Uniformity and inexact version of a proximal method for metrically regular mappings

We study stability properties of a proximal point algorithm for solving the inclusion 0∈T(x) when T is a set-valued mapping that is not necessarily monotone. More precisely we show that the convergence of our algorithm is uniform, in the sense that it is stable under small perturbations whenever the set-valued mapping T is metrically regular at a given solution. We present also an inexact proximal point method for strongly metrically subregular mappings and show that it is super-linearly convergent to a solution to the inclusion 0∈T(x).     

Stability of Mann’s iterates under metric regularity

We consider a Mann-like iteration for solving the inclusion xT(x) where is a set-valued mapping, defined from a Banach space X into itself, which is metrically regular near a point in its graph. We study the behavior of the iterates generated by our method and prove that they inherit the regularity properties of the mapping T. First we consider the case when the mapping T is metrically regular, then the case when it is strongly metrically regular. Finally, we present an inexact version of our method and we study its convergence when the mapping T is strongly metrically subregular.     

A Lyusternik–Graves theorem for the proximal point method

We consider a generalized version of the proximal point algorithm for solving the perturbed inclusion y∈T(x), where y is a perturbation element near 0 and T is a set-valued mapping acting from a Banach space X to a Banach space Y which is metrically regular around some point $({\bar{x}},0)$ in its graph. We study the behavior of the convergent iterates generated by the algorithm and we prove that they inherit the regularity properties of T, and vice versa. We analyze the cases when the mapping T is metrically regular and strongly regular.     

Tikhonov regularization of metrically regular inclusions

We present a Tikhonov regularization method for inclusions of the form where T is a set-valued mapping defined on a Banach space that enjoys metric regularity properties. We investigate, subsequently, the case when the mapping T is metrically regular, strongly metrically regular, strongly subregular and Lipschitz continuous and show the strong convergence of the solutions of regularized problems to a solution to the original inclusion . We also prove that the method has finite termination under some special conditioning assumptions on T and we study its stability with respect to some variational perturbations. These authors are supported by Contract EA3591 (France).     

Stability properties of the Tikhonov regularization for nonmonotone inclusions

We study the Tikhonov regularization for perturbed inclusions of the form T(x) ' y^*{T(x) \ni y^*} where T is a set-valued mapping defined on a Banach space that enjoys metric regularity properties and y* is an element near 0. We investigate the case when T is metrically regular and strongly regular and we show the existence of both a solution x* to the perturbed inclusion and a Tikhonov sequence which converges to x*. Finally, we show that the Tikhonov sequences associated to the perturbed problem inherit the regularity properties of the inverse of T.     

Metrically regular mappings and its application to convergence analysis of a confined Newton-type method for nonsmooth generalized equations

Notion of metrically regular property and certain types of point-based approximations are used for solving the nonsmooth generalized equation f(x)+F(x)?0,where X and Y are Banach spaces,and U is an open subset of X,f:U→Y is a nonsmooth function and F:X■Y is a set-valued mapping with closed graph.We introduce a confined Newton-type method for solving the above nonsmooth generalized equation and analyze the semilocal and local convergence of this method.Specifically,under the point-based approximation of f on U and metrically regular property of f+F,we present quadratic rate of convergence of this method.Furthermore,superlinear rate of convergence of this method is provided under the conditions that f admits p-point-based approximation on U and f+F is metrically regular.An example of nonsmooth functions that have p-point-based approximation is given.Moreover,a numerical experiment is given which illustrates the theoretical result.     

Metric regularity of semi-infinite constraint systems

We obtain a formula for the modulus of metric regularity of a mapping defined by a semi-infinite system of equalities and inequalities. Based on this formula, we prove a theorem of Eckart-Young type for such set-valued infinite-dimensional mappings: given a metrically regular mapping F of this kind, the infimum of the norm of a linear function g such that F+g is not metrically regular is equal to the reciprocal to the modulus of regularity of F. The Lyusternik-Graves theorem gives a straightforward extension of these results to nonlinear systems. We also discuss the distance to infeasibility for homogeneous semi-infinite linear inequality systems. Dedicated to R. T. Rockafellar on his 70th Birthday Research partially supported by grants BFM2002-04114-C02 (01-02) from MCYT (Spain) and FEDER (E.U.), GV04B-648 and GRUPOS04/79 from Generalitat Valenciana (Spain), and Bancaja-UMH (Spain).     

A Sard theorem for tame set-valued mappings

If F is a set-valued mapping from Rn into Rm with closed graph, then y∈Rm is a critical value of F if for some x with y∈F(x), F is not metrically regular at (x,y). We prove that the set of critical values of a set-valued mapping whose graph is a definable (tame) set in an o-minimal structure containing additions and multiplications is a set of dimension not greater than m−1 (respectively a σ-porous set). As a corollary of this result we get that the collection of asymptotically critical values of a set-valued mapping with a semialgebraic graph has dimension not greater than m−1. We also give an independent proof of the fact that a definable continuous real-valued function is constant on components of the set of its subdifferentiably critical points.     

