Subgradient Method Research Articles

Adaptive Resource Allocation in SWIPT-Enabled Cognitive IoT Networks

Integrating simultaneous wireless information and power transfer (SWIPT) and cognitive radio (CR) technologies into Internet-of-Things (IoT) networks, named SWIPT-enabled cognitive IoT networks, has become an effective approach to resolve the short lifetime of battery-constrained IoT Devices (IoDs) and spectrum scarcity. In this type of networks, IoDs are regarded as secondary users (SUs) being charged with wireless power. To improve the sum throughput of IoDs, we allow IoDs to switch among spectrum sensing, SWIPT and information transmission adaptively. Correspondingly, three-dimensional resources, i.e., time (for performing the three actions), power (including power transmitted from an IoT controller to each IoD and power for receiving information and charging at each IoD) and spectrum, are jointly and adaptively allocated to maximize the sum throughput of IoDs. Since the formulated problem is a mixed-integer nonlinear program (MINLP), we adopt an auxiliary variable to convert the original problem into a tractable problem, which is then solved by an efficient algorithm involving the Lagrangian dual method, the subgradient method and the multiple one-dimensional search algorithm. Simulation results show our adaptive design yields superior performance in terms of the sum throughput of IoDs.

Modified mann-type inertial subgradient extragradient methods for solving variational inequalities in real Hilbert spaces

This paper is to investigate the monotone variational inequalities (VIPs) in real Hilbert spaces. We constructed two iterative algorithms based on subgradient extragradient algorithms and Tseng?s algorithms for solving VIPs. Convergence analysis of the suggested methods are proved. Several numerical examples to illustrate the efficiency of the methods are given.

Superiorization with a projected subgradient method

Superiorization with a projected subgradient method

GPU-Accelerated Partially Linear Multiuser Detection for 5G and Beyond URLLC Systems

In this feasibility study, we have implemented a recently proposed partially linear multiuser detection algorithm in reproducing kernel Hilbert spaces (RKHSs) on a GPU-accelerated platform. Partially linear multiuser detection, which combines the robustness of linear detection with the power of nonlinear methods, has been proposed for a massive connectivity scenario with the non-orthogonal multiple access (NOMA). This is a promising approach, but detecting payloads within a received orthogonal frequency-division multiplexing (OFDM) radio frame requires the execution of a large number of inner product operations, which are the main computational burden of the algorithm. Although inner-product operations consist of simple kernel evaluations, their vast number poses a challenge in ultra-low latency (ULL) applications, because the time needed for computing the inner products might exceed the sub-millisecond latency requirement. To address this problem, this study demonstrates the acceleration of the inner-product operations through massive parallelization. The result is a GPU-accelerated real-time OFDM receiver that enables sub-millisecond latency detection to meet the requirements of 5th generation (5G) and beyond ultra-reliable and low latency communications (URLLC) systems. Moreover, the parallelization and acceleration techniques explored and demonstrated in this study can be extended to many other signal processing algorithms in Hilbert spaces, such as those based on projection onto convex sets (POCS) and adaptive projected subgradient method (APSM) algorithms. Experimental results and comparisons with the state-of-art confirm the effectiveness of our techniques.

Multiple-sets split quasi-convex feasibility problems: Adaptive subgradient methods with convergence guarantee

Multiple-sets split quasi-convex feasibility problems: Adaptive subgradient methods with convergence guarantee

Hybrid subgradient method for pseudomonotone equilibrium problem and fixed points of relatively nonexpansive mappings in Banach spaces

In this paper, we propose a new hybrid subgradient algorithm for finding a common point in the set of solutions of pseudomonotone equilibrium problem and the set of fixed points of relatively nonexpansive mapping in a real uniformly convex and uniformly smooth Banach spaces. Weak and strong convergence of the iterative scheme are established. Our results generalizes and improves several recent results in the literature.

