Solving proximal split feasibility problems without prior. Proxregularity of a set c at a point x 2 c is a variational condition related to normal vectors and which is common to many types of sets. These properties of metric projections are considered for possibly nonconvex sets s. To do so, we use the concepts of robinsons normal map and kien and yaos natural map in order to transform the initial multivalued problem into singlevalued. Primal lowernice functions belong to the larger class of proxregular functions. Essentially, our goal is to show how several results in the line of those in the finite dimensional case or those relative to subsets of hilbert spaces can be obtained for functions on hilbert spaces. Moreau envelopes and the related proximal mappings of prox regular functions and pln functions in hilbert space have been established, see 4 and 7. Iterative schemes for nonconvex quasivariational problems.
The aim of the paper is to extend to the setting of uniformly convex banach spaces the results obtained for prox regular sets in hilbert spaces. Proxregular sets and epigraphs in uniformly convex banach. Existence results for second order nonconvex sweeping. In this paper, our aim is to introduce a viscosity type algorithm for solving proximal split feasibility problems and prove the strong convergence of the sequences generated by our iterative schemes in hilbert spaces. The subdifferential characterization allows us to show that some moreau. In particular a set is proxregular if and only if the projection mapping is locally singlevalued. Infimal convolutions and lipschitzian properties of subdifferentials for proxregular functions in hilbert spaces m bacak, jm borwein, a eberhard, bs mordukhovich 30. In particular a set is prox regular if and only if the projection mapping is locally singlevalued. We continue the study of proxregular sets that we began in a previous work in the setting of uniformly convex banach spaces endowed with a norm both uniformly smooth and uniformly convex e. We also study, in a geometrical point of view, the epigraphs of proxregular functions. H r from a hilbert space h can be characterized via the epigraph of f, that is, the set epi f x,s. It is wellknown that the sublevels of f are convex if and only if f is quasiconvex, which is the topic of. Then, we study in section 4 the links between the subdifferential and the proximal mapping of prox regular functions. Regularity and stability for convex multivalued functions.
We show that this notion includes proxregular functions, functions whose subdi. On the metric projection onto proxregular subsets of. Let xand y be topological spaces and f a setvalued mapping. Stability of variational inequalities and proxregularity in. Infimal convolutions and lipschitzian properties of. To do so, we use the concepts of robinsons normal map and kien and yaos natural map in order to transform the initial multivalued. Their result for this kind of behaviour of proxregular functions has been recently extended to hilbert spaces by ba. We essentially generalized in 8 the results of 38 in the more. Here a corresponding local theory is developed for the property of dc being continuously differentiable outside of c on some neighborhood of a point x. The last section establishes thec1 regularity of the moreau envelope of prox regular functions. The following properties are equivalent to f being uniformly proxregular on a neighborhood of x 0 with the same parameter. Stability of variational inequalities and proxregularity.
It is wellknown see 23, 32, 33 that all hilbert spaces h and the banach spaces lp, lp, and wp. Solving proximal split feasibility problems without prior knowledge of operator norms solving proximal split feasibility problems without prior knowledge of operator norms moudafi, a thakur, b. In this paper we study the problem of stability of existence of solutions of a classical variational inequality in a hilbert space. Using proximal analysis techniques, we provide su cient andor necessary conditions for such a generalized equation to have the metric subregularity in hilbert spaces. Proxregular functions in hilbert spaces by frederic bernard and lionel thibault download pdf 158 kb. We also establish results of robinsonursescu theorem type for prox regular multifunctions. We study infimal convolutions of extendedrealvalued functions in hilbert spaces paying a special attention to the rather broad and remarkable class of prox regular functions. The split feasibility problem sfp is important due to its occurrence in signal processing and image reconstruction, with particular progress in intensitymodulated radiation therapy. X5 x has an equivalent norm which is both uniformly convex and uniformly smooth and which has moduli of convexity and smoothness of power type. We continue the study of prox regular sets that we began in a previous work in the setting of uniformly convex banach spaces endowed with a norm both uniformly smooth and uniformly convex e. The study of proxregular functions provides insight on a broad spectrum of important functions. Moreover, while extensions exist to uniformly convex. On properties of differential inclusions with proxregular sets.
