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In this paper we present a subgradient method with non-monotone line search for the minimization of convex functions with simple convex constraints. Different from the standard subgradient method with prefixed step sizes, the new method selects the step sizes in an adaptive way. Under mild conditions asymptotic convergence results and iteration-complexity bounds are obtained. Preliminary numerical results illustrate the relative efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a novel iterative method for finding the minimum-norm solution to a pseudomonotone variational inequality problem in Hilbert spaces. We establish strong convergence of the proposed method and its linear convergence under some suitable assumptions. Some numerical experiments are given to illustrate the performance of our method. Our result improves and extends some existing results in the literature. [ABSTRACT FROM AUTHOR]

We develop a least-mean-squares (LMS) algorithm to identify scattering-sparse systems, where the few significantly large coefficients of the unknown impulse response are dispersed along the full length, instead of grouped into clusters. Many wireless communication channels such as the underwater acoustic channels, cellular communication channels and aviation channels are typical scattering-sparse systems. In the proposed strategy, the difference between the ℓ 1 -norm and ℓ ∞ , 1 -norm of the uniformly-divided tap-weight vector of the adaptive filter is utilized as a penalty and is introduced into the mean-square-error cost function, where the ℓ ∞ , 1 -norm of the tap-weight vector is used to locate the unknown large coefficients in view of the characteristics of dispersed sparsity. A dispersed-sparsity-aware LMS (DS-LMS) algorithm is then proposed by following the stochastic subgradient method. We study the mean and mean-square behaviors of the proposed algorithm. We also provide theoretical guidelines for the parameter settings that ensure that the proposed DS-LMS algorithm converges to a lower mean-square-deviation (MSD) level than the traditional LMS algorithm. Simulation results verify the effectiveness of the proposed DS-LMS algorithm, and correctness of the theoretical findings. • Algorithm identifies systems with dispersed significant signal coefficients. • Penalty based on norm differences locates large coefficients effectively. • Examined mean and mean-square behaviors of the proposed algorithm. • Guidelines ensure algorithm convergence to lower error levels. • Algorithm outperforms traditional methods in simulations. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new inertial self-adaptive parallel subgradient extragradient method for finding common solution of variational inequality problems with monotone and Lipschitz continuous operators. The stepsize of the algorithm is updated self-adaptively at each iteration and does not involve a line search technique nor a prior estimate of the Lipschitz constants of the cost operators. Also, the algorithm does not required finding the farthest element of the finite sequences from the current iterate which has been used in many previous methods. We prove a strong convergence result and provide some applications of our result to other optimization problems. We also give some numerical experiments to illustrate the performance of the algorithm by comparing with some other related methods in the literature. [ABSTRACT FROM AUTHOR]

In this paper, a new three-term spectral conjugate subgradient method is presented to solve an absolute value equation. The new method is constructed based on conjugate gradient method proposed by Dai & Yuan and Barzilai & Borwein technique. In each iteration, a descent direction is generated. The global convergence of the new method is established under some mild assumptions. The numerical results are reported, showing the efficiency of our method. [ABSTRACT FROM AUTHOR]

In this paper, we consider the modification of equilibrium problem (MEP) and new subgradient extragradient algorithm by using the concept of the set of solutions of the modified variational inequality problem introduced by [Kangtunyakarn A. A new iterative scheme for fixed-point problems of infinite family of κ i pseudo contractive mappings, equilibrium problem, variational inequality problems. J Optim Theory Appl. 2013;56:1543–1562.]. Then, we establish and prove weak and strong convergence theorem of the new subgradient extragradient algorithm for finding a common element of the set of solutions of the MEP and two sets of the variational inequality problems under some suitable conditions on α n and β n with α n + β n ≤ 1. Moreover, we apply our main theorem to prove weak and strong convergence theorems to solve the generalized equilibrium problem, the system of equilibrium problem, the variational inequality problem and the general system of variational inequality problems. Finally, we give two numerical examples to support our main result. [ABSTRACT FROM AUTHOR]

In this paper, we propose new methods for finding a common solution to pseudomonotone and Lipschitz-type equilibrium problems, as well as a fixed-point problem for demicontractive mapping in real Hilbert spaces. A novel hybrid technique is used to solve this problem. The method shown here is a hybrid of the extragradient method (a two-step proximal method) and a modified Mann-type iteration. Our methods use a simple step-size rule that is generated by specific computations at each iteration. A strong convergence theorem is established without knowing the operator's Lipschitz constants. The numerical behaviors of the suggested algorithms are described and compared to previously known ones in many numerical experiments. [ABSTRACT FROM AUTHOR]

