arXiv subjects

Mohammed Alshahrani

Publications and source records attributed to Mohammed Alshahrani.

8 recordsLinked to original sources

arXiv2026

Convergence Rates for Variational Inequality Projection Neural Networks with a State-Dependent Metric

We study continuous-time projection neural networks for variational inequalities on closed convex sets. A positive definite matrix that depends on the state preconditions the operator, and its inverse defines the projection metric. Existing convergence analyses of this flow cover Hessian-generated inverse metrics and state-dependent scalar metrics. In the first case, a Bregman distance eliminates metric-derivative terms. We treat a matrix metric whose inverse is of neither kind. We prove joint regularity of the projection in its argument and metric. Under common spectral bounds, the Euclidean Lipschitz estimate improves from the squared bound to the bound itself. For Lipschitz strongly monotone operators and Lipschitz metrics, we prove local exponential convergence with explicit rate and radius. On compact feasible sets, an explicit metric-variation bound yields global exponential convergence. A twice continuously differentiable, uniformly positive definite metric and a strongly monotone linear operator produce an annulus of periodic orbits. The inverse metric violates Hessian integrability throughout the annulus. Deterministic computations confirm the analytic formulas and quantify slack in both convergence certificates.

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arXiv2026

Projection and contraction methods with double inertial steps for variational inclusion problems on Hilbert spaces

In this paper, we propose three projection--contraction algorithms for solving variational inclusion problems in the setting of Hilbert spaces, each incorporating a double inertial technique into the projection--contraction framework. The first algorithm \algoO achieves weak convergence under monotonicity and Lipschitz continuity of the single-valued operator, with an adaptive stepsize rule that does not require prior knowledge of the Lipschitz constant. A modified variant \algoT of \algoO attains $R$-linear convergence under the strong monotonicity assumption. The third algorithm \algoDIM obtains strong convergence to the minimum-norm solution without requiring strong monotonicity. We illustrate the proposed methods on an abstract variational inclusion problem and apply them to the split feasibility problem and the elastic net regularization problem, comparing them with existing methods in the literature. The $R$-linear convergence of \algoT is also verified through numerical simulations.

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arXiv2026

State-Dependent Metric Projection Neural Network for Variational Inequalities

Projection-based dynamical systems and projection neural networks offer a continuous-time approach to solving variational inequalities by driving the state toward its projection onto the feasible set. However, most existing models are built on a fixed Euclidean or constant metric, which can lead to poor conditioning and slow convergence when the operator or feasible set geometry is highly anisotropic. This paper introduces a state-dependent metric projection dynamical system, referred to as a state-dependent scaled projection neural network (SD-SPNN), in which the geometry of the projection operator evolves smoothly with the state. The dynamics generalize classical projected dynamical systems and projection neural networks by embedding continuous-time preconditioning directly into the flow. Under standard monotonicity assumptions on the operator and mild regularity conditions on the metric, we establish existence of solutions, an exact equilibrium-solution correspondence with the underlying variational inequality, and Lyapunov-based stability properties. The framework unifies Euclidean, constant-metric, and state-dependent projection neural network within a single continuous-time model. Numerical experiments illustrate how state-dependent metrics reshape the transient geometry of the projected dynamics while converging to the same equilibrium.

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arXiv2026

Adaptive Metrics for Norm-Minimization-Based Outer Approximation in Convex Vector Optimization

We develop an adaptive-metric framework for norm-minimization-based outer approximation algorithms in bounded convex vector optimization. The key idea is to let the scalarization metric vary across iterations while measuring approximation error in a fixed Euclidean norm. This enables the algorithm to exploit problem geometry dynamically. Our approach rests on two theoretical foundations. First, we prove that the improved Euclidean convergence rate $O(k^{2/(1-q)})$ -- previously known only for the standard $\ell_2$-norm -- extends to all fixed inner-product norms. Second, we establish a dispersion theorem showing that the cut-normals generated by the algorithm naturally spread across all directions when the upper image has a strictly convex boundary with bounded curvature. This geometric condition guarantees that the adaptive metric remains well-conditioned throughout execution. Building on these results, we derive explicit convergence bounds that quantify how metric conditioning influences the Hausdorff error estimates. Numerical experiments on three test problems validate the theoretical convergence rate; on the problems whose Pareto fronts have sufficient curvature, the adaptive metric additionally reduces the iteration count relative to the fixed Euclidean norm. Our results provide a rigorous foundation for adaptive metric selection in convex vector optimization.

