Gradient Projection Algorithm Research Articles

UAV Positioning and Power Control for Two-Way Wireless Relaying

This paper considers an unmanned-aerial-vehicle-enabled (UAV-enabled) wireless network where a relay UAV is used for two-way communications between a ground base station (BS) and a set of distant user equipment (UE). The UAV adopts the amplify-and-forward strategy for two-way relaying over orthogonal frequency bands. The UAV positioning and the transmission powers of all nodes are jointly designed to maximize the sum rate of both uplink and downlink subject to transmission power constraints and the signal-to-noise ratio constraint on the UAV control channel. The formulated joint positioning and power control (JPPC) problem has an intricate expression of the sum rate due to two-way transmissions and is difficult to solve in general. We propose a novel concave surrogate function for the sum rate and employ the successive convex approximation (SCA) technique for obtaining a high-quality approximate solution. We show that the proposed surrogate function has a small curvature and enables a fast convergence of SCA. Furthermore, we develop a computationally efficient JPPC algorithm by applying the FISTA-type accelerated gradient projection (AGP) algorithm to solve the SCA problem as well as one of the projection subproblem, resulting in a double-loop AGP method. Simulation results show that the proposed JPPC algorithms are not only computationally efficient but also greatly outperform the heuristic approaches.

Strong convergence result for solution of stochastic convex minimization problem generated by generalized random iterative scheme

Stochastic approximation procedures have been extensively used in solving stochastic optimization problems. Gradient projection algorithm (GPA) plays an important role in solving constrained convex minimization problems. In this paper, strong convergence result for approximate solution of a constrained stochastic convex minimization problem in Hilbert space is proved. A generalized random iterative scheme based on skewed gradient projection algorithm and a strictly pseudo-contractive mapping is used to generate the solution.

A penalty algorithm for solving convex separable knapsack problems

In this paper, we propose a penalized gradient projection algorithm for solving the continuous convex separable knapsack problem, which is simpler than existing methods and competitive in practice. The algorithm only performs function and gradient evaluations, sums, and updates of parameters. The relatively complex task of the algorithm, which consists in minimizing a function in a compact set, is given by a closed formula. The convergence of the algorithm is presented. Moreover, to demonstrate its efficiency, illustrative computational results are presented for medium-sized problems.

Global convergence via descent modified three-term conjugate gradient projection algorithm with applications to signal recovery

In this article, we propose a three-term conjugate gradient projection algorithm for solving constrained monotone nonlinear equations. The global convergence of the algorithm was established under suitable assumptions. Numerical examples presented indicate that the algorithm has a very good performance in solving monotone nonlinear equations. Finally, the algorithm is applied to solve signal recovery problems.

Regularized Jacobi Iteration for Decentralized Convex Quadratic Optimization With Separable Constraints

We consider multiagent, convex quadratic optimization programs subject to separable constraints, where the constraint function of each agent involves only its local decision vector, while the decision vectors of all agents are coupled via a common objective function. We focus on a regularized variant of the so-called Jacobi algorithm for decentralized computation in such problems. We provide a fixed-point theoretic analysis showing that the algorithm converges to a minimizer of the centralized problem under more relaxed conditions on the regularization coefficient from those available in the literature, and in particular with respect to scaled projected gradient algorithms. The efficacy of the proposed algorithm is illustrated by applying it to the problem of optimal charging of electric vehicles.