In this work, we study the performance of sub-gradient method (SubGM) on a natural nonconvex and nonsmooth formulation of low-rank matrix recovery with l1-loss, where the goal is to recover a low-rank matrix from a limited number of measurements, a subset of which may be grossly corrupted with noise. We study a scenario where the rank of the true solution is unknown and over-estimated instead. The over-estimation of the rank gives rise to an over-parameterized model in which there are more degrees of freedom than needed. Such over-parameterization may lead to overfitting, or adversely affect the performance of the algorithm. We prove that a simple SubGM with small initialization is agnostic to both over-parameterization and noise in the measurements. In particular, we show that small initialization nullifies the effect of over-parameterization on the performance of SubGM, leading to an exponential improvement in its convergence rate. Moreover, we provide the first unifying framework for analyzing the behavior of SubGM under both outlier and Gaussian noise models, showing that SubGM converges to the true solution, even under arbitrarily large and arbitrarily dense noise values, and, perhaps surprisingly, even if the globally optimal solutions do not correspond to the ground truth. At the core of our results is a robust variant of restricted isometry property, called Sign-RIP, which controls the deviation of the sub-differential of the l1-loss from that of an ideal, expected loss. As a byproduct of our results, we consider a subclass of robust low-rank matrix recovery with Gaussian measurements, and show that the number of required samples to guarantee the global convergence of SubGM is independent of the over-parameterized rank.1 [ABSTRACT FROM AUTHOR]

Purpose: The purpose of this paper extends the authors' previous contributions on aircraft system identification, such as open loop identification or closed loop identification, to cascade system identification. Because the cascade system is one special network system, existing in lots of practical engineers, more unknown systems are needed to identify simultaneously within the statistical environment with the probabilistic noises. Consider this problem of cascade system identification, prediction error method is proposed to identify three unknown systems, which are parameterized by three unknown parameter vectors. Then the cascade system identification is transferred as one parameter identification problem, being solved by the online subgradient descent algorithm. Furthermore, the nonparametric estimation is proposed to consider the general case without any parameterized process. To make up the identification mission, model validation process is given to show the asymptotic interval of the identified parameter. Finally, simulation example confirms the proposed theoretical results. Design/methodology/approach: Firstly, aircraft system identification is reviewed through the understanding about system identification and advances in control theory, then cascade system identification is introduced to be one special network system. Secondly, for the problem of cascade system identification, prediction error method and online subgradient decent algorithm are combined together to identify the cascade system with the parameterized systems. Thirdly from the point of more general completeness, another way is proposed to identify the nonparametric estimation, then model validation process is added to complete the whole identification mission. Findings: This cascade system corresponds to one network system, existing in lots of practice, such as aircraft, ship and robot, so it is necessary to identify this cascade system, paving a way for latter network system identification. Parametric and nonparametric estimations are all studied within the statistical environment. Then research on bounded noise is an ongoing work. Originality/value: To the best of the authors' knowledge, research on aircraft system identification only concern on open loop and closed loop system identification, no any identification results about network system identification. This paper considers cascade system identification, being one special case on network system identification, so this paper paves a basic way for latter more advanced system identification and control theory. [ABSTRACT FROM AUTHOR]

In this paper, we studied the equilibrium problem where the bi-function may be quasiconvex with respect to the second variable and the feasible set is the intersection of a finite number of convex sets. We propose a projection algorithm, where the projection can be computed independently onto each component set. The convergence of the algorithm is investigated and numerical examples for a variational inequality problem involving affine fractional operator are provided to demonstrate the behaviour of the algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a self-adaptive subgradient extragradient method for finding a common element in the solution set of a pseudomonotone equilibrium problem and set of fixed points of a quasi-ϕ-nonexpansive mapping in 2-uniformly convex and uniformly smooth Banach spaces. The stepsize of the algorithm is selected self-adaptively which helps to prevent the necessity for prior estimates of the Lipschitz-like constants of the bifunction. A strong convergence result is proved under some mild conditions and some applications to Nash equilibrium problem and contact problem are presented to show the applicability of the result. Furthermore, some numerical examples are given to illustrate the efficiency and accuracy of the proposed method by comparing with other related methods in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a self-adaptive Bregman subgradient extragradient method for solving variational inequalities in the framework of a reflexive Banach space. The step-adaptive strategy avoids the difficult task of choosing a stepsize based on the Lipschitz constant of the cost function of the variational inequalities and improves the performance of the algorithm. Moreover, the use of the Bregman distance technique allows the consideration of a general feasible set for the problem. Under some suitable conditions, we prove some weak and strong convergence results for the sequence generated by the algorithm without prior knowledge of the Lipschitz constant. We further provide an application to contact problems and some numerical experiments to illustrate the performance of the algorithm. [ABSTRACT FROM AUTHOR]

