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

• 成果类型: Article
• 署名作者: Li, G.; Mordukhovich, B. S.; Pham, T. S.
• 署名单位: University of New South Wales Sydney; Wayne State University; King Fahd University of Petroleum & Minerals; Duy Tan University
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-014-0806-9
• 发表日期: 2015
• 页码: 333-362
• 关键词: weak sharp minima; lojasiewicz exponent; linear regularity; tilt stability; INEQUALITY; nonsmooth

摘要

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 Lojasiewicz's gradient inequality to maxima of polynomials with explicitly determined exponents. Our major applications concern quantitative Holderian 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.