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 ?ojasiewicz'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.

http://www.optimization-online.org/DB_FILE//2013/01/3726.pdf

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

This paper sheds new light on several interrelated topics of second-order variational analysis, both in finite and infinite-dimensional settings. We establish new relationships between second-order growth conditions on functions, the basic properties of metric regularity and subregularity of the limiting subdifferential, tilt-stability of local minimizers, and positive-definiteness/semidefiniteness properties of the second-order subdifferential (or generalized Hessian).

Second-order growth, tilt stability, and metric regularity of the subdifferential

The paper concerns a systematic study of full stability in general optimization models including its conventional Lipschitzian version as well as the new Holderian one. We derive various characteri...

Full Lipschitzian and Hölderian Stability in Optimization with Applications to Mathematical Programming and Optimal Control

This paper is devoted to the study of tilt stability of local minimizers for classical nonlinear programs with equality and inequality constraints in finite dimensions described by twice continuous...

Complete Characterizations of Tilt Stability in Nonlinear Programming under Weakest Qualification Conditions

The paper is devoted to the study of tilt-stable local minimizers of general optimization problems in finite-dimensional spaces and its applications to classical nonlinear programs with twice continuously differentiable data. The importance of tilt stability has been well recognized from both theoretical and numerical aspects of optimization, and this notion has been extensively studied in the literature. Based on advanced tools of second-order variational analysis and generalized differentiation, we develop a new approach to tilt stability, which allows us to derive not only qualitative but also quantitative characterizations of tilt-stable minimizers with calculating the corresponding moduli. The implementation of this approach and general results in the classical framework of nonlinear programming provides complete characterizations of tilt-stable minimizers under new second-order qualification and optimality conditions.

http://www.optimization-online.org/DB_FILE/2013/07/3959.pdf

Second-order characterizations of tilt stability with applications to nonlinear programming

This paper is mainly devoted to the study of the so-called full Lipschitzian stability of local solutions to finite-dimensional parameterized problems of constrained optimization, which has been well recognized as a very important property from the viewpoints of both optimization theory and its applications. Based on second-order generalized differential tools of variational anal- ysis, we obtain necessary and sufficient conditions for fully stable local minimizers in general classes of constrained optimization problems, including problems of composite optimization, mathemati- cal programs with polyhedral constraints, as well as problems of extended and classical nonlinear programming with twice continuously differentiable data.

http://www.optimization-online.org/DB_FILE/2012/08/3564.pdf

M. E. Sarabi

Papers

Characterizations of full stability in constrained optimization