The paper deals with a significant extension of the recently proposed class of relatively strongly convex optimization problems in spaces of large dimension. In the present paper, we introduce an analog of the concept of relative strong convexity for variational inequalities (relative strong monotonicity) and study estimates for the rate of convergence of some numerical first-order methods for problems of this type. The paper discusses two classes of variational inequalities depending on the conditions related to the smoothness of the operator. The first of these classes of problems contains relatively bounded operators, and the second, operators with an analog of the Lipschitz condition (known as relative smoothness). For variational inequalities with relatively bounded and relatively strongly monotone operators, a version of the subgradient method is studied and an optimal estimate for the rate of convergence is justified. For problems with relatively smooth and relatively strongly monotone operators, we prove the linear rate of convergence of an algorithm with a special organization of the restart procedure of a mirror prox method for variational inequalities with monotone operators. [ABSTRACT FROM AUTHOR]

In this article, we propose a hybrid iterative scheme with strong convergence property for solving variational inequality problems. The algorithm uses a self-adaptive stepsize defined using a simple updating rule. Therefore, the method does not require prior knowledge of the Lipschitz constant of the underlying operator. We consider a more general set of operators as the underlying operators. Moreover, we derived a fixed stepsize scheme from the proposed method. Under some suitable conditions, we show the strong convergence of the iterates generated by the proposed and the derived algorithms. Furthermore, we present numerical experiments to illustrate the computational performance of the proposed algorithm in comparison with some of the existing algorithms in the literature. Additionally, as an application, we use the proposed algorithm to solve the problem of recovering an original signal from a noisy signal [ABSTRACT FROM AUTHOR]

In this paper, we revisit the subgradient extragradient method for solving a pseudomonotone variational inequality problem with the Lipschitz condition in real Hilbert spaces. A new algorithm based on the subgradient extragradient method with the technique of choosing a new step size is proposed. The weak convergence of the proposed algorithm is established under the pseudomonotonicity and the Lipschitz continuity as well as without using the sequentially weakly continuity of the variational inequality mapping and the nonasymptotic O (1 / n) convergence rate of the proposed algorithm is presented, while the strong convergence theorem of the proposed algorithm is also proved under the strong pseudomonotonicity and the Lipschitz continuity hypotheses. In order to show the computational effectiveness of our algorithm, some numerical results are provided. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new inexact version of the projected subgradient method to solve nondifferentiable constrained convex optimization problems. The method combine ϵ-subgradient method with a procedure to obtain a feasible inexact projection onto the constraint set. Asymptotic convergence results and iteration-complexity bounds for the sequence generated by the method employing the well-known exogenous stepsizes, Polyak's stepsizes, and dynamic stepsizes are established. [ABSTRACT FROM AUTHOR]

Introduction. The problems of optimal planning of multi-product flows on transportation networks have a variety of important practical applications (transportation, logistics, communication networks). The mathematical models of these problems, as a rule, are characterized by a large dimension (hundreds of thousands of variables). Due to the specific features of multi-product transportation problems (pronounced block structure), the use of standard general-purpose optimization packages for their solution is not advisable. In this work of an applied nature, we propose a method for solving one class of such problems, taking into account their specificity. The method is based on the decomposition scheme with respect to a set of binding constraints. The corresponding dual problem is the nonsmooth optimization problem. To solve it, a new version of the subgradient algorithm with space dilation of variables is used. This approach to solving block optimization problems has been tested in solving a wide class of applied problems and, as experience shows, is characterized by a fairly high efficiency. Purpose of the article to develop a solution method and software for the problem of optimal planning of multi-product flows and modernization of the transportation system. Results. A mathematical model of the problem of optimal planning multi-product flows on the transportation network is proposed. The model is based on a variant transportation realization scheme. The software is developed in the C++ language in the style of object-oriented programming. The following algorithms are used as basic elements of the software: an algorithm for finding the shortest paths on a graph, an algorithm for solving a transportation problem on a graph, nonsmooth optimization algorithm. Conclusions. The output data of the developed software determines the following information: rational transportation scheme, critical by capacity sections of the transportation network, recommendations for the rational distribution of funds for the reconstruction of the transportation network. The software is designed to solve the problems of long-term planning of the functioning and development of the transportation system.

Backbone satellites in a space information network (SIN) can be used as air base stations or data relay satellites (DRSs) to realize cross-system, cross-network and long-distance relay transmission. In this paper, a traffic load optimization problem for multi-satellite relay systems in SIN is considered to achieve highly efficient cooperative transmission and improve resource utility. A model of SIN based on a distributed satellite cluster (DSC) is considered, and the characteristics of the model are analyzed. Based on this, a hybrid resource management architecture combining distributed and central resources control schemes is proposed to realize a centrally controllable and distributed optimization of resources to meet various comprehensive service requirements. Two scenarios of multi-satellite relay systems in SIN are given, and traffic load optimization problems with joint bandwidth and power allocation for these two scenarios are formulated based on proportional fairness (PF) criterion to achieve traffic load balancing with considerable system capacity. The optimization problems in these two scenarios are proved to be a convex optimization problem with mathematical analysis, and the closed-form solutions of two problems in their dual domain are derived by dual transformation. With the closed-form solutions, two iterative algorithms based on the subgradient method are designed under the proposed hybrid resource management architecture to solve the problems in this paper. Simulation results show that the proposed schemes can effectively improve the upper bound of system capacity by resource sharing and cooperative relay, and it can balance the traffic load well with guarantees of a reasonable level system capacity compared with existing methods. [ABSTRACT FROM AUTHOR]