Large-Scale Maintenance and Unit Commitment: A Decentralized Subgradient Approach

Unit Commitment (UC) is a fundamental problem in power system operations. When coupled with generation maintenance, the joint optimization problem poses significant computational challenges due to coupling constraints linking maintenance and UC decisions. Obviously, these challenges grow with the size of the network. With the introduction of sensors for monitoring generator health and condition-based maintenance(CBM), these challenges have been magnified. ADMM-based decentralized methods have shown promise in solving large-scale UC problems, especially in vertically integrated power systems. However, in their current form, these methods fail to deliver similar computational performance and scalability when considering the joint UC and CBM problem. This paper provides a novel decentralized optimization framework for solving large-scale, joint UC and CBM problems. Our approach relies on the novel use of the subgradient method to temporally decouple various subproblems of the ADMM-based formulation of the joint problem along the maintenance horizon. By effectively utilizing multithreading, our decentralized subgradient approach delivers superior computational performance and eliminates the need to move sensor data thereby alleviating privacy and security concerns. Using experiments on large scale test cases, we show that our framework can provide a speedup of upto 50x as compared to various state of the art benchmarks without compromising on solution quality.

Assimilation of Images via Dictionary Learning-Based Sparsity Regularization Strategy: An Application for Retrieving Fluid Flows

In this work, we propose a structure sparsity regularization strategy in the framework of 4-D variational data assimilation (4-D Var). In meteorology and oceanography, the number of unknown model variables is far fewer than that of image observations, often leading to solve an underdetermined nonlinear inverse problem. In recent years, the l¹-norm-based sparsity regularization approach has attracted great attention in the field of 4-D Var because of its data structure preservation and noise suppression. To avoid little underlying physical priors considered, we introduce a widely used dictionary learning (DL) method to adaptively derive an efficient sparse approximation via learning a basis from a given dataset. For our target application of estimating sea surface flows, we consider a DL sparsity constraint on the variable of flow vorticity due to its rich spatial variation related to flows evolution. A novel anisotropic regularization method combined with fluid dynamics characteristics could overcome magnitude underestimation and staircase artifacts appearing in the gradient regularization-based 4-D Var method. The split Bregman iteration with fast convergence property is employed to solve the l¹+l² nonsmooth minimization problem. The promising fluid flows estimation performance in real test cases (assimilation of image sequences collected from CORIOLIS experimental turntable) demonstrates the efficiency of our approach.

An inertial parallel CQ subgradient extragradient method for variational inequalities application to signal-image recovery

In this paper, we introduce an inertial parallel CQ subgradient extragradient method for finding a common solutions of variational inequality problems. The novelty of this paper is using linesearch methods to find unknown L constant of L-Lipschitz continuous mappings. Strong convergence theorem has been proved under some suitable conditions in Hilbert spaces. Finally, we show applications to signal and image recovery, and show the good efficiency of our proposed algorithm when the number of subproblems is increasing.

On the Problem of Planning of Multi-Product Flows and Modernization of the Transportation Network

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. Keywords: optimal planning, mathematical model, subgradient methods, dual problem, software.

Inertial hybrid self-adaptive subgradient extragradient method for fixed point of quasi$-\phi-$nonexpansive multivalued mappings and equilibrium problem

In this paper, we propose a new inertial self-adaptive subgradient extragradient algorithm for approximating common solution in the set of pseudomonotone equilibrium problems and the set of fixed point of finite family of quasi$-\phi-$nonexpansive multivalued mappings in real uniformly convex Banach spaces and uniformly smooth Banach spaces. The step size n is chosen self adaptively and estimates of Lipschizt-type constants are dispensed with. Strong convergence of the iterative scheme is established. Our results generalizes and improves several recent results anouced in the literature.

An application of Newton-like algorithm for H∞ proportional–integral–derivative controller synthesis of seesaw-cart system

This paper is concerned with the synthesis of proportional–integral–derivative (PID) controller according to the [Formula: see text] optimality criterion for seesaw-cart system. The equations of dynamics are obtained through modelling a seesaw-cart system actuated by direct-current motor via rack and pinion mechanism using the Euler–Lagrange approach. The obtained model is linearised and synthesis of the PID controller for linear model is performed. An algorithm based on the sub-gradient method, the Newton method, the self-adapting backpropagation algorithm and the Adams method is proposed to calculate the PID controller gains. The proposed control strategy is tested and compared with standard linear matrix inequality (LMI)-based method on computer simulations and experimentally on a laboratory model.

