39 research outputs found
Computing the smallest fixed point of order-preserving nonexpansive mappings arising in positive stochastic games and static analysis of programs
The problem of computing the smallest fixed point of an order-preserving map
arises in the study of zero-sum positive stochastic games. It also arises in
static analysis of programs by abstract interpretation. In this context, the
discount rate may be negative. We characterize the minimality of a fixed point
in terms of the nonlinear spectral radius of a certain semidifferential. We
apply this characterization to design a policy iteration algorithm, which
applies to the case of finite state and action spaces. The algorithm returns a
locally minimal fixed point, which turns out to be globally minimal when the
discount rate is nonnegative.Comment: 26 pages, 3 figures. We add new results, improvements and two
examples of positive stochastic games. Note that an initial version of the
paper has appeared in the proceedings of the Eighteenth International
Symposium on Mathematical Theory of Networks and Systems (MTNS2008),
Blacksburg, Virginia, July 200
Generalized differentiation with positively homogeneous maps: Applications in set-valued analysis and metric regularity
We propose a new concept of generalized differentiation of set-valued maps
that captures the first order information. This concept encompasses the
standard notions of Frechet differentiability, strict differentiability,
calmness and Lipschitz continuity in single-valued maps, and the Aubin property
and Lipschitz continuity in set-valued maps. We present calculus rules, sharpen
the relationship between the Aubin property and coderivatives, and study how
metric regularity and open covering can be refined to have a directional
property similar to our concept of generalized differentiation. Finally, we
discuss the relationship between the robust form of generalization
differentiation and its one sided counterpart.Comment: This submission corrects errors from the previous version after
referees' comments. Changes are in Proposition 2.4, Proposition 4.12, and
Sections 7 and
Complete Characterizations of Local Weak Sharp Minima With Applications to Semi-Infinite Optimization and Complementarity
In this paper we identify a favorable class of nonsmooth functions for which local weak sharp minima can be completely characterized in terms of normal cones and subdifferentials, or tangent cones and subderivatives, or their mixture in finite-dimensional spaces. The results obtained not only significantly extend previous ones in the literature, but also allow us to provide new types of criteria for local weak sharpness. Applications of the developed theory are given to semi-infinite programming and to semi-infinite complementarity problems
A trust region-type normal map-based semismooth Newton method for nonsmooth nonconvex composite optimization
We propose a novel trust region method for solving a class of nonsmooth and
nonconvex composite-type optimization problems. The approach embeds inexact
semismooth Newton steps for finding zeros of a normal map-based stationarity
measure for the problem in a trust region framework. Based on a new merit
function and acceptance mechanism, global convergence and transition to fast
local q-superlinear convergence are established under standard conditions. In
addition, we verify that the proposed trust region globalization is compatible
with the Kurdyka-{\L}ojasiewicz (KL) inequality yielding finer convergence
results. We further derive new normal map-based representations of the
associated second-order optimality conditions that have direct connections to
the local assumptions required for fast convergence. Finally, we study the
behavior of our algorithm when the Hessian matrix of the smooth part of the
objective function is approximated by BFGS updates. We successfully link the KL
theory, properties of the BFGS approximations, and a Dennis-Mor{\'e}-type
condition to show superlinear convergence of the quasi-Newton version of our
method. Numerical experiments on sparse logistic regression and image
compression illustrate the efficiency of the proposed algorithm.Comment: 56 page