Ordinal regression methods are widely used to predict the ordered labels of data, among which support vector ordinal regression (SVOR) methods are popular because of their good generalization. In many realistic circumstances, data are collected by a distributed network. In order to protect privacy or due to some practical constraints, data cannot be transmitted to a center for processing. However, as far as we know, existing SVOR methods are all centralized. In the above situations, centralized methods are inapplicable, and distributed methods are more suitable choices. In this paper, we propose a distributed SVOR (dSVOR) algorithm. First, we formulate a constrained optimization problem for SVOR in distributed circumstances. Since there are some difficulties in solving the problem with classical methods, we used the random approximation method and the hinge loss function to transform the problem into a convex optimization problem with constraints. Then, we propose subgradient-based algorithm dSVOR to solve it. To illustrate the effectiveness, we theoretically analyze the consensus and convergence of the proposed method, and conduct experiments on both synthetic data and a real-world example. The experimental results show that the proposed dSVOR could achieve close performance to that of the corresponding centralized method, which needs all the data to be collected together. [ABSTRACT FROM AUTHOR]

We consider a cross-layer packet scheduling problem with hybrid ARQ (HARQ) in fading channels in which the channel state information at the transmitter (CSIT) is known only after one slot delay. Packets arrive according to a Bernoulli process at the transmitter, and each packet is required to be timely-delivered at the receiver, within a delay of $d$ time-slots, and is dropped, if the delay deadline is not met. Since the transmitter has only a delayed CSIT, a HARQ with Chase combining is employed for error recovery. The problem is to decide the transmit-energy in each time-slot such that the timely-throughput is maximum for a given average transmit-energy constraint. We pose this problem as a constrained Markov decision process, and provide an optimum policy based on Lagrangian relaxation. The optimum Lagrangian multiplier is obtained using a subgradient method. We obtain the structure of the optimum policy, based on which we propose a computationally simple policy READER that requires no CSIT. We show that for large Doppler spread (or mobility), the timely-throughput of READER is close to the optimum policy. We also provide two more policies: an optimum policy assuming perfect CSIT with zero delay, and a naive randomization policy, and compare the throughput performance of the proposed policies. [ABSTRACT FROM AUTHOR]

In this paper, we consider deploying an intelligent reflecting surface (IRS) to enhance the downlink (DL) energy transfer and uplink (UL) information transmission efficiency for secure multiple-input multiple-output (MIMO) wireless powered communication networks (WPCNs). We aim to maximize the secrecy throughput of all users by jointly optimizing the DL/UL time allocation, the energy transmit covariance matrix of hybrid access point (AP), the information transmit beamforming matrix of users and the phase shifts of IRS in DL/UL, subject to constraints of energy/information transmit power at the hybrid AP/users and that of unit-modulus IRS phase shifts for DL/UL. To tackle the non-convex problem, we first transform the original problem into an equivalent form based on the mean-square error (MSE) method given time allocation, and then apply the alternating algorithm to update the optimization variables iteratively. Specifically, the energy covariance matrix and the information beamforming matrix are obtained based on the dual subgradient method. For the IRS phase shifts, we investigate two IRS beamforming reflection setups, namely different DL/UL IRS beamforming and identical DL/UL IRS beamforming. For the former case, the second-order cone programming technique and the Majorization-Minimization algorithm/element by element iterative algorithm are applied to obtain the DL and UL IRS phase shifts, respectively. For the latter case, the IRS phase shifts are obtained by the successive convex approximation technique. To further reduce the computational complexity of the single-user system, we derive the closed-form solutions of IRS phase shifts in each iteration for the two different reflection setups. Simulation results show that all the proposed algorithms can greatly improve the secrecy throughput compared to the conventional system without IRS. [ABSTRACT FROM AUTHOR]

In the context of a low-orbit mega constellation network, we consider the large-scale inter-satellite routing problem with time windows and capacity constraints (ISRPTWC) with the goal of minimizing the total consumption cost, including transmission, resource consumption, and other environmentally impacted costs. Initially, we develop an integer linear programming model for ISRPTWC. However, a difficult issue when solving ISRPTWC is how to deal with complex time window constraints and how to reduce congestion and meet transmission capacity. Along this line, we construct a three-dimensional time-space state network aiming to comprehensively enumerate the satellite network state at any moment in time and a task transmission route at any given time and further propose a time-discretized multi-commodity network flow model for the ISRPTWC. Then, we adopt a dynamic programming algorithm to solve the single-task ISRPTWC. By utilizing a Lagrangian relaxation algorithm, the primal multi-task routing problem is decomposed into a sequence of single-task routing subproblems, with Lagrangian multipliers for individual task route nodes and links being updated by a subgradient method. Notably, we devise a novel idea for constructing the upper bound of the ISRPTWC. Finally, a case study using illustrative and real-world mega constellation networks is performed to demonstrate the effectiveness of the proposed algorithm. [ABSTRACT FROM AUTHOR]