Metrically Regular Mapping and Its Utilization to Convergence Analysis of a Restricted Inexact Newton-Type Method

In the present paper,we study the restricted inexact Newton-type method for solving the generalized equation 0∈f(x)+F(x),where X and Y are Banach spaces,f:X→Y is a Frechet differentiable function and F:X■Y is a set-valued mapping with closed graph.We establish the convergence criteria of the restricted inexact Newton-type method,which guarantees the existence of any sequence generated by this method and show this generated sequence is convergent linearly and quadratically according to the particular assumptions on the Frechet derivative of f.Indeed,we obtain semilocal and local convergence results of restricted inexact Newton-type method for solving the above generalized equation when the Frechet derivative of f is continuous and Lipschitz continuous as well as f+F is metrically regular.An application of this method to variational inequality is given.In addition,a numerical experiment is given which illustrates the theoretical result.     

Iterative solving of variational inclusions under Wijsman perturbations

We prove that the metric regularity of set-valued mappings is stable under some Wijsman-type perturbations. Then, we solve a variational inclusion viewed as a limit-problem using assumptions on a sequence of associated problems. Finally, we apply our results to classical methods for solving variational inclusions.      

Degree of convergence of modified averaged iterations for fixed points problems and operator equations

Let H be a real Hilbert space. We propose a modification for averaged mappings to approximate the unique fixed point of a mapping T:H→H such that T is boundedly Lipschitzian and −T is monotone. We not only prove strong convergence theorems, but also determine the degree of convergence. Using this result, an iteration process is given for finding the unique solution of the equation Ax=f, where A:H→H is strongly monotone and boundedly Lipschitzian.     

A projected subgradient method for solving generalized mixed variational inequalities

We consider the projected subgradient method for solving generalized mixed variational inequalities. In each step, we choose an ε_k-subgradient u^k of the function f and w^k in a set-valued mapping T, followed by an orthogonal projection onto the feasible set. We prove that the sequence is weakly convergent.     

Convergence analysis of a family of Steffensen-type methods for generalized equations

A class of Steffensen-type algorithms for solving generalized equations on Banach spaces is proposed. Using well-known fixed point theorem for set-valued maps [A.L. Dontchev, W.W. Hager, An inverse function theorem for set-valued maps, Proc. Amer. Math. Soc. 121 (1994) 481-489] and some conditions on the first-order divided difference, we provide a local convergence analysis. We also study the perturbed problem and we present a new regula-falsi-type method for set-valued mapping. This study follows the works on the Secant-type method presented in [S. Hilout, A uniparametric Secant-type methods for nonsmooth generalized equations, Positivity (2007), submitted for publication; S. Hilout, A. Piétrus, A semilocal convergence of a Secant-type method for solving generalized equations, Positivity 10 (2006) 673-700] and extends the results related to the resolution of nonlinear equations [M.A. Hernández, M.J. Rubio, The Secant method and divided differences Hölder continuous, Appl. Math. Comput. 124 (2001) 139-149; M.A. Hernández, M.J. Rubio, Semilocal convergence of the Secant method under mild convergence conditions of differentiability, Comput. Math. Appl. 44 (2002) 277-285; M.A. Hernández, M.J. Rubio, ω-Conditioned divided differences to solve nonlinear equations, in: Monogr. Semin. Mat. García Galdeano, vol. 27, 2003, pp. 323-330; M.A. Hernández, M.J. Rubio, A modification of Newton's method for nondifferentiable equations, J. Comput. Appl. Math. 164/165 (2004) 323-330].     

Function spaces in metrically generated theories

The theory of metrically generated constructs provides us with an excellent setting for the study of function spaces. In this paper we develop a function space theory for metrically generated constructs and, by considering different metrically generated constructs, we capture interesting examples. For instance, for uniform spaces we retrieve the uniformity of uniform convergence and its generalization to Σ-convergence and for UG-spaces we obtain a quantified version of these structures. Our theory also allows for many applications, in particular we are able to characterize the complete subspaces of these function spaces and we succeed in producing an appropriate Ascoli theorem.     

Halpern- and Mann-Type Algorithms for Fixed Points and Inclusion Problems on Hadamard Manifolds

In this article, we consider an inclusion problem which is defined by means of a sum of a single-valued vector field and a set-valued vector field defined on a Hadamard manifold. We propose Halpern-type and Mann-type algorithms for finding a common point of the set of fixed points of a nonexpansive mapping and the set of solutions of the inclusion problem defined on a Hadamard manifold. Some particular cases of our problem and algorithm are also discussed. We study the convergence of the proposed algorithm to a common point of the set of fixed points of a nonexpansive mapping and the set of solutions of the inclusion problem defined on a Hadamard manifold. As applications of our results and algorithms, we derive the solution methods and their convergence results for the optimization problems, variational inequality problems and equilibrium problems in the setting of Hadamard manifolds.     