On Mann-Type Subgradient-like Extragradient Method with Linear-Search Process for Hierarchical Variational Inequalities for Asymptotically Nonexpansive Mappings

We propose two Mann-type subgradient-like extra gradient iterations with the line-search procedure for hierarchical variational inequality (HVI) with the common fixed-point problem (CFPP) constraint of finite family of nonexpansive mappings and an asymptotically nonexpansive mapping in a real Hilbert space. Our methods include combinations of the Mann iteration method, subgradient extra gradient method with the line-search process, and viscosity approximation method. Under suitable assumptions, we obtain the strong convergence results of sequence of iterates generated by our methods for a solution to HVI with the CFPP constraint.

Two Nonmonotonic Self-Adaptive Strongly Convergent Projection-Type Methods for Solving Pseudomonotone Variational Inequalities

The primary objective of this study is to introduce two novel extragradient-type iterative schemes for solving variational inequality problems in a real Hilbert space. The proposed iterative schemes extend the well-known subgradient extragradient method and are used to solve variational inequalities involving the pseudomonotone operator in real Hilbert spaces. The proposed iterative methods have the primary advantage of using a simple mathematical formula for step size rule based on operator information rather than the Lipschitz constant or another line search method. Strong convergence results for the suggested iterative algorithms are well-established for mild conditions, such as Lipschitz continuity and mapping monotonicity. Finally, we present many numerical experiments that show the effectiveness and superiority of iterative methods.

Two adaptive modified subgradient extragradient methods for bilevel pseudomonotone variational inequalities with applications

We consider the bilevel variational inequality problem with a pseudomonotone operator in real Hilbert spaces and investigate two modified subgradient extragradient methods with inertial terms. Our first scheme requires the operator to be Lipschitz continuous (the Lipschitz constant does not need to be known) while the second one only requires it to be uniformly continuous. The proposed methods employ two adaptive stepsizes making them work without the prior knowledge of the Lipschitz constant of the mapping. The strong convergence properties of the iterative sequences generated by the proposed algorithms are obtained under mild conditions. Some numerical tests and applications are given to demonstrate the advantages and efficiency of the stated schemes over previously known ones.

Parallel subgradient algorithm with block dual decomposition for large-scale optimization

This paper studies computational approaches for solving large-scale optimization problems using a Lagrangian dual reformulation, solved by parallel sub-gradient methods. Since there are many possible reformulations for a given problem, an important question is: Which reformulation leads to the fastest solution time? One approach is to detect a block diagonal structure in the constraint matrix, and reformulate the problem by dualizing the constraints outside of the blocks; the approach is defined herein as block dual decomposition. Main advantage of such a reformulation is that the Lagrangian relaxation has a block diagonal constraint matrix, thus decomposable into smaller sub-problems that can solved in parallel. We show that the block decomposition can critically affect convergence rate of the sub-gradient method. We propose various decomposition methods that use domain knowledge or apply algorithms using knowledge about the structure in the constraint matrix or the dependence in the decision variables, towards reducing the computational effort to solve large-scale optimization problems. In particular, we introduce a block decomposition approach that reduces the number of dualized constraints by utilizing a community detection algorithm. We present empirical experiments on an extensive set of problem instances including a real application. We illustrate that if the number of the dualized constraints in the decomposition increases, the computational effort within each iteration of the sub-gradient method decreases while the number of iterations required for convergence increases. The key message is that it is crucial to employ prior knowledge about the structure of the problem when solving large scale optimization problems using dual decomposition.