In the modern digital economy, optimal decision support systems, as well as machine learning systems, are becoming an integral part of production processes. Artificial neural network training as well as other engineering problems generate such problems of high dimension that are difficult to solve with traditional gradient or conjugate gradient methods. Relaxation subgradient minimization methods (RSMMs) construct a descent direction that forms an obtuse angle with all subgradients of the current minimum neighborhood, which reduces to the problem of solving systems of inequalities. Having formalized the model and taking into account the specific features of subgradient sets, we reduced the problem of solving a system of inequalities to an approximation problem and obtained an efficient rapidly converging iterative learning algorithm for finding the direction of descent, conceptually similar to the iterative least squares method. The new algorithm is theoretically substantiated, and an estimate of its convergence rate is obtained depending on the parameters of the subgradient set. On this basis, we have developed and substantiated a new RSMM, which has the properties of the conjugate gradient method on quadratic functions. We have developed a practically realizable version of the minimization algorithm that uses a rough one-dimensional search. A computational experiment on complex functions in a space of high dimension confirms the effectiveness of the proposed algorithm. In the problems of training neural network models, where it is required to remove insignificant variables or neurons using methods such as the Tibshirani LASSO, our new algorithm outperforms known methods. [ABSTRACT FROM AUTHOR]

The combination of backscatter communication with non-orthogonal multiple access (NOMA) has the potential to support low-powered massive connections in upcoming sixth-generation (6G) wireless networks. More specifically, backscatter communication can harvest and use the existing RF signals in the atmosphere for communication, while NOMA provides communication to multiple wireless devices over the same frequency and time resources. This paper has proposed a new resource management framework for backscatter-aided cooperative NOMA communication in upcoming 6G networks. In particular, the proposed work has simultaneously optimized the base station's transmit power, relaying node, the reflection coefficient of the backscatter tag, and time allocation under imperfect successive interference cancellation to maximize the sum rate of the system. To obtain an efficient solution for the resource management framework, we have proposed a combination of the bisection method and dual theory, where the sub-gradient method is adopted to optimize the Lagrangian multipliers. Numerical results have shown that the proposed solution provides excellent performance. When the performance of the proposed technique is compared to a brute-forcing search technique that guarantees optimal solution however, is very time-consuming, it was seen that the gap in performance is actually 0%. Hence, the proposed framework has provided performance equal to a cumbersome brute-force search technique while offering much less complexity. The works in the literature on cooperative NOMA considered equal time distribution for cooperation and direct communication. Our results showed that optimizing the time-division can increase the performance by more than 110% for high transmission powers. [ABSTRACT FROM AUTHOR]

We show that the vanishing step size subgradient method—widely adopted for machine learning applications—can display rather messy behavior even in the presence of favorable assumptions. We establish that convergence of bounded subgradient sequences may fail even with a Whitney stratifiable objective function satisfying the Kurdyka-Łojasiewicz inequality. Moreover, when the objective function is path-differentiable, we show that various properties all may fail to occur: criticality of the limit points, convergence of the sequence, convergence in values, codimension one of the accumulation set, equality of the accumulation and essential accumulation sets, connectedness of the essential accumulation set, spontaneous slowdown, oscillation compensation, and oscillation perpendicularity to the accumulation set. [ABSTRACT FROM AUTHOR]

We consider a general decomposable convex optimization problem. By using right-hand side allocation technique, it can be transformed into a collection of small dimensional optimization problems. The master problem is a convex non-smooth optimization problem. We propose to apply the exact non-smooth penalty method, which gives a solution of the initial problem under some fixed penalty parameter and provides the consistency of lower level problems. The master problem can be solved with a suitable non-smooth optimization method. The simplest of them is the custom subgradient projection method using the divergent series step-size rule without line-search, whose convergence may be, however, rather low. We suggest to enhance its step-size selection by using a two-speed rule. Preliminary results of computational experiments confirm efficiency of this technique. [ABSTRACT FROM AUTHOR]