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arXiv2026

A Variable-Metric Non-monotone Line Search Method for Mixed Variational Inequalities and Equilibrium Problems

We propose a scaled gap-function method for variational inequalities, mixed variational inequalities, and equilibrium problems over a closed convex set. The method drives a gap function to zero by combining a variable-metric scaled projection step with a modified non-monotone Armijo line search. The construction rests on a structural identity: the scaled projection step is exactly the maximizer that defines the Fukushima regularized gap, so the search direction is simultaneously the algorithmic step and the gap-defining direction. For variational and mixed variational inequalities with a strongly monotone, Lipschitz operator we establish global convergence and an R-linear rate, under a fixed or a controlled-change variable metric. The rate follows from the strong-monotonicity contraction and the resulting explicit gap error bound. For equilibrium problems we obtain global convergence and a gap error bound (Mastroeni's gap). Numerical experiments on controlled problems confirm the convergence guarantees and the predicted rate, and compare the method with standard extragradient and gap-descent methods. The combination of variable-metric scaling, a modified non-monotone line search, and gap-function descent appears to be new for these problem classes.

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arXiv2026

On Parallel and Batch-Cutting Strategies for Norm-Minimization-Based Convex Vector Optimization

We develop parallel and batch-cutting variants of the norm-minimization-based outer approximation algorithm for convex vector optimization. The standard algorithm solves $N_k$ independent subproblems at each iteration~$k$ to evaluate all vertices of the current polyhedral approximation, but processes only the single best cut. We propose two improvements. First, we parallelize the \revise{subproblem evaluations} across $\nw$ workers, reducing per-iteration wall-clock time. Second, we introduce a batch-cutting strategy that adds up to $K$ supporting halfspaces per iteration, using information from all solved subproblems rather than discarding it. We prove that the batch-cutting variant inherits the convergence rate $O(k^{2/(1-q)})$ of the standard algorithm, where $k$ is the number of outer iterations and $q$ is the number of objectives. Computational experiments on eight test problems with $q \in \{2,3,4,5\}$ show that parallelism on 8 cores \revise{increases the speed by a factor of 1.1 to 4.2}, and batch cutting consistently reduces the iteration count by 62--80\%. However, the wall-clock benefit of batch cutting is problem-dependent: the additional cuts per iteration accelerate vertex count growth, so batch cutting is most effective when per-vertex subproblem cost dominates.

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arXiv2026

Convergence Rates for $\ell_p$ Norm Minimization in Convex Vector Optimization

We analyze convergence rates of norm-minimization-based outer approximation algorithms for convex vector optimization when the scalarization uses an $\ell_p$ norm with $p \in (1,\infty)$. While the Euclidean case ($p=2$) achieves the optimal rate $O(k^{2/(1-q)})$, the behavior under general $\ell_p$ norms has remained open. A direct approach via the modulus of smoothness yields only the weaker exponent $\min(p,2)$, which degrades for $1 < p < 2$. We prove that the Hausdorff approximation error satisfies $δ_H(P_k, A) = O(k^{2/(1-q)})$ for \emph{every} $p \in (1,\infty)$, where $q$ is the number of objectives and $k$ is the iteration count. The proof introduces a Euclidean intermediary technique that exploits the ambient inner product structure of $\R^q$ to obtain a quadratic bound on the hyperplane distance, bypassing the $\ell_p$ smoothness limitation; norm equivalence then converts this to any $\ell_p$ metric at the cost of only a dimension-dependent constant, not a loss of exponent. Numerical experiments confirm the $p$-independent rate predicted by the theory.

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arXiv2026

Spectral conjugate gradient projection methods for large-scale monotone equations without Lipschitz continuity

We introduce two derivative-free projection methods for large-scale systems of nonlinear monotone equations subject to convex constraints. Both methods incorporate an adaptive spectral parameter into established conjugate gradient frameworks: the first generalizes the modified optimal Perry method via an eigenvalue-optimized scaling matrix, and the second generalizes the Hager--Zhang-type conjugate gradient projection method via a spectral Dai--Liao parameter. The resulting search directions satisfy a sufficient descent condition independent of the line search. For the first method, we establish global convergence under monotonicity alone, without requiring Lipschitz continuity of the mapping. For the second, global convergence holds under the standard monotonicity and Lipschitz continuity assumptions. Numerical experiments on 18 test problems across dimensions up to 120{,}000, together with applications to $\ell_1$-regularized signal recovery and regularized logistic regression, confirm the practical effectiveness of the proposed approach.

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