Search Results (8)

This paper proposes a new accelerated fixed-point algorithm based on a double-inertial extrapolation technique for solving structured variational inclusion and convex bilevel optimization problems. The underlying framework leverages fixed-point theory and operator splitting methods to address inclusion problems of the form $0\in (A+B)(x)$, where A is a cocoercive operator and B is a maximally monotone operator defined on a real Hilbert space. The algorithm incorporates two inertial terms and a relaxation step via a contractive mapping, resulting in improved convergence properties and numerical stability. Under mild conditions of step sizes and inertial parameters, we establish strong convergence of the proposed algorithm to a point in the solution set that satisfies a variational inequality with respect to a contractive mapping. Beyond theoretical development, we demonstrate the practical effectiveness of the proposed algorithm by applying it to data classification tasks using Deep Extreme Learning Machines (DELMs). In particular, the training processes of Two-Hidden-Layer ELM (TELM) models is reformulated as convex regularized optimization problems, enabling robust learning without requiring direct matrix inversions. Experimental results on benchmark and real-world medical datasets, including breast cancer and hypertension prediction, confirm the superior performance of our approach in terms of evaluation metrics and convergence. This work unifies and extends existing inertial-type forward–backward schemes, offering a versatile and theoretically grounded optimization tool for both fundamental research and practical applications in machine learning and data science. Full article

This work focuses on split feasibility problems in Hilbert spaces. To accelerate the convergent rate of gradient-CQ algorithms, we introduce an inertial term. Additionally, non-monotone stepsizes are employed to adjust the relaxation parameter applied to the original stepsizes, ensuring that these original stepsizes maintain a positive lower bound. Thereby, the efficiency of the algorithms is improved. Moreover, the weak and strong convergence of the proposed algorithms are established through proofs that exhibit a similar symmetry structure and do not require the assumption of Lipschitz continuity for the gradient mappings. Finally, the LASSO problem is presented to illustrate and compare the performance of the algorithms. Full article

Based on recent perturbative and non-perturbative lattice calculations with almost quark flavors and the thermal contributions from photons, neutrinos, leptons, electroweak particles, and scalar Higgs bosons, various thermodynamic quantities, at vanishing net-baryon densities, such as pressure, energy density, bulk viscosity, relaxation time, and temperature have been calculated up to the TeV-scale, i.e., covering hadron, QGP, and electroweak (EW) phases in the early Universe. This remarkable progress motivated the present study to determine the possible influence of the bulk viscosity in the early Universe and to understand how this would vary from epoch to epoch. We have taken into consideration first- (Eckart) and second-order (Israel–Stewart) theories for the relativistic cosmic fluid and integrated viscous equations of state in Friedmann equations. Nonlinear nonhomogeneous differential equations are obtained as analytical solutions. For Israel–Stewart, the differential equations are very sophisticated to be solved. They are outlined here as road-maps for future studies. For Eckart theory, the only possible solution is the functionality, $H\left(a\right(t\left)\right)$, where $H\left(t\right)$ is the Hubble parameter and $a\left(t\right)$ is the scale factor, but none of them so far could to be directly expressed in terms of either proper or cosmic time t. For Eckart-type viscous background, especially at finite cosmological constant, non-singular $H\left(t\right)$ and $a\left(t\right)$ are obtained, where $H\left(t\right)$ diverges for QCD/EW and asymptotic EoS. For non-viscous background, the dependence of $H\left(a\right(t\left)\right)$ is monotonic. The same conclusion can be drawn for an ideal EoS. We also conclude that the rate of decreasing $H\left(a\right(t\left)\right)$ with increasing $a\left(t\right)$ varies from epoch to epoch, at vanishing and finite cosmological constant. These results obviously help in improving our understanding of the nucleosynthesis and the cosmological large-scale structure. Full article

In each iteration, the projection methods require computing at least one projection onto the closed convex set. However, projections onto a general closed convex set are not easily executed, a fact that might affect the efficiency and applicability of the projection methods. To overcome this drawback, we propose two iterative methods with self-adaptive step size that combines the Halpern method with a relaxed projection method for approximating a common solution of variational inequality and fixed point problems for an infinite family of multivalued relatively nonexpansive mappings in the setting of Banach spaces. The core of our algorithms is to replace every projection onto the closed convex set with a projection onto some half-space and this guarantees the easy implementation of our proposed methods. Moreover, the step size of each algorithm is self-adaptive. We prove strong convergence theorems without the knowledge of the Lipschitz constant of the monotone operator and we apply our results to finding a common solution of constrained convex minimization and fixed point problems in Banach spaces. Finally, we present some numerical examples in order to demonstrate the efficiency of our algorithms in comparison with some recent iterative methods. Full article

The relaxed inertial Tseng-type method for solving the inclusion problem involving a maximally monotone mapping and a monotone mapping is proposed in this article. The study modifies the Tseng forward-backward forward splitting method by using both the relaxation parameter, as well as the inertial extrapolation step. The proposed method follows from time explicit discretization of a dynamical system. A weak convergence of the iterates generated by the method involving monotone operators is given. Moreover, the iterative scheme uses a variable step size, which does not depend on the Lipschitz constant of the underlying operator given by a simple updating rule. Furthermore, the proposed algorithm is modified and used to derive a scheme for solving a split feasibility problem. The proposed schemes are used in solving the image deblurring problem to illustrate the applicability of the proposed methods in comparison with the existing state-of-the-art methods. Full article

We introduce an iterative algorithm which converges strongly to a common element of fixed point sets of nonexpansive mappings and sets of zeros of maximal monotone mappings. Our iterative method is quite general and includes a large number of iterative methods considered in recent literature as special cases. In particular, we apply our algorithm to solve a general system of variational inequalities, convex feasibility problem, zero point problem of inverse strongly monotone and maximal monotone mappings, split common null point problem, split feasibility problem, split monotone variational inclusion problem and split variational inequality problem. Under relaxed conditions on the parameters, we derive some algorithms and strong convergence results to solve these problems. Our results improve and generalize several known results in the recent literature. Full article

In this paper, we consider the generalized nonlinear quasi-variational inequalities problem for set-valued mappings and construct an iterative algorithm for find the approximate solution of this problem by exploiting the projection method and prove the existence of the solution to our problem involving relaxed Lipschitz and relaxed monotone mappings and the convergence of the iterative sequences generated by this algorithm. Full article