In this paper, we consider the nonsmooth convex optimization problems over the fixed point constraint sets of firmly nonexpansive operators. To find an optimal solution of the problem, we present an iterative method based on the hybrid steepest descent method and the idea of a delayed subgradient scheme in which allows the use of staled subgradients from the earlier iteration when updating the next iteration. We start the convergence part by deriving an upper bound for the difference of the best-achieved function values and the optimal value. After that, to ensure the convergence in iterations, we prove that there exists a subsequence of the generated sequence by the proposed method which converges to an optimal solution. Moreover, we subsequently show that the whole generated sequence converges to an optimal solution when the strict convexity of the objective function is imposed. We further extend the presented results to the centralized network system consisting of a finite number of workers and a central server. Finally, we apply the proposed method to image inpainting problems. The numerical results describe the effect of delay in many cases of objective functions.

We consider a class of nonsmooth fractional programming problems with fixed-point constraints, where the numerator is convex and the denominator is concave. To solve this problem, we propose splitting algorithms that compute subgradient steps separately for the convex numerator and the concave denominator. These methods offer a straightforward approach by eliminating the need to solve subproblems at each iteration. By leveraging fixed-point constraints, the proposed algorithms are particularly well-suited for problems with complex constraint structures. Under certain assumptions, we establish the convergence of the proposed methods. Furthermore, to address large-scale optimization, we propose an incremental subgradient algorithm for a class of nonsmooth sum-of-ratios fractional programming problems and analyze its convergence. Finally, we present numerical experiments, including comparative analyses of our algorithms with existing methods, to demonstrate the effectiveness and performance of the proposed approach.

In this paper, we propose an inertial splitting proximal algorithm to solve equilibrium problems in a real Hilbert space, where the governing bifunction is the sum of two other bifunctions. We establish weak and strong convergence results for the proposed algorithm under suitable assumptions. Numerical experiments are presented to illustrate the algorithm’s performance, demonstrating that the splitting approach is particularly effective for strongly coupled problems and image restoration tasks, where it exhibits a significant computational time advantage over non-splitting methods.

In this work, we consider the problem of minimizing a quasi-convex function over a nonempty closed convex constrained set. In order to approximate a solution of the considered problem, we propose delayed star subgradient methods. The main feature of the proposed methods is that it allows us to use the stale star subgradients when updating the next iteration rather than computing the new star subgradient in every iteration. We subsequently investigate the convergence results of sequences generated by the proposed methods. Finally, we present some numerical experiments on the Cobb–Douglas production efficiency problem to illustrate the effectiveness of the proposed method.

In this letter, we consider a bilevel optimization problem in which the outer-level objective function is strongly convex, whereas the inner-level problem consists of a finite sum of convex functions. Bilevel optimization problems arise in situations where the inner-level problem does not have a unique solution. This has led to the idea of introducing an outer-level objective function to select a solution with the specific desired properties. We propose an iterative method that combines an incremental algorithm with a broadcast algorithm, both based on the principles of federated learning. Under appropriate assumptions, we establish the convergence results of the proposed algorithm. To demonstrate its performance, we present two numerical examples related to binary classification and a location problem.

In this paper, we propose an incremental-type subgradient scheme for solving a nonsmooth convex–concave minimax optimization problem in the setting of Euclidean spaces. We investigate convergence results by deriving an upper bound for the absolute value of the difference between the function value of the averaged iterates and the saddle value, provided that the step size is a constant. By assuming that the step-size sequence is diminishing, we prove the convergences of both the averaged sequence of function values and the sequence of function values of averaged iterates to the saddle value. Finally, we also show some numerical examples for illustrating the obtained theoretical result.

In this work, we consider a convex minimization problem over the intersection of a compact convex simple set and a finite intersection of sublevel sets of quasi-convex functions. We propose the quasi-subgradient type method which separately deals with the objective function and a simple constrained set through a subgradient projection scheme and then performs parallel feasibility updates of constrained functions via a quasi-subgradient scheme with the appropriate weight function. This strategy provides a straightforward computation since we need not solve a subproblem to determine the metric projection onto the whole constrained set. Focusing on the convergence results, we prove subsequence convergence to the optimal solution of the considered problem and also establish the convergence rate for functional value to the optimal value. Additionally, by imposing the Hölder error bound property, we prove the convergence of the whole sequences to the optimal solution. We finally perform a numerical example to demonstrate the convergence behaviors of the proposed method for various choices of relating parameters and weight functions.

In this paper, we propose the greedy fixed-point method for the variational inequality problem over the intersection of the fixed-point sets of strongly quasi-nonexpansive operators. At each iteration, the proposed method updates a new operator in which the distance between the current iterate and its image under the operator is the farthest. Under some certain conditions, we obtain the strong convergence of the generated sequence of iterates to the unique solution of the considered problem. In order to demonstrate the effectiveness and performance of our proposed algorithm, we present an application addressing binary classification via support vector machines.

