arXiv subjects

Morteza Rahimi

Publications and source records attributed to Morteza Rahimi.

11 recordsLinked to original sources

arXiv2026

Relative Weak Convexity and Projected Subgradient Methods: Analysis and Convergence

We introduce the class of relatively weakly convex functions, which extends the classical notion of weak convexity by measuring nonconvexity relative to a distance-generating function. We investigate the fundamental properties of this function class, establishing characterization results, calculus rules, and illustrative examples. We further analyze the associated optimization landscape and identify a neighborhood of the set of global minimizers that is free of saddle points. Motivated by this geometric structure, we propose the Projected SubGradient Algorithm (PSGA) with several step-size strategies. Under a sharpness error bound, we prove that, when initialized within this saddle-point-free neighborhood, the iterates generated by PSGA converge to a global minimizer for each of the proposed step-size strategies. Furthermore, linear convergence is established for the geometrically decaying step-size strategy.

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arXiv2025

Minimizing Smooth Kurdyka-{\L}ojasiewicz Functions via Generalized Descent Methods: Convergence Rate and Complexity

This paper introduces a generalized descent algorithm (DEAL) for minimizing smooth nonconvex functions. If the objective function is nonsmooth, a smoothing technique (e.g., forward-backward and high-order Moreau envelopes) is applied to generate a smooth counterpart. The proposed framework unifies several methods, such as gradient-based methods with constant step-sizes and Armijo line search, and several proximal splitting methods. The method is built around a generalized descent inequality that adapts the amount of decrease to the geometry of the objective function. Under the Kurdyka-{\L}ojasiewicz (KL) property, we establish global convergence of the generated sequence to critical points and provide a unified convergence rate analysis. In particular, we show that the convergence behavior depends jointly on the KL exponent and the descent order, and we identify a precise condition under which generalized descent methods achieve linear convergence. By choosing the order of high-order proximal regularization according to the KL exponent, our boosted high-order proximal-point method achieves linear convergence for arbitrary KL exponents. If the objective function satisfies a global KL inequality, we further strengthen the results by proving convergence to global minimizers and deriving explicit iteration-complexity bounds. Numerical experiments validate our theoretical foundation.

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arXiv2025

(Adaptive) Scaled gradient methods beyond locally Holder smoothness: Lyapunov analysis, convergence rate and complexity

This paper addresses the unconstrained minimization of smooth convex functions whose gradients are locally Holder continuous. Building on these results, we analyze the Scaled Gradient Algorithm (SGA) under local smoothness assumptions, proving its global convergence and iteration complexity. Furthermore, under local strong convexity and the Kurdyka-Lojasiewicz (KL) inequality, we establish linear convergence rates and provide explicit complexity bounds. In particular, we show that when the gradient is locally Lipschitz continuous, SGA attains linear convergence for any KL exponent. We then introduce and analyze an adaptive variant of SGA (AdaSGA), which automatically adjusts the scaling and step-size parameters. For this method, we show global convergence, and derive local linear rates under strong convexity.

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arXiv2025

Inexact Levenberg-Marquardt methods under H\"{o}lder metric subregularity

This paper investigates two inexact Levenberg-Marquardt (LM) methods for solving systems of nonlinear equations. Both approaches compute approximate search directions by solving the LM linear system inexactly, subject to specific residual-based conditions. The first method uses an adaptive scheme to update the LM parameter, and we establish its local superlinear convergence under H\"older metric subregularity and local H\"older continuity of the gradient. The second method combines an inexact LM step with a nonmonotone quadratic regularization strategy. For this variant, we prove global convergence under the assumption of Lipschitz continuous gradients and derive a worst-case global complexity bound, showing that an approximate stationary point can be found in $\mathcal{O}(\epsilon^{-2})$ function and gradient evaluations. Finally, we justify the use of the LSQR algorithm for efficiently solving the linear systems involved, which is used in our numerical experiment on several nonlinear systems, including those appearing in real-world biochemical reaction networks, monotone and nonlinear equations, and image deblurring problems.

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arXiv2025

Characterization of Highly Robust Solutions in Multi-Objective Programming in Banach Spaces

This paper delves into the challenging issues in uncertain multi-objective optimization, where uncertainty permeates nonsmooth nonconvex objective and constraint functions. In this context, we investigate highly robust (weakly efficient) solutions, a solution concept defined by efficiency across all scenarios. Our exploration reveals important relationships between highly robust solutions and other robustness notions, including set-based and worst-case notions, as well as connections with proper and isolated efficiency. Leveraging modern techniques from variational analysis, we establish necessary and sufficient optimality conditions for these solutions. Moreover, we explore the robustness of multi-objective optimization problems in the face of various uncertain sets, such as ball, ellipsoidal, and polyhedral sets.

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arXiv2024

Projected subgradient methods for paraconvex optimization: Application to robust low-rank matrix recovery

This paper is devoted to the class of paraconvex functions and presents some of its fundamental properties, characterization, and examples that can be used for their recognition and optimization. Next, the convergence analysis of the projected subgradient methods with several step-sizes (i.e., constant, nonsummable, square-summable but not summable, geometrically decaying, and Scaled Polyak's step-sizes) to global minima for this class of functions is studied. In particular, the convergence rate of the proposed methods is investigated under paraconvexity and the H\"olderian error bound condition, where the latter is an extension of the classical error bound condition. The preliminary numerical experiments on several robust low-rank matrix recovery problems (i.e., robust matrix completion, image inpainting, robust nonnegative matrix factorization, robust matrix compression, and robust image deblurring) indicate promising behavior for these projected subgradient methods, validating our theoretical foundations.

