In this paper, we propose a scaled gradient modified non-monotone line search method for solving constrained minimization problems, and explore several specific properties of this method, namely, its convergence analysis. We discuss the linear convergence rate of the sequence generated by the proposed algorithm to a solution of the constrained minimization problem where the objective function is strongly quasiconvex. We consider numerical examples of large-scale fractional programming and quadratic programming for the function of pseudo convex and strongly quasiconvex and compare the performance of the proposed algorithm with the existing ones for these examples.

In this paper, we study the proximal split fixed point problem (PSFPP) and propose a viscosity-type iterative algorithm that incorporates both inertial effects and a self-adaptive stepsize strategy. The proposed scheme is intended to address limitations of existing methods that typically require prior knowledge of the operator norm, which may be difficult to estimate in practice. We establish strong convergence of the generated sequence to a solution associated with a nonexpansive mapping under standard assumptions in Hilbert spaces. To illustrate the effectiveness and practical applicability of the proposed algorithm, we present numerical experiments on representative inverse problems, including image restoration and sparse signal recovery. The results demonstrate the robustness and computational efficiency of the method in practical settings.

Contractual employment offers a relevant avenue for reducing the unemployment period within a labor force. Additionally, they offer valuable work experience to individuals, thereby enhancing their chances of securing regular employment. In this article, we introduce a four-dimensional stage-structured model, taking the service period of contractual jobs as a delay parameter. This approach enables us to examine the impact of contractual employment on the dynamics of unemployment. The proposed model is examined using the stability theory of delay differential equations, revealing that the system has a unique equilibrium that is globally stable under certain conditions for any value of the delay parameter. Furthermore, we analytically derive specific conditions under which an increase in the contractual job tenure either increases or decreases unemployment. Additionally, we validate our analytical findings numerically by using India's unemployment-employment data and considering the employment opportunities provided under the Indian Agnipath scheme as an illustrative example of contractual employment.

In this paper, we introduce an adaptive primal-dual algorithm (PDAc-A) for solving the structured convex-concave saddle point problems with a generic smooth non-bilinear coupling term. The proposed method incorporates a convex combination and an adaptive step size selection, eliminating the need for additional computations, such as linesearch. We establish the global pointwise convergence and an 𝒪(1/N) ergodic sublinear convergence rate for the proposed algorithm, where N is the iteration counter. Furthermore, we extend PDAc-A to solve non-linear compositional convex optimization problems. We also develop an accelerated algorithm (aPDAc-A) for the strongly convex case, which achieves an 𝒪(1/N^2) ergodic convergence rate. Numerical results on quadratically constrained quadratic programming problems demonstrate the superiority of PDAc-A and aPDAc-A over existing methods.

This paper introduces two new methods, namely the relaxed beta-reflected 2-accelerated normal S-iteration method and the inertial relaxed beta-reflected Mann iteration method. These methods are applied to solve the split feasibility problem with multiple output sets in infinite-dimensional Hilbert spaces. The convergence of the sequence generated by these methods is studied under some mild assumptions. The paper also provides some numerical implementations from sparse signal and image deblurring to show the efficiency of these methods.

Our interest lies in developing some efficient methods for minimizing the sum of two geodesically convex functions on Hadamard manifolds, with the aim of improving the convergence of the Douglas–Rachford algorithm in Hadamard manifolds. Specifically, we propose two types of algorithms: inertial and non-inertial algorithms. The convergence analysis of both algorithms is provided under suitable assumptions on algorithmic parameters and the geodesic convexity of the objective functions. This convergence analysis is based on fixed-point theory for nonexpansive operators. We also study the convergence rates of these two methods. Additionally, we introduce parallel Douglas–Rachford type algorithms for minimizing functionals containing multiple summands with applications to the generalized Heron problem on Hadamard manifolds. To demonstrate the effectiveness of the proposed algorithms, we present some numerical experiments for the generalized Heron problems.

The purpose of this paper is to design a novel iterative algorithm to solve a generalized split feasibility and fixed point problem with multiple output sets (GSFFPPM) in the framework of Hilbert spaces. The proposed algorithm combines inertial extrapolation and the S-iterative methodology to accelerate convergence, Tikhonov regularization to ensure stability, and viscosity approximation to guarantee strong convergence to a solution of the GSFFPPM. Due to the generality of our model, we demonstrate the applicability of our iterative method to important classes of problems, including split variational inclusions and split equilibrium problems. To illustrate the practical relevance and computational efficiency of the proposed method, we present numerical experiments on real-world models such as Nash-Cournot semi-oligopolistic market equilibria and signal recovery tasks. These numerical experiments demonstrate the robustness and effectiveness of the proposed method.