In this paper, we extend a recently established subgradient method for the computation of Riemannian metrics that optimizes certain singular value functions associated with dynamical systems. This extension is threefold. First, we introduce a projected subgradient method which results in Riemannian metrics whose parameters are confined to a compact convex set and we can thus prove that a minimizer exists; second, we allow inexact subgradients and study the effect of the errors on the computed metrics; and third, we analyze the subgradient algorithm for three different choices of step sizes: constant, exogenous, and Polyak. The new methods are illustrated by application to dimension and entropy estimation of the H\'enon map. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new inertial subgradient projection algorithm for finding a solution of an equilibrium problem in a real Hilbert space. The algorithm combines subgradient projection methods with the self-adaptive and inertial techniques to generate an iteration sequence converging weakly to a solution of the equilibrium problem. Based on the proposed algorithm, we develop a modified algorithm for finding a common solution of the equilibrium problem and fixed point problems. Several of fundamental experiments are showed to illustrate our algorithms and compare with some others. [ABSTRACT FROM AUTHOR]

By employing time-varying proximal functions, adaptive subgradient methods (ADAGRAD) have improved the regret bound and been widely used in online learning and optimization. However, ADAGRAD with full matrix proximal functions (ADA-FULL) cannot handle large-scale problems due to the impractical $O(d^3)$ O (d 3) time and $O(d^2)$ O (d 2) space complexities, though it has better performance when gradients are correlated. In this paper, we propose two efficient variants of ADA-FULL via a matrix sketching technique called frequent directions (FD). The first variant named as ADA-FD directly utilizes FD to maintain and manipulate low-rank matrices, which reduces the space and time complexities to $O(\tau d)$ O (τ d) and $O(\tau ^2d)$ O (τ 2 d) respectively, where $d$ d is the dimensionality and $\tau \ll d$ τ ≪ d is the sketching size. The second variant named as ADA-FFD further adopts a doubling trick to accelerate FD used in ADA-FD, which reduces the average time complexity to $O(\tau d)$ O (τ d) while only doubles the space complexity of ADA-FD. Theoretical analysis reveals that the regret of ADA-FD and ADA-FFD is close to that of ADA-FULL as long as the outer product matrix of gradients is approximately low-rank. Experimental results demonstrate the efficiency and effectiveness of our algorithms. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and analyze two new inertial-like algorithms with the Bregman divergences for solving the pseudomonotone variational inequality problem in a real Hilbert space. The first algorithm is inspired by the Halpern-type iteration and the subgradient extragradient method and the second algorithm is inspired by the Halpern-type iteration and Tseng's extragradient method. Under suitable conditions, we prove some strong convergence theorems of the proposed algorithms without assuming the Lipschitz continuity and the sequential weak continuity of the given mapping. Finally, we give some numerical experiments with various types of Bregman divergence to illustrate the main results. In fact, the results presented in this paper improve and generalize the related works in the literature. [ABSTRACT FROM AUTHOR]

Parked vehicle-assisted multi-access edge computing (PVMEC) is a paradigm that exploits the under-utilized resources of parked vehicles (PVs) to assist MEC servers for offloaded task execution. This article investigates a partial offloading strategy for multi-user PVMEC, where each mobile device (MD)’s task can be partially offloaded to the MEC server or a nearby PV. We formulate the system utility maximization problem with joint consideration of offloading decisions, offloading ratio, and resource allocation. Considering the complexity and privacy issues of centralized scheme as well as the inherent resource contention among MDs, we adopt game-theoretic approach to devise a low-complexity distributed offloading scheme, in which the offloading decision-making problem is modeled as an exact potential game, and the optimal offloading ratio and resource allocation are determined using a subgradient method. Finally, simulation results show that our proposed approach can significantly improve the system utility over traditional baselines. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new algorithm for finding a solution of an equilibrium problem in a real Hilbert space. Our paper extends the single projection method to pseudomonotone variational inequalities, from a 2018 paper of Shehu et. al., to pseudomonotone equilibrium problems in a real Hilbert space. On the basis of the given algorithm for the equilibrium problem, we develop a new algorithm for finding a common solution of a equilibrium problem and fixed point problem. The strong convergence of the algorithm is established under mild assumptions. Several of fundamental experiments in finite (infinite) spaces are provided to illustrate the numerical behavior of the algorithm for the equilibrium problem and to compare it with other algorithms. [ABSTRACT FROM AUTHOR]

We consider a distributionally robust formulation of stochastic optimization problems arising in statistical learning, where robustness is with respect to ambiguity in the underlying data distribution. Our formulation builds on risk-averse optimization techniques and the theory of coherent risk measures. It uses mean–semideviation risk for quantifying uncertainty, allowing us to compute solutions that are robust against perturbations in the population data distribution. We consider a broad class of generalized differentiable loss functions that can be non-convex and non-smooth, involving upward and downward cusps, and we develop an efficient stochastic subgradient method for distributionally robust problems with such functions. We prove that it converges to a point satisfying the optimality conditions. To our knowledge, this is the first method with rigorous convergence guarantees in the context of generalized differentiable non-convex and non-smooth distributionally robust stochastic optimization. Our method allows for the control of the desired level of robustness with little extra computational cost compared to population risk minimization with stochastic gradient methods. We also illustrate the performance of our algorithm on real datasets arising in convex and non-convex supervised learning problems. [ABSTRACT FROM AUTHOR]