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arXiv2024

Machine Learning-Assisted Intrusion Detection for Enhancing Internet of Things Security

Attacks against the Internet of Things (IoT) are rising as devices, applications, and interactions become more networked and integrated. The increase in cyber-attacks that target IoT networks poses a considerable vulnerability and threat to the privacy, security, functionality, and availability of critical systems, which leads to operational disruptions, financial losses, identity thefts, and data breaches. To efficiently secure IoT devices, real-time detection of intrusion systems is critical, especially those using machine learning to identify threats and mitigate risks and vulnerabilities. This paper investigates the latest research on machine learning-based intrusion detection strategies for IoT security, concentrating on real-time responsiveness, detection accuracy, and algorithm efficiency. Key studies were reviewed from all well-known academic databases, and a taxonomy was provided for the existing approaches. This review also highlights existing research gaps and outlines the limitations of current IoT security frameworks to offer practical insights for future research directions and developments.

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arXiv2024

Artificial Intelligence for Secured Information Systems in Smart Cities: Collaborative IoT Computing with Deep Reinforcement Learning and Blockchain

The accelerated expansion of the Internet of Things (IoT) has raised critical challenges associated with privacy, security, and data integrity, specifically in infrastructures such as smart cities or smart manufacturing. Blockchain technology provides immutable, scalable, and decentralized solutions to address these challenges, and integrating deep reinforcement learning (DRL) into the IoT environment offers enhanced adaptability and decision-making. This paper investigates the integration of blockchain and DRL to optimize mobile transmission and secure data exchange in IoT-assisted smart cities. Through the clustering and categorization of IoT application systems, the combination of DRL and blockchain is shown to enhance the performance of IoT networks by maintaining privacy and security. Based on the review of papers published between 2015 and 2024, we have classified the presented approaches and offered practical taxonomies, which provide researchers with critical perspectives and highlight potential areas for future exploration and research. Our investigation shows how combining blockchain's decentralized framework with DRL can address privacy and security issues, improve mobile transmission efficiency, and guarantee robust, privacy-preserving IoT systems. Additionally, we explore blockchain integration for DRL and outline the notable applications of DRL technology. By addressing the challenges of machine learning and blockchain integration, this study proposes novel perspectives for researchers and serves as a foundational exploration from an interdisciplinary standpoint.

cs.AI↗

arXiv2024

Comparative Analysis of Gradient-Based Optimization Techniques Using Multidimensional Surface 3D Visualizations and Initial Point Sensitivity

This study examines several renowned gradient-based optimization techniques and focuses on their computational efficiency and precision. In the study, the steepest descent, conjugate gradient (Fletcher-Reeves and Polak-Ribiere variants), Newton-Raphson, quasi-Newton (BFGS), and Levenberg-Marquardt techniques were evaluated. These methods were benchmarked using Rosenbrock's, Spring Force Vanderplaats', Ackley's, and Himmelblau's functions. We emphasize the critical role that initial point selection plays in optimizing optimization outcomes in our analysis. It is also important to distinguish between local and global optima since gradient-based methods may have difficulties dealing with nonlinearity and multimodality. We illustrate optimization trajectories using 3D surface visualizations in order to increase understanding. While gradient-based methods have been demonstrated to be effective, they may be limited by computational constraints and by the nature of the objective functions, necessitating the use of heuristic and metaheuristic algorithms in more complex situations.

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arXiv2020

Fog Computing Approaches in Smart Cities: A State-of-the-Art Review

These days, the development of smart cities, specifically in location-aware, latency-sensitive, and security-crucial applications (such as emergency fire events, patient health monitoring, or real-time manufacturing) heavily depends on a more advance computing paradigms that can address these requirements. In this regard, fog computing, a robust cloud computing complement, plays a preponderant role by virtue of locating closer to the end-devices. Nonetheless, utilized approaches in smart cities are frequently cloud-based, which causes not only the security and time-sensitive services to suffer but also its flexibility and reliability to be restricted. So as to obviate the limitations of cloud and other related computing paradigms such as edge computing, this paper proposes a systematic literature review (SLR) for the state-of-the-art fog-based approaches in smart cities. Furthermore, according to the content of the reviewed researches, a taxonomy is proposed, falls into three classes, including service-based, resource-based, and application-based. This SLR also investigates the evaluation factors, used tools, evaluation methods, merits, and demerits of each class. Types of proposed algorithms in each class are mentioned as well. Above all else, by taking various perspectives into account, comprehensive and distinctive open issues and challenges are provided via classifying future trends and issues into practical sub-classes.

cs.NI↗

arXiv2019

Characterization of the norm-based robust solutions in vector optimization

In this paper, we study the norm-based robust (efficient) solutions of a Vector Optimization Problem (VOP). We define two kinds of non-ascent directions in terms of Clarke's generalized gradient and characterize norm-based robustness by means of the newly-defined directions. This is done under a basic Constraint Qualification (CQ). We extend the provided characterization to VOPs with conic constraints. Moreover, we derive a necessary condition for norm-based robustness utilizing a nonsmooth gap function.

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