Outer approximation methods for solving variational inequalities in Hilbert space

In this paper, we study variational inequalities in a real Hilbert space, which are governed by a strongly monotone and Lipschitz continuous operator F over a closed and convex set C. We assume that the set C can be outerly approximated by the fixed point sets of a sequence of certain quasi-nonexpansive operators called cutters. We propose an iterative method, the main idea of which is to project at each step onto a particular half-space constructed using the input data. Our approach is based on a method presented by Fukushima in 1986, which has recently been extended by several authors. In the present paper, we establish strong convergence in Hilbert space. We emphasize that to the best of our knowledge, Fukushima’s method has so far been considered only in the Euclidean setting with different conditions on F. We provide several examples for the case where C is the common fixed point set of a finite number of cutters with numerical illustrations of our theoretical results.     

Finite-dimensional approximations for the equation of nonlinear filtering derived in mild form

The Zakai equation for the unnormalized conditional density is derived as a mild stochastic bilinear differential equation on a suitableL ² space. It is assumed that the Markov semigroup corresponding to the state process isC ₀ on such space. This allows the establishment of the existence and uniqueness of the solution by means of general theorems on stochastic differential equations in Hilbert space. Moreover, an easy treatment of convergence conditions can be given for a general class of finite-dimensional approximations, including Galerkin schemes. This is done by using a general continuity result for the solution of a mild stochastic bilinear differential equation on a Hilbert space with respect to the semigroup, the forcing operator, and the initial state, within a suitable topology.     

Error estimate for a symmetric scheme of the projection-difference method for an abstract hyperbolic-parabolic system of the type of systems of thermoelasticity equations

We study the convergence of the projection-difference method for a system of abstract differential equations in Hilbert spaces generalizing a number of linear systems of coupled thermoelasticity equations. We consider a computational scheme that combines a scheme of the Galerkin method with respect to space and a symmetric three-layer difference scheme with weights in time. We impose no special conditions on the projection subspaces of the Galerkin method. For the error, we obtain an energy estimate of second order in time.     

A generalized contraction proximal point algorithm with two monotone operators

Abstract

In this work, we introduce a generalized contraction proximal point algorithm and use it to approximate common zeros of maximal monotone operators A and B in a real Hilbert space setting. The algorithm is a two step procedure that alternates the resolvents of these operators and uses general assumptions on the parameters involved. For particular cases, these relaxed parameters improve the convergence rate of the algorithm. A strong convergence result associated with the algorithm is proved under mild conditions on the parameters. Our main result improves and extends several results in the literature.     

Hybrid Methods for Solving Simultaneously an Equilibrium Problem and Countably Many Fixed Point Problems in a Hilbert Space

This paper presents a framework of iterative methods for finding a common solution to an equilibrium problem and a countable number of fixed point problems defined in a Hilbert space. A general strong convergence theorem is established under mild conditions. Two hybrid methods are derived from the proposed framework in coupling the fixed point iterations with the iterations of the proximal point method or the extragradient method, which are well-known methods for solving equilibrium problems. The strategy is to obtain the strong convergence from the weak convergence of the iterates without additional assumptions on the problem data. To achieve this aim, the solution set of the problem is outer approximated by a sequence of polyhedral subsets.     

Aveiro method in reproducing kernel Hilbert spaces under complete dictionary

Aveiro method is a sparse representation method in reproducing kernel Hilbert spaces, which gives orthogonal projections in linear combinations of reproducing kernels over uniqueness sets. It, however, suffers from determination of uniqueness sets in the underlying reproducing kernel Hilbert space. In fact, in general spaces, uniqueness sets are not easy to be identified, let alone the convergence speed aspect with Aveiro method. To avoid those difficulties, we propose an new Aveiro method based on a dictionary and the matching pursuit idea. What we do, in fact, are more: The new Aveiro method will be in relation to the recently proposed, the so‐called pre‐orthogonal greedy algorithm involving completion of a given dictionary. The new method is called Aveiro method under complete dictionary. The complete dictionary consists of all directional derivatives of the underlying reproducing kernels. We show that, under the boundary vanishing condition bring available for the classical Hardy and Paley‐Wiener spaces, the complete dictionary enables an efficient expansion of any given element in the Hilbert space. The proposed method reveals new and advanced aspects in both the Aveiro method and the greedy algorithm.     