The purpose of the paper is to introduce a novel second‐order PD ‐type fractional‐order iterative learning control algorithm for a broad class of fractional‐order linear continuous‐time switched systems with time delay. The convergence of the proposed second‐order PD ‐type fractional‐order iterative learning control algorithm is analyzed in the absence of external noise. The robustness of the system is examined under bounded measurement noise. The proposed second‐order PD ‐type fractional‐order iterative learning control algorithm outperforms the first‐order counterpart studied by Zhang and Peng (2020) and classic PD‐type ILC. Simulation results demonstrate the effectiveness and feasibility of the proposed algorithm.

We propose and study a variant of the Dai-Liao spectral conjugate gradient method, developed through an analysis of eigenvalues and inspired by a modified secant condition. We show that our proposed method is globally convergent for general nonlinear functions under standard assumptions. By incorporating the new secant condition and a quasi-Newton direction, we introduce updated spectral parameters. These changes ensure that the resulting search direction satisfies the sufficient descent property without relying on any line search. Numerical experiments show that the proposed algorithm performs better than several existing methods in terms of convergence speed and computational efficiency. Its effectiveness is further demonstrated through an application to signal processing.

This paper introduces a novel class of nonlinear singular switched systems that incorporate state delays and noise effects. A D-type iterative learning control (ILC) algorithm is developed based on the system's structure. The study examines the effects of constant time delay and external noise on tracking performance. The findings reveal that time delay has negligible impact on learning performance when it is smaller than the dwell time. Furthermore, the output remains within an acceptable neighborhood of the desired trajectory during the entire operational interval, provided the noise stays within a finite boundary. The paper establishes sufficient conditions for the convergence of the proposed algorithm, noting that while system switching may reduce the convergence speed of the ILC algorithm, it does not affect its convergence in the iterative domain. To validate the theoretical analysis, a numerical example is presented, demonstrating and confirming the effectiveness of the D-type ILC algorithm for the considered class of switched systems.

This paper aims to solve the monotone inclusion problem, minimization problem of multiple summands and the generalized Heron problem. We present an innovative approach, the modified normal S-iteration method, designed to approximate common fixed points of nearly nonexpansive sequences and families of operators via the property $ (\mathscr {A}) $ (A). Some deductions of our results improve some existing results in the literature. To show the applicability of our result, we give application to the inclusion problem via forward-backward splitting method version of our algorithm and minimization problem via Douglas-Rachford splitting method version of our algorithm. To demonstrate the practical utility of the algorithm, we apply it to the generalized Heron problem.

This article addresses the issue of iterative learning control for a specific category of discrete linear singular time-delay systems. A new iterative learning control algorithm based on the p-accelerated normal S-iteration method is proposed, and convergence analysis of the corresponding learning control algorithm is studied. With certain assumptions, the proposed algorithm guarantees that the output of the system converges to the desired output trajectory within a finite time interval. The theoretical analysis is supported by numerical examples. The results indicate that the p-accelerated normal S-iterative learning control algorithm outperforms both the first-order PD-type iterative learning control and second-order PD-type iterative learning control algorithms for discrete linear singular systems theoretically and numerically.

In this paper, we introduce a new iterative technique with a variable anchoring operator for reckoning the solution of a variational inequality problem over the set of the common fixed points of a nearly nonexpansive sequence of operators in the framework of Hadamard manifolds. We also establish a convergence result on the proposed algorithm for approximating a solution of the problem, under suitable assumptions. We apply our results for finding the solutions of a system of nonlinear equations, and of inclusion problems to support their utility. Our work improves results in the recent literature. Numerical simulations are given for a better understanding of the effectiveness of our outcomes.

The objective of this paper is to introduce a derivative free projection method designed to find the singularities of pseudomonotone vector fields with convex constraints on Hadamard manifolds. This innovative approach combines the hyperplane projection method with a novel search direction. The global convergence of the proposed method is established under certain conditions. Our method improves some existing results in the literature on Hadamard manifolds. Additionally, illustrative numerical examples are provided to demonstrate the practical efficacy of our method.