Strong metric subregularity of mappings in variational analysis and optimization

Although the property of strong metric subregularity of set-valued mappings has been present in the literature under various names and with various (equivalent) definitions for more than two decades, it has attracted much less attention than its older “siblings”, the metric regularity and the strong (metric) regularity. The purpose of this paper is to show that the strong metric subregularity shares the main features of these two most popular regularity properties and is not less instrumental in applications. We show that the strong metric subregularity of a mapping F acting between metric spaces is stable under perturbations of the form $f+F$, where f is a function with a small calmness constant. This result is parallel to the Lyusternik–Graves theorem for metric regularity and to the Robinson theorem for strong regularity, where the perturbations are represented by a function f with a small Lipschitz constant. Then we study perturbation stability of the same kind for mappings acting between Banach spaces, where f is not necessarily differentiable but admits a set-valued derivative-like approximation. Strong metric q-subregularity is also considered, where q is a positive real constant appearing as exponent in the definition. Rockafellar's criterion for strong metric subregularity involving injectivity of the graphical derivative is extended to mappings acting in infinite-dimensional spaces. A sufficient condition for strong metric subregularity is established in terms of surjectivity of the Fréchet coderivative, and it is shown by a counterexample that surjectivity of the limiting coderivative is not a sufficient condition for this property, in general. Then various versions of Newton's method for solving generalized equations are considered including inexact and semismooth methods, for which superlinear convergence is shown under strong metric subregularity. As applications to optimization, a characterization of the strong metric subregularity of the KKT mapping is obtained, as well as a radius theorem for the optimality mapping of a nonlinear programming problem. Finally, an error estimate is derived for a discrete approximation in optimal control under strong metric subregularity of the mapping involved in the Pontryagin principle.     

<Emphasis Type="Italic">ε</Emphasis>-subgradient algorithms for locally lipschitz functions on Riemannian manifolds

This paper presents a descent direction method for finding extrema of locally Lipschitz functions defined on Riemannian manifolds. To this end we define a set-valued mapping \(x\rightarrow \partial _{\varepsilon } f(x)\) named ε-subdifferential which is an approximation for the Clarke subdifferential and which generalizes the Goldstein- ε-subdifferential to the Riemannian setting. Using this notion we construct a steepest descent method where the descent directions are computed by a computable approximation of the ε-subdifferential. We establish the global convergence of our algorithm to a stationary point. Numerical experiments illustrate our results.     

Existence and iterative algorithm of solutions for a system of generalized set-valued strongly nonlinear mixed variational-like inequalities problems in Banach spaces

In this paper, we introduce and study a new system of generalized set-valued strongly nonlinear mixed variational-like inequalities problems and its related auxiliary problems in reflexive Banach spaces. The auxiliary principle technique is applied to study the existence and iterative algorithm of solutions for the system of generalized set-valued strongly nonlinear mixed variational-like inequalities problems. Firstly, we prove the existence and uniqueness of solutions of the auxiliary problems for the system of generalized set-valued strongly nonlinear mixed variational-like inequalities problems. Secondly, an iterative algorithm for solving the system of generalized set-valued strongly nonlinear mixed variational-like inequalities problems is constructed by using this existence and uniqueness result. Finally, we show the existence of solutions of the system of generalized set-valued strongly nonlinear mixed variational-like inequalities problems and discuss the convergence analysis of this algorithm. These results improve, unify and generalize many corresponding known results given in literatures.     

Iterative approximation of a solution of multi-valued variational-like inclusion in Banach spaces: A P-η-proximal-point mapping approach

In this paper, we introduce a class of P-η-accretive mappings, an extension of η-m-accretive mappings [C.E. Chidume, K.R. Kazmi, H. Zegeye, Iterative approximation of a solution of a general variational-like inclusion in Banach spaces, Int. J. Math. Math. Sci. 22 (2004) 1159-1168] and P-accretive mappings [Y.-P. Fang, N.-J. Huang, H-accretive operators and resolvent operator technique for solving variational inclusions in Banach spaces, Appl. Math. Lett. 17 (2004) 647-653], in real Banach spaces. We prove some properties of P-η-accretive mappings and give the notion of proximal-point mapping, termed as P-η-proximal-point mapping, associated with P-η-accretive mapping. Further, using P-η-proximal-point mapping technique, we prove the existence of solution and discuss the convergence analysis of iterative algorithm, for multi-valued variational-like inclusions in real Banach space. The theorems presented in this paper extend and improve many known results in the literature.