Iterative algorithms with the regularization for the constrained convex minimization problem and maximal monotone operators

In this paper, we prove a strong convergence theorem for finding a common element of the solution set of a constrained convex minimization problem and the set of solutions of a finite family of variational inclusion problems in Hilbert space. A strong convergence theorem for finding a common element of the solution set of a constrained convex minimization problem and the solution sets of a finite family of zero points of the maximal monotone operator problem in Hilbert space is also obtained. Using our main result, we have some additional results for various types of non-linear problems in Hilbert space.     

Global attracting sets and stability of neutral stochastic functional differential equations driven by Rosenblatt process

We are concerned with a class of neutral stochastic partial differential equations driven by Rosenblatt process in a Hilbert space. By combining some stochastic analysis techniques, tools from semigroup theory, and stochastic integral inequalities, we identify the global attracting sets of this kind of equations. Especially, some sufficient conditions ensuring the exponent p-stability of mild solutions to the stochastic systems under investigation are obtained. Last, an example is given to illustrate the theory in the work.     

Extragradient method for solving quasivariational inequalities

We study methods for solving a class of the quasivariational inequalities in Hilbert space when the changeable set is described by translation of a fixed, closed and convex set. We consider one variant of the gradient-type projection method and an extragradient method. The possibilities of the choice of parameters of the gradient projection method in this case are wider than in the general case of a changeable set. The extragradient method on each iteration makes one trial step along the gradient, and the value of the gradient at the obtained point is used at the first point as the iteration direction. In the paper, we establish sufficient conditions for the convergence of the proposed methods and derive a new estimate of the rates of the convergence. The main result of this paper is contained in the convergence analysis of the extragradient method.     

A splitting algorithm for a class of bilevel equilibrium problems involving nonexpansive mappings

We propose splitting, parallel algorithms for solving strongly equilibrium problems over the intersection of a finite number of closed convex sets given as the fixed-point sets of nonexpansive mappings in real Hilbert spaces. The algorithm is a combination between the gradient method and the Mann-Krasnosel’skii iterative scheme, where the projection can be computed onto each set separately rather than onto their intersection. Strong convergence is proved. Some special cases involving bilevel equilibrium problems with inverse strongly monotone variational inequality, monotone equilibrium constraints and maximal monotone inclusions are discussed. An illustrative example involving a system of integral equations is presented.     

A Strongly Convergent Combined Relaxation Method in Hilbert Spaces

We consider a combined relaxation method for variational inequalities in a Hilbert space setting. Methods of this class are known to solve finite-dimensional variational inequalities under mild monotonicity type assumptions, whereas in Hilbert space strong monotonicity is the standard assumption for strong convergence. Here, we relax this condition and show strong convergence of such a method, when strong monotonicity holds only on a subspace of finite co-dimension. Thus, the method applies to semi-coercive unilateral boundary value problems in mathematical physics.     

Sourcewise representability conditions and power estimates of convergence rate in Tikhonov’s scheme for solving ill-posed extremal problems

We study the rate of the convergence of approximations generated by the Tikhonov scheme for solving ill-posed optimization problems with smooth functionals given in a general form in a Hilbert space. We establish sourcewise representability conditions which are necessary and sufficient for the convergence of approximations at a power rate. Sufficient conditions are related to the estimate of the discrepancy with respect to the objective functional, while the necessary ones are formulated for the estimate with respect to the argument. We specify certain cases when sufficient and necessary conditions coincide in essence.     