By the way, in 7, some restrictive assumptions are made so that the existence and stability results therein depend heavily on the decomposition of f g f. The proximal normal formula in banach space request pdf. However, it has been studied, in hilbert spaces, in, when is a uniformly prox regular set which is not necessarily a convex set see also 1, 2, 4. Sep 17, 2016 in this paper we study the problem of stability of existence of solutions of a classical variational inequality in a hilbert space. The last section establishes thec1 regularity of the moreau envelope of proxregular functions. Uniform proxregularity of functions and epigraphs in. If is a hilbert space, is a convex closed set in, is a convex bifunction, and 0,then hindawi journal of function spaces. Here and in the sequel, x,h,i is a separable real hilbert space, with norm k k.
Obviously, this class extends the class of uniformly proxregular sets 11, 12 from hilbert spaces to banach spaces since in hilbert spaces we have and the generalized proximal normal cone coincides with the usual proximal subdifferential see for more details on and. Ams transactions of the american mathematical society. Proxregularity and stability of the proximal mapping. Jul, 2006 2005 uniform prox regularity of functions and epigraphs in hilbert spaces. The present paper aims at studying properties of proxregular functions in the setting of infinite dimensional hilbert space. Proxregular sets and epigraphs in uniformly convex banach spaces. We also establish results of robinsonursescu theorem type for proxregular multifunctions. We also refer to rolewicz 29 and references therein for some similar convexlike functions. Parametrically proxregular paraproxregular functions are a further extension of this family, produced by adding a parameter. In the context of uniformly convex banach spaces, the. Iterative methods for nonconvex equilibrium problems in. Hilbert spaces settings see for instance 10 and the references therein. For a closed subset cof a hilbert space h, and any point x 2c, the following properties are equivalent. This paper introduces and characterizes new notions of lipschitzian and holderian full stability of solutions to general parametric variational systems defined via partial subdifferential of proxregular functions acting in finitedimensional and hilbert spaces.
Any closed convex set is uniformly generalized proxregular w. First, we prove strong convergence result for a problem of finding a point which minimizes a convex function f such that its image under a bounded linear. Regular sets and functions on riemannian manifolds. More precisely, we extend some stability results to an infinite dimensional and nonconvex setting. The study of prox regular functions provides insight on a broad spectrum of important functions. For a closed subset c of a hilbert space h, and any point. Researcharticle iterative schemes for nonconvex quasivariational problems with proxregular data in banach spaces m. The present paper aims at studying properties of prox regular functions in the setting of infinite dimensional hilbert space. Unlike a hilbert space, a manifold in general does not have a linear structure, and therefore new techniques are needed for dealing with the concepts of metric projection and distance function from sets in manifolds. The sets whose indicator functions belong to that class are proxregular sets. In this paper we study in mal convolutions of extendedrealvalued functions in hilbert spaces paying a special attention to.
Do, generalized second derivatives of convex functions in reflexive banach spaces, trans. Recently clarke, stern and wolenski characterized, in a hilbert space, the closed subsets c for which the distance function dc is continuously differentiable everywhere on an open tube of uniform thickness around c. Weak regularity of functions and sets in asplund spaces. Prox regularity of a set c at a point x 2 c is a variational condition related to normal vectors and which is common to many types of sets. In this paper we study in mal convolutions of extendedrealvalued functions in hilbert spaces paying a special attention to the rather broad and. The aim of the paper is to extend to the setting of uniformly convex banach spaces the results obtained for proxregular sets in hilbert spaces. These properties are applied to develop a perturbation theory for convex inequalities and to extend results on. Iterative approximation of solutions for proximal split. Parametrically proxregular functions ubc library open.
