A conjecture implying the existence of non-convex Chebyshev sets in infinite-dimensional Hilbert spaces

In this paper, we propose the study of a conjecture whose affirmative solution would provide an example of a non-convex Chebyshev set in an infinite-dimensional real Hilbert space

A Cyclic Douglas-Rachford Iteration Scheme

In this paper we present two Douglas-Rachford inspired iteration schemes which can be applied directly to N-set convex feasibility problems in Hilbert space. Our main results are weak convergence of the methods to a point whose nearest point projections onto each of the N sets coincide. For affine subspaces, convergence is in norm. Initial results from numerical experiments, comparing our methods to the classical (product-space) Douglas-Rachford scheme, are promising.Comment: 22 pages, 7 figures, 4 table

Geometric Conditions of Regularity in Some Kind of Minimal Time Problems

The work is devoted to the problem of reaching a closed subset of a Hilbert space in minimal time from a point situated pear the target subject to a constant convex dynamics. Two types of geometric conditions guaranteeing existence and uniqueness of the end point of an optimal trajectory are given. We study the mapping, which associates to each initial state Chis end point, and under some supplementary assumptions prove its Holder continuity outside the target. Then we estabilish the (Holder) continuous dilferentiability of the value function in an open neighbourhood of the target set and give explicit formulas for its derivative. From the same point of view we treat the close problem with a nonlinear Lipschitzean perturbation and obtain some regularity results for viscosity solutions of a kind of Hamilton-Jacobi equations with non trivial boundary data. / RESUMO - O trabalho é dedicado ao problema de se atingir um subconjunto de um espaço de Hilbert em tempo mínimo a partir de um ponto situado próximo do conjunto-alvo cora uma dinâmica convexa constante. São dados dois tipos de condições geométricas que garantem existência e unicidade do ponto final de uma trajectória óptima. Estudamos a aplicação que associa a cada estado inicial esse ponto final, e sob algumas condições suplementares provamos continuidade de Holder da respectiva aplicação fora do conjunto-alvo. Depois mostramos a diferenciabilidade contínua (de Holder) da função valor também numa vizinhança do alvo apresentando fórmulas explicitas para a sua derivada. Do mesmo ponto de vista tratamos o problema com uma perturbação não linear Lipschitzeana e obtemos alguns resultados de regularidade para soluções viscosas de um certo tipo de equações de Hamilton-Jacobi com dados na fronteira não triviais

Towards Machine Wald

The past century has seen a steady increase in the need of estimating and predicting complex systems and making (possibly critical) decisions with limited information. Although computers have made possible the numerical evaluation of sophisticated statistical models, these models are still designed \emph{by humans} because there is currently no known recipe or algorithm for dividing the design of a statistical model into a sequence of arithmetic operations. Indeed enabling computers to \emph{think} as \emph{humans} have the ability to do when faced with uncertainty is challenging in several major ways: (1) Finding optimal statistical models remains to be formulated as a well posed problem when information on the system of interest is incomplete and comes in the form of a complex combination of sample data, partial knowledge of constitutive relations and a limited description of the distribution of input random variables. (2) The space of admissible scenarios along with the space of relevant information, assumptions, and/or beliefs, tend to be infinite dimensional, whereas calculus on a computer is necessarily discrete and finite. With this purpose, this paper explores the foundations of a rigorous framework for the scientific computation of optimal statistical estimators/models and reviews their connections with Decision Theory, Machine Learning, Bayesian Inference, Stochastic Optimization, Robust Optimization, Optimal Uncertainty Quantification and Information Based Complexity.Comment: 37 page

Discrete Approximations, Relaxation, and Optimization of One-Sided Lipschitzian Differential Inclusions in Hilbert Spaces

We study discrete approximations of nonconvex differential inclusions in Hilbert spaces and dynamic optimization/optimal control problems involving such differential inclusions and their discrete approximations. The underlying feature of the problems under consideration is a modi- fied one-sided Lipschitz condition imposed on the right-hand side (i.e., on the velocity sets) of the differential inclusion, which is a significant improvement of the conventional Lipschitz continuity. Our main attention is paid to establishing efficient conditions that ensure the strong approximation (in the W^1,p-norm as p greater than or equal to 1) of feasible trajectories for the one-sided Lipschitzian differential inclusions under. consideration by those for their discrete approximations and also the strong con- vergence of optimal solutions to the corresponding dynamic optimization problems under discrete approximations. To proceed with the latter issue, we derive a new extension of the Bogolyubov-type relaxation/density theorem to the case of differential inclusions satisfying the modified one-sided Lipschitzian condition. All the results obtained are new not only in the infinite-dimensional Hilbert space framework but also in finite-dimensional spaces

Generalized Differentiation and Characterizations for Differentiability of Infimal Convolutions

This paper is devoted to the study of generalized differentiation properties of the infimal convolution. This class of functions covers a large spectrum of nonsmooth functions well known in the literature. The subdifferential formulas obtained unify several known results and allow us to characterize the differentiability of the infimal convolution which plays an important role in variational analysis and optimization

When some variational properties force convexity

Abstract The notion of adequate (resp. strongly adequate) function has been recently introduced to characterize the essentially strictly convex (resp. essentially firmly subdifferentiable) functions among the weakly lower semicontinuous (resp. lower semicontinuous) ones. In this paper we provide various necessary and sufficient conditions in order that the lower semicontinuous hull of an extended real-valued function on a reflexive Banach space is essentially strictly convex. Some new results on nearest (farthest) points are derived from this approach. keywords Convex duality, well posed optimization problem, essential strict convexity, essential smoothness, best approximation