Global Convergence of Algorithms Based on Unions of Nonexpansive Maps

In his recent research M. K. Tam (2018) considered a framework for the analysis of iterative algorithms which can be described in terms of a structured set-valued operator. At each point in the ambient space, the value of the operator can be expressed as a finite union of values of single-valued paracontracting operators. He showed that the associated fixed point iteration is locally convergent around strong fixed points. This result generalizes a theorem due to Bauschke and Noll (2014). In the present paper we generalize the result of Tam and show the global convergence of his algorithm for an arbitrary starting point. An analogous result is also proved for the Krasnosel'ski-Mann iterations

Union Averaged Operators with Applications to Proximal Algorithms for Min-Convex Functions

In this paper we introduce and study a class of structured set-valued operators which we call union averaged nonexpansive. At each point in their domain, the value of such an operator can be expressed as a finite union of single-valued averaged nonexpansive operators. We investigate various structural properties of the class and show, in particular, that is closed under taking unions, convex combinations, and compositions, and that their fixed point iterations are locally convergent around strong fixed points. We then systematically apply our results to analyze proximal algorithms in situations where union averaged nonexpansive operators naturally arise. In particular, we consider the problem of minimizing the sum two functions where the first is convex and the second can be expressed as the minimum of finitely many convex functions

Universal envelopes of discontinuous functions

This thesis is a contribution to computable analysis in the tradition of Grzegorczyk, Lacombe, and Weihrauch. The main theorem of computable analysis asserts that any computable function is continuous. The solution operators for many interesting problems encountered in practice turn out to be discontinuous, however. It hence is a natural question how much partial information may be obtained on the solutions of a problem with discontinuous solution operator in a continuous or computable way. We formalise this idea by introducing the notion of continuous envelopes of discontinuous functions. The envelopes of a given function can be partially ordered in a natural way according to the amount of information they encode. We show that for any function between computably admissible represented spaces this partial order has a greatest element, which we call the universal envelope. We develop some basic techniques for the calculation of a suitable representation of the universal envelope in practice. We apply the ideas we have developed to the problem of locating the fixed point set of a continuous self-map of the unit ball in finite-dimensional Euclidean space, and the problem of locating the fixed point set of a nonexpansive self-map of the unit ball in infinite-dimensional separable real Hilbert space

Definable Zero-Sum Stochastic Games

International audienceDefinable zero-sum stochastic games involve a finite number of states and action sets, reward and transition functions that are definable in an o-minimal structure. Prominent examples of such games are finite, semi-algebraic or globally subanalytic stochastic games. We prove that the Shapley operator of any definable stochastic game with separable transition and reward functions is definable in the same structure. Definability in the same structure does not hold systematically: we provide a counterexample of a stochastic game with semi-algebraic data yielding a non semi-algebraic but globally subanalytic Shapley operator. %Showing the definability of the Shapley operator in full generality appears thus as a complex and challenging issue. } Our definability results on Shapley operators are used to prove that any separable definable game has a uniform value; in the case of polynomially bounded structures we also provide convergence rates. Using an approximation procedure, we actually establish that general zero-sum games with separable definable transition functions have a uniform value. These results highlight the key role played by the tame structure of transition functions. As particular cases of our main results, we obtain that stochastic games with polynomial transitions, definable games with finite actions on one side, definable games with perfect information or switching controls have a uniform value. Applications to nonlinear maps arising in risk sensitive control and Perron-Frobenius theory are also given