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In this paper, four modified subgradient extragradient algorithms are proposed for solving bilevel pseudomonotone variational inequality problems in real Hilbert spaces. The proposed algorithms can work adaptively without the prior knowledge of the Lipschitz constant of the pseudomonotone mapping. Strong convergence theorems for the suggested algorithms are established under suitable and mild conditions. Finally, some numerical experiments and applications are performed to verify the efficiency of the proposed algorithms with respect to some previously known ones. [ABSTRACT FROM AUTHOR]

In this paper, we propose an iterative algorithm by combining the selective projection method and the stochastic approximation method for solving a stochastic pseudomonotone variational inequality problem. Compared with many known related algorithms, the proposed algorithm uses a simpler adaptive rule to deal with the unknown Lipschitz constant of the involved mapping. The convergence is proved and convergence rate is estimated for the proposed algorithm. Finally, a numerical example is given to illustrate the effectiveness of the proposed algorithm. [ABSTRACT FROM AUTHOR]

In this work, we introduce two new iterative methods for finding the common element of the set of fixed points of a demicontractive mapping and the set of solutions of a pseudomonotone variational inequality problem in real Hilbert spaces. It is shown that the proposed algorithms converge strongly under mild conditions. The advantage of the proposed algorithms is that it does not require prior knowledge of the Lipschitz-type constants of the variational inequality mapping and only requires computing one projection onto a feasible set per iteration as well as without the sequentially weakly continuity of the variational inequality mapping. The novelty of the proposed algorithm may improve the efficiency of algorithms. Our results improve and extend some known results existing in the literature. [ABSTRACT FROM AUTHOR]

To address the resource allocation (RA) challenge in heterogeneous visible light communication (VLC) networks, this study develops an indoor VLC hybrid system comprising multiple VLC access points (APs), a single radio frequency (RF) access point, and a single infrared (IR) AP. A joint load balancing and power allocation strategy is proposed for the VLC/RF downlink, accompanied by an iterative algorithm and optimization framework tailored for the power allocation subproblem, which allocates power to each AP to maximize data throughput. The algorithm determines optimal power distribution through alternating solutions of dual variables. Furthermore, the effective capacity of a single user with varying quality of service (QoS) requirements in the IR uplink is examined, employing four distinct power control strategies: equal power allocation and Water-Filling algorithm, sub-channel independent optimization algorithm, and sub-channel joint optimization algorithm within simulations. Results indicate that this approach is agnostic to step size or initial variable values while offering enhanced convergence speed and performance compared to traditional subgradient methods. System capacity and fairness are notably improved alongside rapid convergence rates. Considering performance metrics based on system capacity, the sub-channel joint optimization algorithm is the optimal power control strategy. [ABSTRACT FROM AUTHOR]

We examine a coordinated capacitated lot-sizing problem for multiple product families, where demand is deterministic and time-varying. The problem considers set-up and holding costs, where capacity constraints limit the number of individual item and family set-up times and the amount of production in each period. Using a strong reformulation and relaxing the demand constraints, we improve both the upper and lower bounds using a combination of Benders decomposition and an evolutionary algorithm, followed by subgradient optimisation. Through computational experiments, we show that our method consistently achieves better bounds, reducing the duality gap compared to other single-family methods studied in the literature. [ABSTRACT FROM AUTHOR]

Effective online portfolio selection necessitates seamless integration of three key properties: diversity, sparsity, and risk control. However, existing algorithms often prioritize one property at the expense of the others due to inherent conflicts. To address this issue, we propose an asset subset-constrained minimax (ASCM) optimization framework, which generates optimal portfolios from diverse investment strategies represented as asset subsets. ASCM consists of: (i) a minimax optimization model that focuses on risk control by considering a set of loss functions constrained by different asset subsets; (ii) the construction of asset subsets via price-feature clipping, which effectively reduces redundant assets in the portfolio; (iii) a state-based estimation of price trends that guides all ASCM loss functions, facilitating the generation of sparse solutions. We solve the ASCM minimax model using an efficient iterative updating formula derived from the projected subgradient method. Furthermore, we achieve near O (1) time complexity through a novel initialization scheme. Experimental results demonstrate that ASCM outperforms eight representative algorithms, including the best constant rebalanced portfolio in hindsight (BCRP) on five out of the six real-world financial datasets. Notably, ASCM achieves a 67-fold improvement over BCRP in cumulative wealth on the TSE dataset. • An extensible framework for integrating diversity, sparsity, and risk control. • A minimax model for risk control of loss functions constrained by asset subsets. • Construct asset subsets by price-feature clipping to reduce redundant assets. • State-based estimation of price trends to guide all loss functions. [ABSTRACT FROM AUTHOR]