Flexible Array Response Control via Oblique Projection

This paper presents a flexible array response control algorithm via oblique projection, abbreviated as FARCOP, and its application to array pattern synthesis. The proposed FARCOP algorithm stems from the adaptive array theory, and it can flexibly, precisely and simultaneously adjust the array response levels at multiple angles based on an arbitrarily given weight vector. Different from the existing approaches, the proposed FARCOP algorithm controls multi-point responses by linearly transferring the given weight vector, with a transformation matrix containing a set of parameters, each of which can be very easily determined by the desired response level (at the control angle). Owing to the fact that those parameters are independent of each other, the response levels at the control angles can be either individually or jointly and, therefore, flexibly adjusted. Since the parameter phases can be arbitrary, we take the beampattern into account and propose to uniquely choose the optimal parameters under the typical criterion of maximum white noise gain (WNG). Accordingly, a gradient projection (GP) algorithm is devised to achieve the optimal solution. Moreover, a closed-form solution is derived for the centro-symmetric array. In addition, the application of the FARCOP algorithm to pattern synthesis is discussed. Comparing to the state-of-the-art methods like multi-point accurate array response control (MA <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> RC), the proposed FARCOP algorithm controls the array responses more flexibly with lower computational complexity. Representative examples are presented to demonstrate the effectiveness and superiority of the FARCOP algorithm under various situations.

Strong convergence results for convex minimization and monotone variational inclusion problems in Hilbert space

In this paper, we propose a new modification of the Gradient Projection Algorithm and the Forward–Backward Algorithm. Using our proposed algorithms, we establish two strong convergence theorems for solving convex minimization problem, monotone variational inclusion problem and fixed point problem for demicontractive mappings in a real Hilbert space. Furthermore, we apply our results to solve split feasibility and optimal control problems. We also give two numerical examples of our algorithm in real Euclidean space of dimension 4 and in an infinite dimensional Hilbert space, to show the efficiency and advantage of our results.

New iterative methods for equilibrium and constrained convex minimization problems

The gradient-projection algorithm (GPA) plays an important role in solving constrained convex minimization problems. In this paper, we combine the GPA and averaged mapping approach to propose implicit and explicit composite iterative schemes for finding a common solution of an equilibrium problem and a constrained convex minimization problem. Then, we prove some strong convergence theorems which improve and extend some recent results.

A Priori Error Analysis for Time-Stepping Discontinuous Galerkin Finite Element Approximation of Time Fractional Optimal Control Problem

In this paper a priori error analysis for time-stepping discontinuous Galerkin finite element approximation of optimal control problem governed by time fractional diffusion equation is presented. A time-stepping discontinuous Galerkin finite element method and variational discretization approach are used to approximate the state and control variables respectively. Regularity of the optimal control problem is discussed. Since the time fractional derivative is nonlocal, in order to reduce the computational cost a fast gradient projection algorithm is designed for the control problem based on the block triangular Toeplitz structure of the discretized state equation and adjoint state equation. Numerical examples are carried out to illustrate the theoretical findings and fast algorithm.

A new local path planning approach based on improved dual covariant Hamiltonian optimization for motion planning method

We propose a new local path planning approach based on optimization methods with probabilistic completeness in this article. This approach adds a linear constraint to the original covariant Hamiltonian optimization for motion planning problem with a new cost function. By deducing the dual form, the path planning problem is described as a box-constrained quadratic programming problem. The nonmonotone gradient projection algorithm is introduced to solve the dual problem, which makes the algorithm adaptable to non-convex cost functions. In order to prevent early convergence at local minima that can occur when applying optimization methods, this article introduces Hamiltonian Monte Carlo to the modification, which constantly forces the initial path to jump out of the local extremum, thus improving the robustness and success rate of the path planning approach. Compared with other methods through simulations, this approach is proven to provide balanced planning efficiency and path quality. The feasibility in a real environment is experimentally validated by applying the approach to a wheeled mobile robot.

A new optimization-driven path planning method with probabilistic completeness for wheeled mobile robots

Wheeled mobile robots are widely utilized for environment-exploring tasks both on earth and in space. As a basis for global path planning tasks for wheeled mobile robots, in this study we propose a method for establishing an energy-based cost map. Then, we utilize an improved dual covariant Hamiltonian optimization for motion planning method, to perform point-to-region path planning in energy-based maps. The method is capable of efficiently handling high-dimensional path planning tasks with non-convex cost functions through applying a robust active set algorithm, that is, non-monotone gradient projection algorithm. To solve the problem that the path planning process is locked in weak minima or non-convergence, we propose a randomized variant of the improved dual covariant Hamiltonian optimization for motion planning based on simulated annealing and Hamiltonian Monte Carlo methods. The results of simulations demonstrate that the final paths generated can be time efficient, energy efficient and smooth. And the probabilistic completeness of the method is guaranteed.