In this paper, we investigate the distributed approximate subgradient-type method for minimizing a sum of differentiable and non-differentiable convex functions subject to nondifferentiable convex functional constraints in a Euclidean space. We establish the convergence of the sequence generated by our method to an optimal solution of the problem under consideration. Moreover, we derive a convergence rate of order $ \mathcal{O}(N^{1-a}) $ for the objective function values, where $ a\in (0.5, 1) $. Finally, we provide a numerical example illustrating the effectiveness of the proposed method.

In this work, we focused on minimizing a strongly convex smooth function over the common fixed-point constraints. We proposed an extrapolated fixed-point optimization method, which is a modified version of the extrapolated sequential constraint method with conjugate gradient direction. We proved the convergence of the generated sequence to the unique solution to the considered problem without boundedness assumption. We also investigated some numerical experiments to underline the effectiveness and performance of the proposed method.

In this paper, we propose a strongly convergent simultaneous cutter method for finding the min-imal norm solution over the intersection of fixed point sets of cutters. The proposed method is the combination of the simultaneous cutter method and the strongly variance of Krasnosel'skii-Mann method. We show a strong convergence result of the sequence generated by the proposed method to the unique minimal norm solution. We finally present the numerical experiments on the minimal norm solution over a finite number of linear feasibility problem.

We consider the problem of minimizing a finite sum of differentiable and nondifferentiable convex functions in the setting of finite-dimensional Euclidean space. We propose and analyze a distributed proximal gradient method with computational delays. The occurrence of local delays when computing local gradient of each differentiable cost function allows the use of out-of-date iterates when generating the next estimates, which benefits a situation where the cost of gradient computation is expensive so that it cannot be done within a limited time constraints. We provide a condition on control parameter to guarantee that the sequences generated by the proposed method converge to the unique solution. We finally illustrate the presented theoretical results by performing some numerical experiments on binary image classification.

In this paper, we consider additive convex optimization problems in which the objective function is the sum of a large number of convex nondifferentiable cost functions. We assume that each cost function is specifically written as the sum of two convex nondifferentiable functions in which one function is appropriate for the subgradient method, and another one is not. To this end, we propose a distributed optimization algorithm based on the subgradient and proximal methods. The proposed method is also governed by an asynchronous feature that allows time-varying delays when computing the subgradients. We prove the convergences of function values of iterates to the optimal value. To demonstrate the efficiency of the presented theoretical result, we investigate the binary classification problem via support vector machine learning.

In this paper, we aim to solve a convex-concave min–max optimization problem, where the convex-concave coupling function is nonsmooth in both variables. We propose a simple subgradient method which is simultaneously updating the iterates through their delayed subgradients at stale iterates of their own variable together with the current iterates of the other variable. We investigate the convergence behavior of the proposed method by providing an upper bound of the function value at the coupling averaged iterates to the saddle value. Specifically, the obtained result refers to the function value at the coupling averaged iterates as the function value of an approximate saddle point at the current iteration. The numerical results demonstrate that, with a suitable strategy of delays’ selections, the delayed subgradient method can achieve better convergence behaviors than its non-delayed counterpart.

In this paper, we propose a dynamic distributed conjugate gradient method for solving the strongly monotone variational inequality problem over the intersection of fixed-point sets of firmly nonexpansive operators. The proposed method allows the independent computation of a firmly nonexpansive operator along with the dynamic weight which is updated at each iteration. This strategy aims to speed up the convergence behavior of the algorithm by updating control factors to drive each iterative step. Under some suitable control conditions on corresponding parameters, we show a strong convergence of the iterate to the unique solution of the considered variational inequality problem. We consider the numerical experiments and discuss some observation points by applying the model to solve the image classification problem via the support vector machine learning.

In this study, we focus on solving the nonlinear fractional optimization problem in which the numerator is smooth convex and the denominator is smooth concave. To achieve this goal, we develop an algorithm called the adaptive projection gradient method. The main advantage of this method is that it allows the computations for the gradients of the considered functions and the metric projection to take place separately. Moreover, an interesting property that distinguishes the proposed method from some of the existing methods is the nonincreasing property of its step-size sequence. In this study, we also prove that the sequence of iterates that is generated by the method converges to a solution for the considered problem and we derive the rate of convergence. To illustrate the performance and efficiency of our algorithm, some numerical experiments are performed.

In this paper, we consider the solving of an equilibrium problem over the common fixed set of cutter mappings in a real Hilbert space. To this end, we present a subgradient-type extrapolation cyclic method. The proposed method is generated based on the ideas of a subgradient method and an extrapolated cyclic cutter method. We prove a strong convergence of the method provided that some suitable assumptions of step-size sequences are assumed. We finally show the numerical behavior of the proposed method.

We propose a modified extragradient method for solving the variational inequality problem in a Hilbert space. The method is a combination of the well-known subgradient extragradient with the Mann's mean value method in which the updated iterate is picked in the convex hull of all previous iterates. We show weak convergence of the mean value iterate to a solution of the variational inequality problem, provided that a condition on the corresponding averaging matrix is fulfilled. Some numerical experiments are given to show the effectiveness of the obtained theoretical result.