In this paper, we consider distributed quantile regression over a decentralized network, which is an example of a popularly used non-smooth statistical model. Somewhat different from the viewpoint of the optimization literature, we jointly consider numerical convergence and statistical convergence when the objective function is the loss function constructed from i.i.d. data, and the convergence to the true population parameter (instead of the minimizer of the empirical risk) is emphasized. Distributed sub-gradient methods with and without gradient tracking are considered, with an emphasis on the former. With gradient tracking, the estimate linearly converges to the true parameter up to the statistical precision based on all data, when the local sample size is sufficiently large. The distinction of our result compared to the existing optimization literature is that linear convergence can be established despite the fact that the function is not strongly convex. Numerical illustrations are given to compare with distributed descent algorithm without gradient tracking, and compare the ATC (adapt-then-combine) version with the non-ATC one. • Establish linear convergence of decentralized estimation for nonsmooth quantile regression. • Several variants of decentralized gradients methods are compared. • Numerical demonstrate the performance in comparison with ADMM approach. [ABSTRACT FROM AUTHOR]

In this paper, we introduce an efficient subgradient extragradient (SE) based method for solving variational inequality problems with monotone operator in Hilbert space. In many existing SE methods, two values of operator are needed over each iteration and the Lipschitz constant of the operator or linesearch is required for estimating step sizes, which are usually not practical and expensive. To overcome these drawbacks, we present an inertial SE based algorithm with adaptive step sizes, estimated by using an approximation of the local Lipschitz constant without running a linesearch. Each iteration of the method only requires a projection on the feasible set and a value of the operator. The numerical experiments illustrate the efficiency of the proposed algorithm. [ABSTRACT FROM AUTHOR]

Heterogeneous cellular networks are a viable solution in response to the growing demand for broadband services in the new-generation wireless networks. The dense deployment of small cell networks is a key feature of next-generation heterogeneous networks aimed at providing the necessary capacity increase. However, the approach to apply green networks is very important especially in the downlink because uncontrolled deployment of too many small-cells may increase operational costs and emit more carbon dioxide. In addition, given the novel services and resource limitation of the user layer, energy efficiency and fairness assurance are critical issues in the uplink. Considering the uplink fairness criterion, this paper proposes a dynamic optimization model which maximizes the total uplink/downlink energy efficiency in addition to providing the essential coverage and capacity of heterogeneous cellular networks. Based on the non-convex characteristics of the energy efficiency maximization model, the mathematical model can be formulated to two subproblems, i.e., resource optimization and user association. So that, a subgradient method is applied for fair resource management and also successive convex approximation and dual decomposition methods are adopted to solve the proportional fairness problem. The simulation results exhibit considerable throughput increase by 30% and 22% on average for random and hotspot user distributions, respectively. It also proved that the proposed approach managed to significantly improve the total network energy efficiency by up to 35%. [ABSTRACT FROM AUTHOR]

In this paper, several extragradient algorithms with inertial effects and adaptive non-monotonic step sizes are proposed to solve pseudomonotone variational inequalities in real Hilbert spaces. The strong convergence of the proposed methods is established without the prior knowledge of the Lipschitz constant of the mapping. Some numerical experiments are given to illustrate the advantages and efficiency of the proposed schemes over previously known ones. [ABSTRACT FROM AUTHOR]

The paper develops a modified inertial subgradient extragradient method to find a solution to the variational inequality problem over the set of common solutions to the variational inequality and null point problems. The proposed method adopts a nonmonotonic stepsize rule without any linesearch procedure. We describe how to incorporate the regularization technique and the subgradient extragradient method; then, we establish the strong convergence of the proposed method under some appropriate conditions. Several numerical experiments are also provided to verify the efficiency of the introduced method with respect to previous methods. [ABSTRACT FROM AUTHOR]

This study studies the problem of efficient multimedia data acquisition and decreasing whole energy expenditure of wireless multimedia sensor networks and proposes a three-step multimedia data acquisition and wireless energy supplement strategy. Firstly, for network partition, this study proposes a network partition scheme based on vicinity likeness and distance of sensor nodes (VLD), which divides the whole sensor network into multiple regions. The physical links inside the region are dense and concentrated, while the link connections between regions are sparse. Disconnecting the connections between regions hardly affects the data transmission of sensor nodes. Secondly, this study proposes an efficient data acquisition and processing scheme for wireless multimedia sensor network ASS. Compared with other anchor selection schemes, this scheme has obvious performance advantages. Then, the problem of minimizing network energy expenditure is defined, and the optimal sensor node data perception rate and network link transmission rate of the optimization function are obtained by dual decomposition and sub-gradient method. Finally, in the case of a given network energy threshold, the performance of the overall strategy in this study is verified by comparing the amount of data collected by the base station. [ABSTRACT FROM AUTHOR]