Iteration–discretization methods for variational inequalities over fixed point sets

The multiple-sets split feasibility problem (MSFP) is to find a point belongs to the intersection of a family of closed convex sets in one space, such that its image under a linear transformation belongs to the intersection of another family of closed convex sets in the image space. Many iterative methods can be employed to solve the MSFP. Jinling Zhao et al. proposed a modification for the CQ algorithm and a relaxation scheme for this modification to solve the MSFP. The strong convergence of these algorithms are guaranteed in finite-dimensional Hilbert spaces. Recently López et al. proposed a relaxed CQ algorithm for solving split feasibility problem, this algorithm can be implemented easily since it computes projections onto half-spaces and has no need to know a priori the norm of the bounded linear operator. However, this algorithm has only weak convergence in the setting of infinite-dimensional Hilbert spaces. In this paper, we introduce a new relaxed self-adaptive CQ algorithm for solving the MSFP where closed convex sets are level sets of some convex functions such that the strong convergence is guaranteed in the framework of infinite-dimensional Hilbert spaces. Our result extends and improves the corresponding results.

Finite dimensional approximation and Newton-based algorithm for stochastic approximation in Hilbert space

We prove the existence of solutions of a differential variational inequality involving a prox-regular set in an infinite dimensional Hilbert space via a new existence result of a non-convex state-dependent sweeping process.

Sequences of [formula omitted]-term deviations in Hilbert space

In this paper we consider an augmented Lagrangian method for the minimization of a nonlinear functional in the presence of an equality constraint whose image space is in a Hilbert space, an inequality constraint whose image space is finite dimensional, and an affine inequality constraint whose image space is in an infinite dimensional Hilbert space. We obtain local convergence of this method without imposing strict complementarity conditions when the equality, as well as the inequality constraint with finite dimensional image space are augmented. To the author's knowledge this result even generalizes the convergence results which are known when all spaces are finite dimensional.

Metric projection and compatibly parameterized families of prox-regular sets in Hilbert space

We study a class of stochastic optimization problems in which the state as well as the observation spaces are permitted to be (Hilbert spaces) of non-finite dimension. Although there have been previous attempts in the Hilbert space setting, our results, techniques, as well as applications, are totally different. We initiate the use of Gauss measure on a Hilbert space even though it is only finitely additive; and an associated theory of white noise, in contrast to the Wiener process theory, which is novel even in the finite dimensional case. We only treat time-invariant systems, but no strong ellipticity or coercivity conditions are used; we exploit the theory of semigroups of operators in contrast to the Lions-Magenes theory. A key result involves a far-reaching generalization of the Factorization theorem of Krein. We apply the results to the problem of boundary observation and control for partial differential equations. By the creation of a special state space, we can apply the theory to problems in which the state equations are finitedimensional but the noise does not have a rational spectrum. In a final section, we present a stochastic theory for inverse problems (System Identification) in the Hilbert space setting. The basic theoretical problem is the calculation of R-N derivatives for finitely additive measures. A fundamental result concerns Identifiability; in particular the identifiability of diffusion coefficients from boundary data is treated here for the first time.

We present a parallel iterative method to find the shortest distance projection of a given point onto the intersection of a finite number of closed convex sets in a real Hilbert space. In the method use is made of weights and a relaxation coefficient which may vary at each iteration step, and which are determined at each step by geometrical conditions.

We study classes of mappings between finite and infinite dimensional Banach spaces that are monotone and mappings which are differences of monotone mappings (DM). We prove a Radó–Reichelderfer estimate for monotone mappings in finite dimensional spaces that remains valid for DM mappings. This provides an alternative proof of the Fréchet differentiability a.e. of DM mappings. We establish a Morrey-type estimate for the distributional derivative of monotone mappings. We prove that a locally DM mapping between finite dimensional spaces is also globally DM. We introduce and study a new class of the so-called UDM mappings between Banach spaces, which generalizes the concept of curves of finite variation.

We are concerned with a variational inclusion problem with two monotone mappings, and fixed-point problems of asymptotically nonexpansive mappings and pseudocontractive mappings. A viscosity method is introduced and studied for the two problems. A convergence theorem in norm of common solutions is established in the setting of infinite dimensional real Hilbert spaces.

In this paper we give the solvability class of generalized strongly nonlinear variational inequalities modified by the use of the new concept of admissible perturbation operator on nonempty closed convex sets in Hilbert spaces.

Projections on Convex Sets in Hilbert Space and Spectral Theory: Part I. Projections on Convex Sets: Part II. Spectral Theory

In this paper, we shall generalize the unicellularity of operators on finite-dimensional spaces to that of the contraction of class Co on Hilbert spaces. We prove: (1) Each nilpotent operator on Hilbert space is Banach reducible (Theorem 3). (2) A contraction T of class CO on Hilbert space is unicellular if and only if T has one-point spectrum and every invariant subspace for T is cyclic (Theorem 6). (3) A contraction T of class Co on Hilbert space is unicellular if and only if T has one-point spectrum and all invariant subspaces of T are hyperinvariant subspaces of T (Theorem 8). The aim of this paper is to discuss the unicellularity of contractions of class Co. We shall generalize the unicellularity of operators acting on finitedimensional spaces to that on infinite-dimensional spaces. Throughout this paper, we denote by H a complex, separable, and infinitedimensional Hilbert space. The related concepts and symbols can be found in [4]. Definition 1. Let H1 be a nontrivial invariant subspace of T. H1 is called a Banach reducible subspace of T if there exists another nontrivial invariant subspace H2 of T such that

In this paper, we investigate the problem of solving strongly monotone variational inequalities over the solution set of the split variational inequality problem with multiple output sets in real Hilbert spaces and establish a new iterative algorithm for it. Our algorithms are accelerated by the inertial technique and eliminate the dependence on the norm of the transformation operators and the strongly monotone and Lipschitz continuous constants of the involved operator by employing a self-adaptive step size criterion. The strong convergence result is given under some mild conditions widely used in the convergence analysis. Two corollaries for the solutions of the split variational inequality problem and the split feasibility problem with multiple output sets are also obtained using our main result. Finally, some numerical experiments have been conducted to illustrate the effectiveness of the proposed algorithms and compare them with the related ones.

The split common fixed problem (SCFPP) has been intensively studied by numerous author due to its various applications in many physical problem. However, to employ the algorithm for solving such a problem, one needs to know the prior information on the normed of bounded linear operator. Recently, Cui and Wang introduced the new algorithm for solving such a problem which does not needs any prior information on the normed on bounded linear operator and they established the weak convergence results under some mild conditions. It is well-known that in setting of infinite dimensional Hilbert space, the weak convergence does not implies strong convergence. It is the aims of this article to continue studying this problem (SCFPP) and establish the strong convergence result based on the result of Cui and Wang, this will be done for the class of demicontractive mappings. The results presented in this paper, not only extend and improve the result of Cui and Wang, but also extend, improve and generalize several well-known results announced.