The Combined Distribution and Assignment Model: A New Solution Algorithm and Its Applications in Travel Demand Forecasting for Modern Urban Transportation

With the development of the advanced Intelligent Transportation System (ITS) in modern cities, it is of great significance to upgrade the forecasting methods for travel demand with the impact of ITS. The widespread use of ITS clearly changes the urban travelers’ behavior at present, in which case it is difficult for the conventional four-step travel demand forecasting model to have good performance. In this study, we apply the combined distribution and assignment (CDA) model to forecasting travel demand for modern urban transportation, in which travelers may choose the destination and path simultaneously. Furthermore, we present a new solution algorithm for solving the CDA model. With the network representation method that converts the CDA model into a standard traffic assignment problem (TAP), we develop a new path-based algorithm based on the gradient projection (GP) algorithm to solve the converted CDA model. The new solution algorithm is designed to find a more accurate solution compared with the widely used algorithm, the Evans’ two-stage algorithm. Two road networks, Sioux Falls and Chicago Sketch, are used to verify the performance of the new algorithm. Also, we conduct some experiments on the Sioux Falls network to illustrate several applications of the CDA model in consideration of the influences of ITS.

Linear convergence of the relaxed gradient projection algorithm for solving the split equality problems in Hilbert spaces

In this paper, we consider the relaxed gradient projection algorithm to solve the split equality problem in Hilbert spaces, and we investigate its linear convergence. In particular, we use the concept of the bounded linear regularity property for the split equality problem to prove the linear convergence property for the above algorithm. Furthermore, we conclude the linear convergence rate of the relaxed gradient projection algorithm. Finally, some numerical experiments are given to test the validity of our results.

Fractional spectral collocation method for optimal control problem governed by space fractional diffusion equation

In this paper, we mainly investigate the fractional spectral collocation discretization of optimal control problem governed by a space-fractional diffusion equation. Existence and uniqueness of the solution to optimal control problem is proved. The continuous first order optimality condition is derived. The eigenfunctions of two classes of fractional Strum–Liouville problems are used as basis functions to approximate state variable and adjoint state variable, respectively. The fractional spectral collocation scheme for the control problem is constructed based on ‘first optimize, then discretize’ approach. Note that the solutions of fractional differential equations are usually singular near the boundary, a generalized fractional spectral collocation scheme for the control problem is proposed based on ‘first optimize, then discretize’ approach. A projected gradient algorithm is designed based on the discrete optimality condition. Numerical experiments are carried out to verify the effectiveness of the proposed numerical schemes and algorithm.

A new nonconvex approach for image restoration with Gamma noise

In this paper, a new total generalized variational(TGV) model for restoring images with multiplicative noise is proposed, which contains a nonconvex fidelity term and a TGV term. We use a difference of convex functions algorithm (DCA) to deal with the proposed model. For multiplicative noise removal, there exist many models and algorithms, most of which focus on convex approximation so that numerical algorithms with guaranteed convergence can be designed. Unlike these algorithms, we use the DCA algorithm to remove multiplicative noise. By numerical experiments, it is shown that the proposed approach leads to a better solution compared with the gradient projection algorithm for solving the classic multiplicative noise removal models. We prove that the sequence generated by the DCA algorithm converges to a stationary point, which satisfies the first order optimality condition. Finally, we demonstrate the performance of our whole scheme by numerical examples. A comparison with other methods is provided as well. Numerical results demonstrate that the proposed algorithm significantly outperforms some previous methods for multiplicative Gamma noise removal.

