New fractional error bounds for polynomial systems with applications to Hölderian stability in optimization and spectral theory of tensors

By Guoyin Li, Boris S. Mordukhovich, Tien Son Pham. 2015

73 citations recorded by the source corpus.

In this paper we derive new fractional error bounds for polynomial systems with exponents explicitly determined by the dimension of the underlying space and the number/degree of the involved polynomials. Our major result extends the existing error bounds from the system involving only a single polynomial to a general polynomial system and do not require any regularity assumptions. In this way we resolve, in particular, some open questions posed in the literature. The developed techniques are largely based on variational analysis and generalized differentiation, which allow us to establish, e.g., a nonsmooth extension of the seminal Łojasiewicz’s gradient inequality to maxima of polynomials with explicitly determined exponents. Our major applications concern quantitative Hölderian stability of solution maps for parameterized polynomial optimization problems and nonlinear complementarity systems with polynomial data as well as high-order semismooth properties of the eigenvalues of symmetric tensors.

Subfields: Computational Mathematics, Modeling and Simulation, Numerical Analysis

Citations in the corpus

• Calculus of the Exponent of Kurdyka–Łojasiewicz Inequality and Its Applications to Linear Convergence of First-Order Methods (2016, unknown): “In addition, the result, together with results in [23] that give the exponent of the maximum of finitely many convex polynomials, can be used to study theKLproperty of a large class of convex polynomial optimization problems with multiple convex polynomial constraints.”
• Error Bounds and Hölder Metric Subregularity (2014, unknown): “[4, 8, 9, 16,17,34,46,47,58].”