Convergence of Directional Methods under mild differentiability and applications

A semilocal convergence analysis for Directional Methods under mild differentiability conditions is provided in this study. Using our idea of recurrent functions, we provide sufficient convergence conditions as well as the corresponding errors bounds. The results are extended to hold in a Hilbert space setting and a favorable comparison is provided with earlier works [6], [7], [8], [9], [10], [11] and [20]. Numerical examples are also provided in this study.     

A relaxed self-adaptive CQ algorithm for the multiple-sets split feasibility problem

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.     

关于投影算子随机乘积序列的收敛性

投影算子随机乘积的收敛性奠定广泛应用数学方法的理论基础,本文将在某些宽松条件下研究并肯定回答有关投影算子随机乘积序列收敛性的两个问题．     

Gradient learning in a classification setting by gradient descent

Learning gradients is one approach for variable selection and feature covariation estimation when dealing with large data of many variables or coordinates. In a classification setting involving a convex loss function, a possible algorithm for gradient learning is implemented by solving convex quadratic programming optimization problems induced by regularization schemes in reproducing kernel Hilbert spaces. The complexity for such an algorithm might be very high when the number of variables or samples is huge. We introduce a gradient descent algorithm for gradient learning in classification. The implementation of this algorithm is simple and its convergence is elegantly studied. Explicit learning rates are presented in terms of the regularization parameter and the step size. Deep analysis for approximation by reproducing kernel Hilbert spaces under some mild conditions on the probability measure for sampling allows us to deal with a general class of convex loss functions.     

On the Lorentz Cone Complementarity Problems in Infinite-Dimensional Real Hilbert Space

In this article, we consider the Lorentz cone complementarity problems in infinite-dimensional real Hilbert space. We establish several results that are standard and important when dealing with complementarity problems. These include proving the same growth of the Fishcher–Burmeister merit function and the natural residual merit function, investigating property of bounded level sets under mild conditions via different merit functions, and providing global error bounds through the proposed merit functions. Such results are helpful for further designing solution methods for the Lorentz cone complementarity problems in Hilbert space.     

A new self-adaptive CQ algorithm with an application to the LASSO problem

In this paper, we introduce a new self-adaptive CQ algorithm for solving split feasibility problems in real Hilbert spaces. The algorithm is designed, such that the stepsizes are directly computed at each iteration. We also consider the corresponding relaxed CQ algorithm for the proposed method. Under certain mild conditions, we establish weak convergence of the proposed algorithm as well as strong convergence of its hybrid-type variant. Finally, numerical examples illustrating the efficiency of our algorithm in solving the LASSO problem are presented.     

The method of alternating projections and the method of subspace corrections in Hilbert space

A new identity is given in this paper for estimating the norm of the product of nonexpansive operators in Hilbert space. This identity can be applied for the design and analysis of the method of alternating projections and the method of subspace corrections. The method of alternating projections is an iterative algorithm for determining the best approximation to any given point in a Hilbert space from the intersection of a finite number of subspaces by alternatively computing the best approximations from the individual subspaces which make up the intersection. The method of subspace corrections is an iterative algorithm for finding the solution of a linear equation in a Hilbert space by approximately solving equations restricted on a number of closed subspaces which make up the entire space. The new identity given in the paper provides a sharpest possible estimate for the rate of convergence of these algorithms. It is also proved in the paper that the method of alternating projections is essentially equivalent to the method of subspace corrections. Some simple examples of multigrid and domain decomposition methods are given to illustrate the application of the new identity.

     

Fischer decomposition of the space of entire functions for the convolution operator

It is known that any function in a Hilbert Bargmann–Fock space can be represented as the sum of a solution of a given homogeneous differential equation with constant coefficients and a function being a multiple of the characteristic function of this equation with conjugate coefficients. In the paper, a decomposition of the space of entire functions of one complex variable with the topology of uniform convergence on compact sets for the convolution operator is presented. As a corollary, a solution of the de la Vallée Poussin interpolation problem for the convolution operator with interpolation points at the zeros of the characteristic function with conjugate coefficient is obtained.