Infimal convolutions and lipschitzian properties of subdifferentials for proxregular functions in hilbert spaces, miroslav bacak, jonathan m. Existence and differentiability of metric projections in hilbert spaces. We establish also that weak regularity is equivalent to mordukhovich. Proxregularity is a generalization of convexity that includes all c2, lowerc2, strongly amenable, and primallowernice functions. In particular, a subdifferential characterization is established as well as several.
Linear nonlinear analysis algebra and its applications. Since 2005, there have been a number of papers published that extend particular properties of proxregular functions from finitedimensional spaces to hilbert spaces 1, 5, 6, 21, 29, so it is likely the techniques and results seen in those works would be of use in progressing in this direction for paraproxregular functions. The proof of the assertions 1 and 2 in the following example are given in 8. If the set s is convex, then it is well known that the corresponding metric projections always exist, unique and directionally differentiable at boundary points of s. It is wellknown that the sublevels of f are convex if and only if f. This is done with the help of the moreau local envelopes and local proximal mappings. Mathematics research reports mathematics wayne state.
Metric subregularity for proximal generalized equations in. Research article iterative methods for nonconvex equilibrium problems in uniformly convex and uniformly smooth banach spaces messaoudbounkhel department of mathematics, college of science, p. Planiden september 17, 2019 abstract proxregularity is a generalization of convexity that includes all c2, lowerc2, strongly amenable, and primallowernice functions. Zlateva, characterizations of proxregular sets in uniformaly convex banach spaces, j. We study infimal convolutions of extendedrealvalued functions in hilbert spaces paying a special attention to the rather broad and remarkable class of proxregular functions. Siam journal on optimization siam society for industrial. In particular, a subdifferential characterization is established.
Multivalued functions with convex graphs are shown to exhibit certain desirable regularity properties when their ranges have internal points. Characterizations of proxregular sets in uniformly convex. These include most of the functions that arise in the framework of nonlinear programming and its extensions e. In the more complex case, we consider the class of parametrically prox regular functions. Iterative schemes for nonconvex quasivariational problems with proxregular data in banach spaces m. We study their properties and find several characterizations of such. In this paper, we define the prox regularity for functions on banach spaces by adapting the original definition in r n. Proxregular functions carma university of newcastle.
However, it has been studied, in hilbert spaces, in bounkhel et al. First, we prove strong convergence result for a problem of finding a point which minimizes a convex function f such that its image under a bounded linear operator a minimizes. In this paper, we define the proxregularity for functions on banach spaces by adapting the original definition in r n. This paper considers metric projections onto a closed subset s of a hilbert space. Proxregular functions in hilbert spaces sciencedirect. Siam journal on optimization society for industrial and. Prox regularity is a generalization of convexity that includes all c2, lowerc2, strongly amenable, and primallowernice functions. The class of continuously proxregular functions includes all convex functions, all lowerc2 functions a function is lowerc2 if the function plus a multiple of the norm square is locally convex and all strongly amenable functions. Q, where a is a bounded linear operator, c and q are subsets of two hilbert spaces h. These properties are applied to develop a perturbation theory for convex inequalities and to extend results on the continuity of convex functions. Uniform proxregularity of functions and epigraphs in hilbert.
When and the last inequality becomes which is known as the classical variational inequality introduced and studied in stampacchia 9. Then, we study in section 4 the links between the subdifferential and the proximal mapping of proxregular functions. Mathematics research reports wayne state university. Spectral theory proxregularity of spectral functionssets a. This paper studies the proxregularity concept for functions in the general context of hilbert space. We prove normal and tangential regularity properties for these sets, and in particular the equality between mordukhovich and. In the more complex case, we consider the class of parametrically proxregular functions. Banach space 9, the results below seem most useful and quite possibly valid only in hilbert spaces. The next proposition gives several characterizations of local uniform proxregularity of functions. Bounekhel2 1departmentofmathematics,collegeofscience,kingsauduniversity,p. Second order sufficient conditions for stable wellposedness of. In this paper, we study a new concept of weak regularity of functions and sets in asplund spaces. Proxregularity of functions and sets in banach spaces. Research article iterative methods for nonconvex equilibrium.
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