## Abstract

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.

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## Notes

After submitting the paper we have become familiar with the manuscript [11], where some ideas of applying Hölder error bound to study complexity of some known first-order algorithms in the convex setting are of similar flavors with ours.

The presented simplified proof of this result follows from the suggestions of both referees while incorporating some ideas of [7].

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## Acknowledgments

The authors are indebted to both anonymous referees for their careful reading the paper and valuable remarks that allowed us to improve the original presentation.

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## Additional information

*Dedicated to Terry Rockafellar in honor of his 80th birthday.*

G. Li: Research of this author was partly supported by Australian Research Council (Project Number: FT130100038).

B. S. Mordukhovich: Research of this author was partly supported by the USA National Science Foundation under Grants DMS-1007132 and DMS-1512846 and by the USA Air Force Office of Scientific Research under Grant No. 15RT0462.

T. S. Phạm: Research of this author was partly supported by the Vietnam National Foundation for Science and Technology Development (NAFOSTED).

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Li, G., Mordukhovich, B.S., Nghia, T.T.A. *et al.* Error bounds for parametric polynomial systems with applications to higher-order stability analysis and convergence rates.
*Math. Program.* **168**, 313–346 (2018). https://doi.org/10.1007/s10107-016-1014-6

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DOI: https://doi.org/10.1007/s10107-016-1014-6

### Keywords

- Polynomial optimization
- Error bounds
- Variational analysis
- Semialgebraic functions
- Higher-order stability analysis
- Convergence rate of algorithms