Quantitative stability of variational systems: III.ε-approximate solutions

We prove that the-optimal solutions of convex optimization problems are Lipschitz continuous with respect to data perturbations when these are measured in terms of the epi-distance. A similar property is obtained for the distance between the level sets of extended real valued functions. We also show that these properties imply that the-subgradient mapping is Lipschitz continuous.Research supported in part by the National Science Foundation and the Air Force Office of Scientific Research.     

Epi-convergence almost surely, in probability and in distribution

The paper deals with an epi-convergence of random real functions defined on a topological space. We follow the idea due to Vogel (1994) to split the epi-convergence into the lower semicontinuous approximation and the epi-upper approximation and localize them onto a given set. The approximations are shown to be connected to the miss- resp. hit-part of the ordinary Fell topology on sets. We introduce two procedures, called “localization”, separately for the miss-topology and the hit-topology on sets. Localization of the miss- resp. hit-part of the Fell topology on sets allows us to give a suggestion how to define the approximations in probability and in distribution. It is shown in the paper that in case of the finite-dimensional Euclidean space, the suggested approximations in probability coincide with the definition from Vogel and Lachout (2003). The research has been partially supported by Deutsche Forschungsgemeinschaft under grant No. 436TSE113/40, by the Ministry of Education, Youth and Sports of the Czech Republic under Project MSM 113200008 and by the Grant Agency of the Czech Republic under grant No. 201/03/1027.     

Epi-convergent discretization of the generalized Bolza problem in dynamic optimization

The paper is devoted to well-posed discrete approximations of the so-called generalized Bolza problem of minimizing variational functionals defined via extended-real-valued functions. This problem covers more conventional Bolza-type problems in the calculus of variations and optimal control of differential inclusions as well of parameterized differential equations. Our main goal is find efficient conditions ensuring an appropriate epi-convergence of discrete approximations, which plays a significant role in both the qualitative theory and numerical algorithms of optimization and optimal control. The paper seems to be the first attempt to study epi-convergent discretizations of the generalized Bolza problem; it establishes several rather general results in this direction. Research of B. S. Mordukhovich was partially supported by the USA National Science Foundation under grants DMS-0304989 and DMS-0603846 and by the Australian Research Council under grant DP-0451168. Research of T. Pennanen was supported by the Finnish Academy of Sciences under contract No. 3385.     

Epigraphical and Uniform Convergence of Convex Functions

We examine when a sequence of lsc convex functions on a Banach space converges uniformly on bounded sets (resp. compact sets) provided it converges Attouch-Wets (resp. Painlevé-Kuratowski). We also obtain related results for pointwise convergence and uniform convergence on weakly compact sets. Some known results concerning the convergence of sequences of linear functionals are shown to also hold for lsc convex functions. For example, a sequence of lsc convex functions converges uniformly on bounded sets to a continuous affine function provided that the convergence is uniform on weakly compact sets and the space does not contain an isomorphic copy of .

     

Metrization of Epi-Convergence: An Application to the Strong Consistency of M-Estimators

Strong consistency in the class of M-estimators is examined here as an application of epi-convergence, a functional convergence which is particularly suited for the study of convergence of the functions' minimizing values and arguments. Starting from a 1988 paper by J. Dupaova and R. Wets, which contains a thorough account of the relations between consistency and epi-convergence, a quantitative approach of the same topic is pursued here. Epi-convergence is compared with two definitions introduced in 1980 by one of the authors. The results are merged in order to define a distance between lower semicontinuous functions that is compatible with epi-convergence and bounds the distance between the minimizing arguments. These results applied to the statistical problem allow the definition of a bound of the distance between the estimator and the parameter.     

Approximating infinite horizon stochastic optimal control in discrete time with constraints

Traditional approaches to solving stochastic optimal control problems involve dynamic programming, and solving certain optimality equations. When recast as stochastic programming problems, structural aspects such as convexity are retained, and numerical solution procedures based on decomposition and duality may be exploited. This paper explores a class of stationary, infinite-horizon stochastic optimization problems with discounted cost criterion. Constraints on both states and controls are permitted, and modeled in the objective function by allowing it to take infinite values. Approximating techniques are developed using variational analysis, and intuitive lower bounds are obtained via averaging the future. These bounds could be used in a finite-time horizon stochastic programming setting to find solutions numerically. Research supported in part by a grant of the National Science Foundation. AMS Classification 46N10, 49N15, 65K10, 90C15, 90C46     

The splitting game and applications

First we define the splitting operator, which is related to the Shapley operator of the splitting game introduced by Sorin (2002). It depends on two compact convex sets C and D and associates to a function defined on C×D a saddle function, extending the usual convexification or concavification operators. We first prove general properties on its domain and its range. Then we give conditions on C and D allowing to preserve continuity or Lipschitz properties, extending the results in Laraki (2001a) obtained for the convexification operator. These results are finally used, through the analysis of the asymptotic behavior of the splitting game, to prove the existence of a continuous solution for the Mertens-Zamir system of functional equations (Mertens and Zamir (1971–72) and (1977)) in a quite general framework. Revised November 2001     

Monotonic analysis: convergence of sequences of monotone functions

In this article we examine various kinds of convergence of sequences of increasing positively homogeneous (IPH) functions and nonnegative decreasing functions defined on the interior of a pointed closed solid convex cone K. We show that five different types of convergency (including pointwise and epi-convergence) coincide for IPH functions. If the space under consideration is finite dimensional then the sixth type can be added: uniform convergence on bounded subsets of itn K. Using IPH functions, we study epi-convergence of sequences of lower semi-continuous (lsc) nonnegative decreasing functions.