arXiv subjects

Mohammad S. Alkousa

Publications and source records attributed to Mohammad S. Alkousa.

8 recordsLinked to original sources

arXiv2026

Iterative Methods for Computing the Moore--Penrose Inverse of Split-Quaternion Matrices with Applications

We study iterative methods for computing the Moore--Penrose inverse of split-quaternion matrices. We first establish a consistent framework based on a $2\times2$ real representation and the associated $i$-conjugate transpose. This representation gives a direct definition of the Moore--Penrose inverse and clarifies the treatment of nonzero zero divisors. We then analyze Newton--Schulz iterations for rectangular and rank-deficient matrices. Using a thin singular value decomposition of the real representative, we derive the convergence conditions, the evolution of the projector residuals, and an exact relation between the residual and the error. The resulting analysis applies both to an embedded real implementation and to a native split-quaternion iteration, which are shown to be equivalent under the representation. We also propose a low-degree polynomial initialization inspired by Souriau's inverse recursion. The polynomial is obtained by a least-squares approximation of the inverse Gram factor and is combined with a spectral acceptance test and a safe fallback initialization. Finally, we apply the proposed methods to cross and CUR approximations of split-quaternion matrices. We characterize the optimal middle factor for fixed sampled rows and columns and give conditions under which the cheaper cross factor gives an exact reconstruction. Numerical experiments illustrate the accuracy of the methods and the practical benefit of the polynomial initialization.

math.NA↗

arXiv2026

Mirror Descent-Type Algorithms for the Variational Inequality Problem with Functional Constraints

Variational inequalities play a key role in machine learning research, such as generative adversarial networks, reinforcement learning, adversarial training, and generative models. This paper is devoted to the constrained variational inequality problems with functional constraints (inequality-type constraints). We propose some mirror descent-type algorithms that switch between productive and non-productive steps depending on the values of the functional constraints at iterations, with many different step size rules and stopping criteria. We analyze the proposed algorithms and prove their optimal convergence rate to achieve a solution with desired accuracy, for problems with bounded and monotone operators and Lipschitz convex functional constraints. In addition, we propose a modification of the proposed algorithms by considering each functional constraint in the calculation when we have a productive step, as well as the first constraint that violates the feasibility. This modification can save the running time of algorithms when we have many functional constraints. In addition, we provide an analysis of the proposed algorithms for $δ$-monotone operators, allowing us to apply the proposed algorithms, as a special case, to constrained minimization problems when we do not have access to the exact information about the subgradient of the objective function. Numerical experiments that illustrate the work and performance of the proposed algorithms are also given.

cs.LG↗

arXiv2026

Lipschitz-Free Mirror Descent Methods for Relatively Strongly Convex Functions with/without Absolute and Relative Inexactness

In this paper, we analyze the mirror descent algorithm for non-smooth optimization problems in which the objective function is relatively strongly convex, without relying on the standard Lipschitz continuity assumption commonly used in the literature. We provide convergence analyses for both exact and inexact subgradient information. Furthermore, through numerical experiments, we compare the derived bounds on the quality of the approximate solutions with existing estimates in the literature and demonstrate the effectiveness of the proposed results.

math.OC↗

arXiv2025

Mirror Descent Methods with Weighting Scheme for Outputs for Constrained Variational Inequality Problems

This paper is devoted to the variational inequality problems. We consider two classes of problems, the first is classical constrained variational inequality and the second is the same problem with functional (inequality type) constraints. To solve these problems, we propose mirror descent-type methods with a weighting scheme for the generated points in each iteration of the algorithms. This scheme assigns smaller weights to the initial points and larger weights to the most recent points, thus it improves the convergence rate of the proposed methods. For the variational inequality problem with functional constraints, the proposed method switches between adaptive and non-adaptive steps in the dependence on the values of the functional constraints at iterations. We analyze the proposed methods for the time-varying step sizes and prove the optimal convergence rate for variational inequality problems with bounded and monotone operators. The results of numerical experiments of the proposed methods for classical constrained variational inequality problems show a significant improvement over the modified projection method.

math.OC↗

arXiv2025

Lipschitz-Free Mirror Descent Methods for Non-Smooth Optimization Problems

The part of the analysis of the convergence rate of the mirror descent method that is connected with the adaptive time-varying step size rules due to Alkousa et al. (MOTOR 2024, pp. 3-18) is corrected. Moreover, a Lipschitz-free mirror descent method that achieves weak ergodic convergence is presented, generalizing the convergence results of the mirror descent method in the absence of the Lipschitz assumption.

math.OC↗

arXiv2022

Gradient-Type Methods for Optimization Problems with Polyak-Łojasiewicz Condition: Early Stopping and Adaptivity to Inexactness Parameter

Due to its applications in many different places in machine learning and other connected engineering applications, the problem of minimization of a smooth function that satisfies the Polyak-Łojasiewicz condition receives much attention from researchers. Recently, for this problem, the authors of recent work proposed an adaptive gradient-type method using an inexact gradient. The adaptivity took place only with respect to the Lipschitz constant of the gradient. In this paper, for problems with the Polyak-Łojasiewicz condition, we propose a full adaptive algorithm, which means that the adaptivity takes place with respect to the Lipschitz constant of the gradient and the level of the noise in the gradient. We provide a detailed analysis of the convergence of the proposed algorithm and an estimation of the distance from the starting point to the output point of the algorithm. Numerical experiments and comparisons are presented to illustrate the advantages of the proposed algorithm in some examples.

math.OC↗

arXiv2020

On Modification of an Adaptive Stochastic Mirror Descent Algorithm for Convex Optimization Problems with Functional Constraints

This paper is devoted to a new modification of a recently proposed adaptive stochastic mirror descent algorithm for constrained convex optimization problems in the case of several convex functional constraints. Algorithms, standard and its proposed modification, are considered for the type of problems with non-smooth Lipschitz-continuous convex objective function and convex functional constraints. Both algorithms, with an accuracy $\varepsilon$ of the approximate solution to the problem, are optimal in the terms of lower bounds of estimates and have the complexity $O\left( \varepsilon^{-2} \right)$. In both algorithms, the precise first-order information, which connected with (sub)gradient of the objective function and functional constraints, is replaced with its unbiased stochastic estimates. This means that in each iteration, we can still use the value of the objective function and functional constraints at the research point, but instead of their (sub)gradient, we calculate their stochastic (sub)gradient. Due to the consideration of not all functional constraints on non-productive steps, the proposed modification allows saving the running time of the algorithm. Estimates for the rate of convergence of the proposed modified algorithm is obtained. The results of numerical experiments demonstrating the advantages and the efficient of the proposed modification for some examples are also given.

math.OC↗

arXiv2018

Adaptive algorithms for mirror descent in convex programming problems with Lipschitz constraints

The paper is devoted to new modifications of recently proposed adaptive methods of Mirror Descent for convex minimization problems in the case of several convex functional constraints. Methods for problems of two classes are considered. The first type of problems with Lipschitz-continuous objective (generally speaking, nonsmooth) functional. The second one is for problems with a Lipschitz-continuous gradient of the objective smooth functional. We consider the class of problems with a non-smooth objective functional equal to the maximum of smooth functionals with a Lipschitz-continuous gradient. Note that functional constraints, generally speaking, are non-smooth and Lipschitz-contionuous. The proposed modifications allow saving the algorithm running time due to consideration of not all functional constraints on non-productive steps. Estimates for the rate of convergence of the methods under consideration are obtained. The methods proposed are optimal from the point of view of lower oracle estimates. The results of numerical experiments illustrating the advantages of the proposed procedure for some examples are given.

math.OC↗