Distributed real-time optimal control of central air-conditioning systems

Real-time optimal control (RtOpt) is an efficient tool to improve the energy efficiency of heating, ventilating and air-conditioning (HVAC) systems. RtOpt is always centralized and solved entirely in building energy management systems (BEMSs). However, centralized optimization is not be an efficient solution because real-time operation data in local loops need to be sent to the central computation unit in BEMS for optimization. This paper, therefore, proposes a distributed RtOpt scheme for typical centralized HVAC systems, which uses a dual decomposition mechanism to break the centralized problem into one mater problem and several sub-problems that can be solved in parallel and independently. The mater problem is converted to a simple mean value calculation using a projected sub-gradient method, thus distributing the computation load from the central computation unit into local computation units. Unlike available decomposition schemes in literature, the proposed method considered the functionality and spatial distribution of the major components of HVAC systems, which could enhance the physical meaning and facilitate the application of the decomposition scheme. The proposed distributed RtOpt scheme was tested in case studies, with numerical analysis being used to demonstrate its potential benefits.

QoS Violation Probability Minimization in Federating Vehicular-Fogs With Cloud and Edge Systems

Due to the fact that most user equipments (UEs) do not have enough computation power, time-sensitive tasks should be offloaded to other computation resources. In such cases, a cloud is an ideal destination to which computation tasks can be offloaded because of its huge data processing power. However, the distance between a UE and the cloud may increase the latency of data transmission. At the same time, edges and vehicular- fogs (consisting of electric vehicles with computing resources) are closer to UEs and mostly remain under-utilized. To address this issue, a federated architecture with UEs, vehicular-fogs (parking lot fog and traffic intersection fog), edges, and cloud is proposed along with a probabilistic offloading strategy. The quality of service (QoS) violation probability can be minimized by finding the optimal offloading probabilities while satisfying the delay constraint through an optimal offloading probability estimation algorithm based on the subgradient method proposed in this pa- per. Our results show that the QoS violation probability and the average waiting time can be decreased by 10%12% and 45%, respectively, for a federated architecture with two vehicular-fogs as compared to an architecture without any fogs. As offloading to edges and cloud has longer communication delays than offloading to vehicular-fogs, the offloading probabilities to the edges in the architecture with two vehicular-fogs in the case of heavy traffic are reduced by nearly 35% as compared to the architecture without any fogs.

Modified inertial subgradient extragradient method for equilibrium problems

Abstract The subgradient extragradient method with inertial extrapolation step x n + θ n (x n − x n−1) (also known as inertial subgradient extragradient method) has been studied extensively in the literature for solving variational inequalities and equilibrium problems. Most of the inertial subgradient extragradient methods in the literature for both variational inequalities and equilibrium problems have not considered the special case when the inertial factor θ n = 1. The convergence results have always been obtained when the inertial factor θ n is assumed 0 ≤ θ n < 1. This paper considers the relaxed inertial version of subgradient extragradient method for equilibrium problems with 0 ≤ θ n ≤ 1. We give both weak and strong convergence results using this inertial subgradient extragradient method and also give some numerical illustrations.

Decentralized blockchain-based peer-to-peer energy-backed token trading for active prosumers

This paper designs and models a fully decentralized peer-to-peer (P2P) energy token market for small-scale prosumers using blockchain technology in a smart grid environment in the presence of the demand response (DR) program and demurrage mechanism. As the market players, prosumers in the local distribution network are considered in two groups, producers (sellers) and consumers (buyers). Using smart contracts, all sellers and buyers in the proposed market can engage in bilateral energy token transactions (P2P) with each other under an agreed price and with the retail market at a certain price. Furthermore, the local consumers can participate in price-based DR programs and shift their consuming load to the periods with high local generation. Demurrage mechanism is applied to avoid energy token accumulation and enhance attraction for local transactions. With demurrage in place, the redemption value of energy-backed tokens reduces with time. A fully decentralized approach called the primal-dual sub-gradient method is developed to clear this fully decentralized energy market in the presence of DR programs and demurrage. The proposed market-clearing scheme guarantees the global and feasible solution without requiring the players' private information. Numerical studies demonstrated the feasibility and effectiveness of the proposed energy token market and the decentralized approach for its clearing.