Real Hilbert Space Research Articles

Lipschitz continuity of the metric projection operator and convergence of gradient methods

Various support conditions for a closed subset of a real Hilbert space $\mathcal H$ at a boundary point of this set are considered. These conditions ensure certain local Lipschitz continuity of the metric projection operator as a function of a point. The local Lipschitz continuity of the metric projection as a function of the set in the Hausdorff metric is also proved. This Lipschitz property is used to verify the linear convergence of some gradient methods (the gradient projection method and the conditional gradient method) without assuming that the function must be strongly convex (or even convex) and for not necessarily convex sets. The function is assumed to be differentiable with Lipschitz continuous gradient. Bibliography: 29 titles.

Strong convergence for split variational inclusion problems under hybrid algorithms with applications

In this article, we present two inertial modifications of regularized algorithms for the split variational inclusion problem (SVIP, for short) in real Hilbert spaces (RHSs). When the circumstances are right, strong convergence theorems are demonstrated. The major findings are used to solve the split minimization, split common fixed point problem (SCFPP), split minimization problem (SMP), and split feasibility problems (SFP) in applications. The proposed algorithms are contrasted with a number of other existing algorithms in the literature in order to test their numerical performances. Finally, the computer tests demonstrate that the suggested algorithms outperform alternative strategies in terms of speed and efficiency.

Forward-backward splitting algorithm with self-adaptive method for finite family of split minimization and fixed point problems in Hilbert spaces

In this paper, we introduce an inertial forward-backward splitting method together with a Halpern iterative algorithm for approximating a common solution of a finite family of split minimization problem involving two proper, lower semicontinuous and convex functions and fixed point problem of a nonexpansive mapping in real Hilbert spaces. Under suitable conditions, we proved that the sequence generated by our algorithm converges strongly to a solution of the aforementioned problems. The stepsizes studied in this paper are designed in such a way that they do not require the Lipschitz continuity condition on the gradient and prior knowledge of operator norm. Finally, we illustrate a numerical experiment to show the performance of the proposed method. The result discussed in this paper extends and complements many related results in literature.

A PROJECTION SUBGRADIENT ALGORITHM FOR FINDING A COMMON SOLUTION OF EQUILIBRIUM AND FIXED POINT PROBLEMS

In this paper, we design a new projection subgradient algorithm for finding a common solution of equilibrium and fixed point problems in a real Hilbert space. The proposed algorithm is a combination of the projection method, subgradient method and Man iterative technique. The convergent theorem is established under mild conditions.

CONTINUOUS REGULARIZATION METHOD FOR A COMMON MINIMUM POINT OF A FINITE SYSTEM OF CONVEX FUNCTIONALS

The concept of the ill-posed problem was introduced by Hadamard, a French mathematician in 1932 when he studied the effect of the boundary value problem on differential equations. Due to the unstability of the ill-posed problems, the numerical computation is difficult to do. Therefore, one of the main study directions for ill-posed problems is construct stable methods to solve this problems such that when the error of the input data is smaller, the approximate solution is closer to the correct solution of the original problem. Although there are some known important results obtained in studying the regularization method for solving the ill-posed problems, the improvement of the methods to increase their effectiveness always attracts the attention of many researchers. In this paper, we present a regularization method for a common minimum point of a finite system of Gateau differentiable weakly lower semi-continuous and properly convex functionals on real Hilbert spaces. And then, an application our theoretical results to convex feasibility problems and common fixed points of nonexpansive mappings.

A modified Tseng's extragradient method for solving variational inequality problems

In this paper, we introduce a modified Tseng's extragradient method with a new step-length rule to solve pseudo-monotone variational inequalities in real Hilbert spaces. Under suitable conditions, the sequence generated by this algorithm strongly converges to the common elements of the solution set of pseudo-monotone variational inequality problems and the fixed point set of k-demicontractive mappings. Finally, we give some numerical experiments to illustrate the effectiveness of the proposed algorithm. The main results of this paper generalize and improve some known results in the literature.

Two regularized inertial Tseng methods for solving inclusion problems with applications to convex bilevel programming

Combining the inertial technique, regularization, and the Tseng method for approximating a zero of the sum of monotone operators in a real Hilbert space, we propose two regularized inertial Tseng methods for approximating a zero of the sum of two maximal monotone operators in the case where one of them is monotone and Lipschitz continuous. We establish strong convergence of the sequences generated by our algorithms and derive several interesting consequences of our results. We then apply our results to solving convex bilevel programming problems and illustrate the cost efficiency of our methods by numerically comparing them with other methods in the literature.