The purpose of this article is to investigate the second-order P-type iterative learning control (ILC) scheme in the presence of data loss for a class of linear discrete-time switched systems with disturbances. Employing the super-vector representation technique, the discrete-time linear switched system is reformulated as an input-output transmission equation. The robustness of the resulting switched system driven by a second-order P-type ILC scheme is guaranteed through the use of the super-vector representation technique. Importantly, the article also explores cases of data loss occurring during data transmission. The proposed methodology exhibits significantly improved convergence performance compared to the P-type ILC scheme (Yang et al., 2022, Robust finite-iteration tracking of discrete-time systems in repetitive process setting via ILC scheme. International Journal of Robust and Nonlinear Control, 32(5), 2585-2602. https://doi.org/10.1002/rnc.5782). Simulation examples are provided to demonstrate the effectiveness of the proposed scheme.

The second-order ILC algorithm is studied for a class of linear discrete-time switched systems with model uncertainties, external noises and the time-delay for tracking reference trajectory by taking advantage of super-vector representation. This study is based on the assumption that the systems operate under finite intervals. A sufficient condition for convergence of the algorithms is deduced when the model uncertainties and external noises are absent. Then the convergence is analysed, when the model uncertainties are present but the external noises are absent and the robustness against the bounded external noises is discussed. The analysis manifests that the second-order ILC algorithm is feasible and effective when it is imposed on the linear switched systems specified by the arbitrarily present switching rules when they are imposed on the system. We have presented a simulation example to illustrate effectiveness of the proposed second-order ILC algorithm in the study of robustness of the considered SISO linear discrete time-invariant time delay switched system.

The main strategy of this paper is intended to speed up the convergence of the inertial Mann iterative method and further speed up it through the normal S-iterative method for a certain class of nonexpansive-type operators that are linked with variational inequality problems. Our new convergence theory permits us to settle down the difficulty of unification of Korpelevich’s extragradient method, Tseng’s extragardient method, and subgardient extragardient method for solving variational inequality problems through an auxiliary algorithmic operator, which is associated with the seed operator. The paper establishes an interesting the fact that the relaxed inertial normal S-iterative extragradient methods do influence much more on convergence behaviour. Finally, the numerical experiments are carried out to illustrate that the relaxed inertial iterative methods; in particular, the relaxed inertial normal S-iterative extragradient methods may have a number of advantages over other methods in computing solutions to variational inequality problems in many cases.

The mathematical formulation of the equilibrium problem is to find an element x̄ of a set K such that

The objective of this work is to design a new iterative method based on Armijo’s type-modified extragradient method for solving the inclusion problem (A+B)^-1(0) , where A is a maximal monotone vector field and B is a continuous monotone vector field. The proposed method requires one projection at each iteration, reducing the cost of computational viewpoint and improving the convergence rate. A convergence theorem is established for the proposed extragradient method, significantly improving existing results. We provide concrete examples of Hadamard manifolds and convergency for numerical confirmation. Moreover, we demonstrate convergence results for the variational inequality problems in which the vector field’s monotonicity can be removed.

We consider the monotone inclusion problems in real Hilbert spaces. Proximal splitting algorithms are very popular technique to solve it and generally achieve weak convergence under mild assumptions. Researchers assume the strong conditions like strong convexity or strong monotonicity on the considered operators to prove strong convergence of the algorithms. Mann iteration method and normal S-iteration method are popular methods to solve fixed point problems. We propose a new common fixed point algorithm based on normal S-iteration method using Tikhonov regularization to find common fixed point of non-expansive operators and prove strong convergence of the generated sequence to the set of common fixed points without assuming strong convexity and strong monotonicity. Based on proposed fixed point algorithm, we propose a forward-backward-type algorithm and a Douglas-Rachford algorithm in connection with Tikhonov regularization to find the solution of monotone inclusion problems. Further, we consider the complexly structured monotone inclusion problems which are very popular these days. We also propose a strongly convergent forward-backward-type primal-dual algorithm and a Douglas-Rachford-type primal-dual algorithm to solve the monotone inclusion problems. Finally, we conduct a numerical experiment to solve image deblurring problems.