmooth

摘要

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 Aojasiewicz gradient inequality to the general nonsmooth class of supremum marginal functions as well as higher-order (Holder 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.