An Inertial Iterative Algorithm for Approximating Common Solutions to Split Equalities of Some Nonlinear Optimization Problems

AbstractIn this paper, we introduce a new inertial Tseng’s extragradient method with self-adaptive step sizes for approximating a common solution of split equalities of equilibrium problem (EP), non-Lipschitz pseudomonotone variational inequality problem (VIP) and fixed point problem (FPP) of nonexpansive semigroups in real Hilbert spaces. We prove that the sequence generated by our proposed method converges strongly to a common solution of the EP, pseudomonotone VIP and FPP of nonexpansive semigroups without any linesearch procedure nor the sequential weak continuity condition often assumed by authors when solving non-Lipschitz VIPs. Finally, we provide some numerical experiments for the proposed method in comparison with related methods in the literature. Our result improves, extends and generalizes several of the existing results in this direction.

Fast Optimistic Gradient Descent Ascent (OGDA) Method in Continuous and Discrete Time

AbstractIn the framework of real Hilbert spaces, we study continuous in time dynamics as well as numerical algorithms for the problem of approaching the set of zeros of a single-valued monotone and continuous operator V. The starting point of our investigations is a second-order dynamical system that combines a vanishing damping term with the time derivative of V along the trajectory, which can be seen as an analogous of the Hessian-driven damping in case the operator is originating from a potential. Our method exhibits fast convergence rates of order $$o \left( \frac{1}{t\beta (t)} \right) $$ o 1 t β ( t ) for $$\Vert V(z(t))\Vert $$ ‖ V ( z ( t ) ) ‖ , where $$z(\cdot )$$ z ( · ) denotes the generated trajectory and $$\beta (\cdot )$$ β ( · ) is a positive nondecreasing function satisfying a growth condition, and also for the restricted gap function, which is a measure of optimality for variational inequalities. We also prove the weak convergence of the trajectory to a zero of V. Temporal discretizations of the dynamical system generate implicit and explicit numerical algorithms, which can be both seen as accelerated versions of the Optimistic Gradient Descent Ascent (OGDA) method for monotone operators, for which we prove that the generated sequence of iterates $$(z_k)_{k \ge 0}$$ ( z k ) k ≥ 0 shares the asymptotic features of the continuous dynamics. In particular we show for the implicit numerical algorithm convergence rates of order $$o \left( \frac{1}{k\beta _k} \right) $$ o 1 k β k for $$\Vert V(z^k)\Vert $$ ‖ V ( z k ) ‖ and the restricted gap function, where $$(\beta _k)_{k \ge 0}$$ ( β k ) k ≥ 0 is a positive nondecreasing sequence satisfying a growth condition. For the explicit numerical algorithm, we show by additionally assuming that the operator V is Lipschitz continuous convergence rates of order $$o \left( \frac{1}{k} \right) $$ o 1 k for $$\Vert V(z^k)\Vert $$ ‖ V ( z k ) ‖ and the restricted gap function. All convergence rate statements are last iterate convergence results; in addition to these, we prove for both algorithms the convergence of the iterates to a zero of V. To our knowledge, our study exhibits the best-known convergence rate results for monotone equations. Numerical experiments indicate the overwhelming superiority of our explicit numerical algorithm over other methods designed to solve monotone equations governed by monotone and Lipschitz continuous operators.

An Algorithm That Adjusts the Stepsize to Be Self-Adaptive with an Inertial Term Aimed for Solving Split Variational Inclusion and Common Fixed Point Problems

In this research paper, we present a new inertial method with a self-adaptive technique for solving the split variational inclusion and fixed point problems in real Hilbert spaces. The algorithm is designed to choose the optimal choice of the inertial term at every iteration, and the stepsize is defined self-adaptively without a prior estimate of the Lipschitz constant. A convergence theorem is demonstrated to be strong even under lenient conditions and to showcase the suggested method’s efficiency and precision. Some numerical tests are given. Moreover, the significance of the proposed method is demonstrated through its application to an image reconstruction issue.

On Bilevel Monotone Inclusion and Variational Inequality Problems

In this article, the problem of solving a strongly monotone variational inequality problem over the solution set of a monotone inclusion problem in the setting of real Hilbert spaces is considered. To solve this problem, two methods, which are improvements and modifications of the Tseng splitting method, and projection and contraction methods, are presented. These methods are equipped with inertial terms to improve their speed of convergence. The strong convergence results of the suggested methods are proved under some standard assumptions on the control parameters. Also, strong convergence results are achieved without prior knowledge of the operator norm. Finally, the main results of this research are applied to solve bilevel variational inequality problems, convex minimization problems, and image recovery problems. Some numerical experiments to show the efficiency of our methods are conducted.

A Regularized Tseng Method for Solving Various Variational Inclusion Problems and Its Application to a Statistical Learning Model

We study three classes of variational inclusion problems in the framework of a real Hilbert space and propose a simple modification of Tseng’s forward-backward-forward splitting method for solving such problems. Our algorithm is obtained via a certain regularization procedure and uses self-adaptive step sizes. We show that the approximating sequences generated by our algorithm converge strongly to a solution of the problems under suitable assumptions on the regularization parameters. Furthermore, we apply our results to an elastic net penalty problem in statistical learning theory and to split feasibility problems. Moreover, we illustrate the usefulness and effectiveness of our algorithm by using numerical examples in comparison with some existing relevant algorithms that can be found in the literature.

