Set-Valued and Variational Analysis最新文献

We establish the weak convergence of inertial Krasnoselskii-Mann iterations towards a common fixed point of a family of quasi-nonexpansive operators, along with estimates for the non-asymptotic rate at which the residuals vanish. Strong and linear convergence are obtained in the quasi-contractive setting. In both cases, we highlight the relationship with the non-inertial case, and show that passing from one regime to the other is a continuous process in terms of the hypotheses on the parameters. Numerical illustrations are provided for an inertial primal-dual method and an inertial three-operator splitting algorithm, whose performance is superior to that of their non-inertial counterparts.

In this paper we give several existence results for solutions of equilibrium problems in topological spaces without linear structure. To this end we introduce a new concept of convexity for maps and multivalued maps in spaces without linear structure. The discussion on convexity is enriched with some example useful to compare the new conditions with the existing one in literature. Finally, we apply the existence results obtained to a Nash equilibrium problem and to a maximization of a binary relation.

We provide new characterizations of the (varepsilon )-subdifferential of the supremum of an arbitrary family of convex functions. The resulting formulas only involve approximate subdifferentials of adequate convex combinations of the data functions. Families of convex functions with a concavity-like property are introduced and their relationship with affine models is studied. The role of the lower semi-continuity of the data functions is also analyzed.

The main purpose of the present paper is the characterization, in the finite dimensional setting, of weak and strong invariance of closed sets with respect to a differential inclusion governed by time-dependent maximal monotone operators and multi-valued perturbation, by the use of the corresponding Hamiltonians.

Probability functions appear in constraints of many optimization problems in practice and have become quite popular. Understanding their first-order properties has proven useful, not only theoretically but also in implementable algorithms, giving rise to competitive algorithms in several situations. Probability functions are built up from a random vector belonging to some parameter-dependent subset of the range of that given random vector. In this paper, we investigate first order information of probability functions specified through a convex-valued set-valued application. We provide conditions under which the resulting probability function is indeed locally Lipschitzian. We also provide subgradient formulæ. The resulting formulæ are made concrete in a classic optimization setting and put to work in an illustrative example coming from an energy application.

The aim of the present paper is to state and discuss the well-posedness of a new evolution inclusion governed by the subdifferential of a function (varphi ) perturbed both by a Carathéodory mapping and by an integral forcing term. The integrand of the forcing term depends on two time-variables. We prove the existence and uniqueness of local and global solution, assuming that (varphi ) is primal lower regular and the perturbation terms satisfy some standard conditions. The results are applied to the study of non-regular electrical circuits with tranmission line, Zener diode and inductor. A second application is concerned with semilinear heat equation with memory. These applications illustrate an additional novelty of our paper.

We develop a new epi-convergence based on the use of bounded convergent nets on the product topology of the strong topology on the primal space and weak star topology on the dual space of a general real Banach space. We study the propagation of the associated variational convergences through conjugation of convex functions defined on this product space. These results are then applied to the problem of construction of a bigger-conjugate representative function for the recession operator associated with a maximal monotone operator on this real Banach space. This is then used to study the relationship between the recession operator of a maximal monotone operator and the normal–cone operator associated with the closed, convex hull of the domain of that monotone operator. This allows us to show that the strong closure of the domain of any maximal monotone operator is convex in a general real Banach space.

Existence results for a Cauchy problem driven by a semilinear differential Sturm-Liouville inclusion are achived by proving, in a preliminary way, an existence theorem for a suitable integral inclusion. In order to obtain this proposition we use a recent fixed point theorem that allows us to work with the weak topology and the De Blasi measure of weak noncompactness. So we avoid requests of compactness on the multivalued terms. Then, by requiring different properties on the map p involved in the Sturm-Liouville inclusion, we are able to establish the existence of both mild solutions and strong ones for the problem examinated. Moreover we focus our attention on the study of controllability for a Cauchy problem governed by a suitable Sturm-Liouville equation. Finally we precise that our results are able to study problems involving a more general version of a semilinear differential Sturm-Liouville inclusion.

In this paper we study the existence and properties of solutions for a discontinuous sweeping process involving prox-regular sets in a Hilbert spaces. The variation of the moving set is controlled by a positive Radon measure and the perturbation is the sum of two multivalued mappings. The values of the first one are closed, bounded, not necessarily convex sets. It is measurable in the time variable, Lipschitz continuous in the phase variable, and it satisfies a conventional growth condition. The values of the second one are closed, convex, not necessarily bounded sets. We assume that this mapping has a closed with respect to the phase variable graph.

Other assumptions concern the intersection of the second mapping with the multivalued mapping defined by the growth conditions. We suppose that this intersection has a measurable selector and it possesses some compactness properties.

We prove the existence of right-continuous solutions of bounded variation for our inclusion. If the values of the first inclusion are closed convex sets, then the solution set is a closed subset of the space of right-continuous functions of bounded variation with sup-norm. If, in addition, the values of the moving sets are compact sets, then the solution set is compact in the space of right-continuous functions of bounded variation endowed with the topology of uniform convergence on an interval.

The proofs are based on the author’s theorem on continuous with respect to a parameter selectors passing through fixed points of contraction multivalued maps with closed, nonconvex, decomposable values depending on the parameter and some compactness criteria (an analog of the Arzelà–Ascoli theorem) for sets in the space of right-continuous functions of bounded variation with sup-norm. The classical Ky Fan fixed point theorem is also used. The results that we obtain are new.

Various types of variational convergence of functions and bifunctions are applied to study global approximations of a quasi-equilibrium problem and a Nash quasi-game (generalized noncooperative game), two typical and important optimization models. We prove that when the data of approximating problems of a problem under consideration converge in the sense of suitable types of epi-, epi/hypo-, lop-convergence, or their weak variants, approximate solutions of the above approximating problems set converge to solutions of the original problem. Some results are new and others improve certain known ones since the applied types of variational convergence are weaker than the usually used classical types.