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New Fractional Error Bounds for Nonconvex Polynomial Systems with Applications to Holderian Stability in Optimization and Spectral Theory of Tensors

G Li (g.li***at***unsw.edu.au)
B. S. Mordukhovich (boris***at***math.wayne.edu)
T. S. Pham (sonpt***at***dlu.edu.vn)

Abstract: In this paper we derive new fractional error bounds for nonconvex polynomial systems with exponents explicitly determined by the dimension of the underlying space and the number/degree of the involved polynomials. The results obtained do not require any regularity assumptions and 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 Lojasiewicz’s gradient in equality to maxima of nonconvex polynomials with explicitly determined exponents. Our major applications concern quantitative Ho ̈lderian 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.

Keywords: Error Bounds, Nonconvex Polynomials, Variational Analysis, Generalized Differentiation, Lojasiewicz’s Inequality, Holderian Stability, PolynomialOptimization and Complementarity

Category 1: Convex and Nonsmooth Optimization

Citation: To appear in Math. Program. DOI: 10.1007/s10107-014-0806-9.

Download: [PDF]

Entry Submitted: 01/07/2013
Entry Accepted: 01/07/2013
Entry Last Modified: 08/31/2014

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