/pdf/convex-analysisnoer-san-nojin-zhan-nituite-5dl2rbmoyr.pdf

Convex Analysisの二,三の進展について

关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解

Differential-Difference Equations. By Richard Bellman and Kenneth L. Cooke. Pp. xvi, 465. 1963. (Academic Press)

General Topology-I

We introduce a proximal alternating linearized minimization (PALM) algorithm for solving a broad class of nonconvex and nonsmooth minimization problems. Building on the powerful Kurdyka---?ojasiewicz property, we derive a self-contained convergence analysis framework and establish that each bounded sequence generated by PALM globally converges to a critical point. Our approach allows to analyze various classes of nonconvex-nonsmooth problems and related nonconvex proximal forward---backward algorithms with semi-algebraic problem's data, the later property being shared by many functions arising in a wide variety of fundamental applications. A by-product of our framework also shows that our results are new even in the convex setting. As an illustration of the results, we derive a new and simple globally convergent algorithm for solving the sparse nonnegative matrix factorization problem.

/pdf/proximal-alternating-linearized-minimization-for-nonconvex-lbqbrhi298.pdf

Proximal alternating linearized minimization for nonconvex and nonsmooth problems

Applications.- Constrained Optimization and Equilibria.- Optimal Control of Evolution Systems in Banach Spaces.- Optimal Control of Distributed Systems.- Applications to Economics.

Variational Analysis and Generalized Differentiation II

Variational analysis and generalized differentiation I : basic theory

Variational Analysis and Generalized Differentiation II: Applications

We consider some basic properties of nonsmooth and set-valued mappings (multifunctions) connected with open and inverse mapping principles, distance estimates to the level sets (metric regularity), and a locally Lipschitzian behavior. These properties have many important applications to various problems in nonlinear analysis, optimization, control theory, etc., especially for studying sensitivity and stability questions with respect to perturbations of initial data and parameters. We establish interrelations between these properties and prove effective criteria for their fulfillment stated in terms of robust generalized derivatives for multifunctions and nonsmooth mappings

/pdf/complete-characterization-of-openness-metric-regularity-and-4fj8jn67w7.pdf

Complete characterization of openness, metric regularity, and Lipschitzian properties of multifunctions

We develop a generalized differentiation theory for nonsmooth functions and sets with nonsmooth boundaries defined in Asplund spaces. This broad subclass of Banach spaces provides a convenient framework for many important applications to optimization, sensitivity, variational inequalities, etc. Our basic normal and subdifferential constructions are related to sequential weak-star limits of Frechet normals and subdifferentials. Using a variational approach, we establish a rich calculus for these nonconvex limiting objects which turn out to be minimal among other set-valued di erential constructions with natural properties. The results obtained provide new developments in infinite dimensional nonsmooth analysis and have useful applications to optimization and the geometry of Banach spaces.

/pdf/nonsmooth-sequential-analysis-in-asplund-spaces-1d4v4r0if0.pdf

Boris S. Mordukhovich

Papers

Variational Analysis and Generalized Differentiation II

Variational analysis and generalized differentiation I : basic theory

Variational Analysis and Generalized Differentiation II: Applications

Complete characterization of openness, metric regularity, and Lipschitzian properties of multifunctions

Nonsmooth sequential analysis in Asplund spaces