Hölder Metric Subregularity with Applications to Proximal Point Method

This paper is mainly devoted to the study and applications of Holder metric subregularity (or metric $q$-subregularity of order $q\in(0,1]$) for general set-valued mappings between infinite-dimensional spaces. Employing advanced techniques of variational analysis and generalized differentiation, we derive neighborhood and point-based sufficient conditions as well as necessary conditions for $q$-metric subregularity with evaluating the exact subregularity bound, which are new even for the conventional (first-order) metric subregularity in both finite and infinite dimensions. In this way we also obtain new fractional error bound results for composite polynomial systems with explicit calculating fractional exponents. Finally, metric $q$-subregularity is applied to conduct a quantitative convergence analysis of the classical proximal point method (PPM) for finding zeros of maximal monotone operators on Hilbert spaces.

[1]  R. Rockafellar.  Monotone Operators and the Proximal Point Algorithm , 1976 .

[2]  E. R. Avakov.  Theorems on estimates in the neighborhood of a singular point of a mapping , 1990 .

[3]  Osman Güer.  On the convergence of the proximal point algorithm for convex minimization , 1991 .

[4]  B. Mordukhovich.  Complete characterization of openness, metric regularity, and Lipschitzian properties of multifunctions , 1993 .

[5]  Jong-Shi Pang,et al.  Error bounds in mathematical programming , 1997, Math. Program..

[6]  A. Lewis,et al.  Error Bounds for Convex Inequality Systems , 1998 .

[7]  Heinz H. Bauschke,et al.  A new proximal point iteration that converges weakly but not in norm , 2005 .

[8]  René Henrion,et al.  Calmness of constraint systems with applications , 2005, Math. Program..

[9]  Xi Yin Zheng,et al.  Metric Subregularity and Constraint Qualifications for Convex Generalized Equations in Banach Spaces , 2007, SIAM J. Optim..

[10]  Alexander D. Ioffe,et al.  On Metric and Calmness Qualification Conditions in Subdifferential Calculus , 2008 .

[11]  Boris S. Mordukhovich,et al.  Failure of Metric Regularity for Major Classes of Variational Systems , 2008 .

[12]  Jen-Chih Yao,et al.  Covering properties at positive-order rates of multifunctions and some related topics , 2008 .

[13]  Boris S. Mordukhovich,et al.  Coderivative calculus and metric regularity for constraint and variational systems , 2009 .

[14]  Huynh van Ngai,et al.  Error bounds for systems of lower semicontinuous functions in Asplund spaces , 2008, Math. Program..

[15]  Guoyin Li,et al.  Error Bounds of Generalized D-Gap Functions for Nonsmooth and Nonmonotone Variational Inequality Problems , 2009, SIAM J. Optim..

[16]  Didier Aussel,et al.  Single-Directional Property of Multivalued Maps and Variational Systems , 2009, SIAM J. Optim..

[17]  D. Leventhal.  Metric subregularity and the proximal point method , 2009, 0902.4200.

[18]  R. Rockafellar,et al.  Implicit Functions and Solution Mappings , 2009 .

[19]  Boris Polyak,et al.  B.S. Mordukhovich. Variational Analysis and Generalized Differentiation. I. Basic Theory, II. Applications , 2009 .

[20]  Michaël Gaydu,et al.  Metric subregularity of order q and the solving of inclusions , 2011 .

[21]  Aram V. Arutyunov,et al.  Optimality Conditions: Abnormal and Degenerate Problems , 2010 .

[22]  A. Kruger,et al.  Error Bounds: Necessary and Sufficient Conditions , 2010 .

[23]  Xi Yin Zheng,et al.  Metric Subregularity and Calmness for Nonconvex Generalized Equations in Banach Spaces , 2010, SIAM J. Optim..

[24]  Guoyin Li.  On the Asymptotically Well Behaved Functions and Global Error Bound for Convex Polynomials , 2010, SIAM J. Optim..

[25]  Boris S. Mordukhovich,et al.  Metric regularity and Lipschitzian stability of parametric variational systems , 2010 .

[26]  R. Henrion,et al.  On calmness conditions in convex bilevel programming , 2011 .

[27]  Boris S. Mordukhovich,et al.  Enhanced metric regularity and Lipschitzian properties of variational systems , 2011, J. Glob. Optim..

[28]  Heinz H. Bauschke,et al.  Convex Analysis and Monotone Operator Theory in Hilbert Spaces , 2011, CMS Books in Mathematics.

[29]  Helmut Gfrerer,et al.  First Order and Second Order Characterizations of Metric Subregularity and Calmness of Constraint Set Mappings , 2011, SIAM J. Optim..

[30]  Alexander Y. Kruger,et al.  From convergence principles to stability and optimality conditions , 2012 .

[31]  Marc Quincampoix,et al.  Hölder metric regularity of set-valued maps , 2012, Math. Program..

[32]  Marius Durea,et al.  Calculus of tangent sets and derivatives of set-valued maps under metric subregularity conditions , 2011, Journal of Global Optimization.

[33]  Guoyin Li,et al.  Global error bounds for piecewise convex polynomials , 2013, Math. Program..

[34]  Bastian Goldlücke,et al.  Variational Analysis , 2014, Computer Vision, A Reference Guide.