Two-step inertial forward–reflected–anchored–backward splitting algorithm for solving monotone inclusion problems

The main purpose of this paper is to propose and study a two-step inertial anchored version of the forward–reflected–backward splitting algorithm of Malitsky and Tam in a real Hilbert space. Our proposed algorithm converges strongly to a zero of the sum of a set-valued maximal monotone operator and a single-valued monotone Lipschitz continuous operator. It involves only one forward evaluation of the single-valued operator and one backward evaluation of the set-valued operator at each iteration; a feature that is absent in many other available strongly convergent splitting methods in the literature. Finally, we perform numerical experiments involving image restoration problem and compare our algorithm with known related strongly convergent splitting algorithms in the literature.

Characterizations of additive local Jordan *-derivations by action at idempotents

Let H be a real or complex Hilbert space and B ( H ) the algebra of all bounded linear operators on H. Recall that a map δ : B ( H ) → B ( H ) is called an inner Jordan ∗ -derivation if there exists some T ∈ B ( H ) such that δ ( A ) = AT − T A ∗ for all A ∈ B ( H ) . In this paper, it is proved that inner Jordan ∗ -derivations are the only additive maps δ of B ( H ) with the property that δ ( P ) = δ ( P ) P ∗ + Pδ ( P ) for all idempotent operators P ∈ B ( H ) if dim ⁡ H = ∞ , which is satisfied by additive local Jordan ∗ -derivations. For the finite dimensional case, additional conditions are required for δ to be an inner Jordan ∗ -derivation. As applications, it is shown that, for any given C , D ∈ B ( H ) , δ satisfies δ ( A ) B ∗ + Bδ ( A ) + δ ( B ) A ∗ + Aδ ( B ) = D for all A , B ∈ B ( H ) with AB + BA = C if and only if δ is an inner Jordan ∗ -derivation and D = δ ( C ) . Also, several known results are generalized.

An Inertial-Like Algorithm for Solving Common Fixed Point Problems of a Family of Continuous Pseudocontractive Mappings

In this paper, it is our purpose to introduce an inertial-like algorithm for approximating common fixed points of continuous pseudocontractive mappings in the framework of real Hilbert spaces. We prove strong convergence theorems for the sequence generated by the algorithm under certain conditions on the control sequences. We also provide numerical examples to demonstrate the efficiency of our algorithm.

An algorithm for approximating solutions of the split variational inclusion problem

We introduce and study an iterative algorithm to approximate a solution to the split variational inclusion problem in real Hilbert spaces. The strong convergence of the proposed algorithm is proved via some suitable conditions. As applications, we derive consequences for several special cases such as the split variational inequality problem, the split minimum point problem, and the split common fixed point problem. Finally, we present some numerical experiments to demonstrate the performance of the proposed algorithm.

Strong Convergence of Trajectories via Inertial Dynamics Combining Hessian-Driven Damping and Tikhonov Regularization for General Convex Minimizations

Let be a real Hilbert space, and be a convex twice differentiable function whose solution set is nonempty. We investigate the long time behavior of the trajectories of the vanishing damped dynamical system with Tikhonov regularizing term and Hessian-driven damping where are three positive constants, and the time scale parameter β is a positive nondecreasing function such that . Under some assumptions on the parameter , we will show rapid convergence of values, strong convergence toward the minimum norm element of , and rapid convergence of the gradients toward zero. Note that the Hessian-driven damping significantly reduces the oscillatory aspects, and the time scale parameter β improves the rate of convergences mentioned above. As particular cases of , we set , for , and , for and . The manuscript concludes with two numerical examples and comments on their performance.

The variable metric three-operator algorithms for solving monotone inclusions

In this paper, we propose an inexact variable metric three-operator algorithm and a viscous version of it for finding a zero of the sum of three monotone operators in a real Hilbert space. Under some appropriate conditions, we establish the weak or strong convergence of the proposed algorithms. As a special case of the proposed problem, we present a variable metric Douglas–Rachford splitting algorithm and obtain a strong convergence result. The algorithms and the results in this paper are further extensions of the algorithms and results for solving the zero inclusion problems of monotone operators. To illustrate the efficiency of the proposed algorithms, we construct the numerical experiments and give some choices for the variable metric matrix.

An algorithmic approach to solving split common fixed point problems for families of demicontractive operators in Hilbert spaces

In this article, we propose a new iterative algorithm for solving split common fixed point problems for finite families of single-valued demicontractive mappings in the setting of real Hilbert spaces. In order to solve this problem, we introduce a Halpern method with with an Armijo-line search for approximating the solution of the aforementioned problems. We establish a strong convergence result of the proposed method and display some numerical examples to illustrate the performance of our main result. Our result complements and generalizes some related results in literature.

An inertial-type algorithm for a class of bilevel variational inequalities with the split variational inequality problem constraints

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.