Energy Efficiency Beamforming Design for UAV Communications With Broadband Hybrid Polarization Antenna Arrays

Stringent physical resource constraints in unmanned aerial vehicle (UAV) communications bring new challenges to energy efficient transmission. In this paper, we study energy efficient beamforming for UAV communications, where a wideband hybrid polarized antenna array with tapped-delay-lines (TDLs) structure is deployed on the UAV. By exploiting hybrid structured, TDLs, and polarized antenna arrays, the strict requirements of hardware size and cost on UAV can be satisfied. It is shown that the energy efficient beamforming with the adopted array can be formulated as a WMMSE-based parameter optimization problem, which however is still a challenging nonconvex problem due to unit modulus constraints in analog beamformer. To overcome the nonconvex nature, an iterative algorithm is proposed, in which the optimization problem is iteratively solved through three convex quadratic programming problems with convex equality or inequality constraints. Finally, the numerical results demonstrate that the proposed wideband beamforming algorithm can significantly improve the energy efficiency, compared with the gradient projection algorithm.

Projected Gradient and Model Predictive Control : Optimal Energy and Pollutants Management for Hybrid Electric Vehicle

Abstract This article aims to propose a real-time compatible strategy for the energy management of a compression-ignition Hybrid Electric Vehicle (HEV), under pollutants emissions constraints. First, an offline method is applied in order to obtain a performance reference. Then, an online-oriented approach based on Model Predictive Control (MPC) is described, and a projected gradient algorithm used to solve the successive optimal control problems is explicitly presented. Both approaches are compared through simulation and the results are validated through experimentation.

A fast gradient projection algorithm for time fractional optimal control problem

In this paper a fast gradient projection algorithm for optimal control problems governed by time fractional diffusion equation is developed. The state equation is discretized by piecewise linear FE for space variable and L1 scheme for time variable, while the control variable is approximated by variational discretization. Based on the block triangular Toeplitz structure of the coefficient matrix of the discretized state equation and adjoint state equation, a fast gradient projection algorithm is designed for the control problem. Numerical examples are carried out to illustrate the effectiveness of the algorithms.

Nonblind Image Deblurring Based on Bi-Composition Decomposition by Local Smoothness and Nonlocal Self-Similarity Priors

Image deblurring is a classical inverse problem in image processing and computer vision. The vital task is to construct the proper image prior model to obtain the high-quality restored image with salient edges and rich details. A new nonblind image deblurring method by combining local smoothness and nonlocal self-similarity of natural images in the regularization framework is proposed. First, the observed image is decomposed into two components: structure component and detail component by a global gradient extraction scheme. Second, the four-directional anisotropic total variation regularization satisfying the local smoothness property is adopted for the structure component, and a new nonlocal statistical modeling for self-similarity is used for the detail component, respectively. At last, the split Bregman-based iteration algorithm and four-directional fast gradient projection algorithm are introduced to optimize the proposed L 1 -regularized problem. The extensive experiments demonstrate the efficiency and viability of the proposed method for preserving salient edges and texture details while alleviating the artifacts.

Stochastic Block-Coordinate Gradient Projection Algorithms for Submodular Maximization

We consider a stochastic continuous submodular huge-scale optimization problem, which arises naturally in many applications such as machine learning. Due to high-dimensional data, the computation of the whole gradient vector can become prohibitively expensive. To reduce the complexity and memory requirements, we propose a stochastic block-coordinate gradient projection algorithm for maximizing continuous submodular functions, which chooses a random subset of gradient vector and updates the estimates along the positive gradient direction. We prove that the estimates of all nodes generated by the algorithm converge to some stationary points with probability 1. Moreover, we show that the proposed algorithm achieves the tight (pmin/2F⁎-ϵ) approximation guarantee after O(1/ϵ2) iterations for DR-submodular functions by choosing appropriate step sizes. Furthermore, we also show that the algorithm achieves the tight (γ2/1+γ2pminF⁎-ϵ) approximation guarantee after O(1/ϵ2) iterations for weakly DR-submodular functions with parameter γ by choosing diminishing step sizes.