Hyperplane Projection Method Research Articles

A derivative free projection method for the singularities of vector fields with convex constraints on Hadamard manifolds

ABSTRACT The objective of this paper is to introduce a derivative free projection method designed to find the singularities of pseudomonotone vector fields with convex constraints on Hadamard manifolds. This innovative approach combines the hyperplane projection method with a novel search direction. The global convergence of the proposed method is established under certain conditions. Our method improves some existing results in the literature on Hadamard manifolds. Additionally, illustrative numerical examples are provided to demonstrate the practical efficacy of our method.

An inertial hyperplane projection method for split feasibility problems and applications

ABSTRACT In this paper, we propose a new projection algorithm to solve the split feasibility problem in Hilbert spaces. This method involves a new strategy that projects the inertial term onto the intersection of the feasible set and a half-space. The strong and weak convergence of the new algorithms under standard assumptions are established. We examine the performance of our method on sparse signal processing and image deblurring problems. The reported numerical results demonstrate the effectiveness of the proposed algorithm.

An efficient hierarchical electric vehicle charging control strategy

AbstractElectric vehicle (EV) has emerged as a crucial component in addressing both energy and environmental problems, and has become an essential part of nowadays' intelligent transportation systems. However, the charging demands of large amounts of EVs can put substantial pressure on power grid systems and cause potential grid congestion problems. In this article, we propose a hierarchical EV charging control strategy that considers the network‐wide communication overheads, computational complexity, total energy cost, EV user preferences, and data privacy protection. The hierarchical charging structure contains two phases, that is, a centralized control for EV aggregators and a distributed control for EVs within an aggregator. We prove that the centralized control with the objective to minimize the total energy cost of the power system constitutes a convex quadratic programming problem. Then a unique global optimum for the energy consumption profiles of EV aggregators can be achieved. The distributed charging control for EVs within an aggregator is studied using a generalized Nash equilibrium problem (GNEP). We show that the solution of the GNEP can be obtained via a variational inequality. Then the Solodov and Svaiter hyperplane projection method is employed to iteratively approach the variational equilibrium while ensuring the protection of EV users' data privacy. Extensive simulation studies are conducted to verify the correctness and effectiveness of our proposed hierarchical charging control algorithm for EVs.

A derivative-free memoryless BFGS hyperplane projection method for solving large-scale nonlinear monotone equations

A derivative-free memoryless BFGS hyperplane projection method for solving large-scale nonlinear monotone equations

An improved hyperplane projection method for generalized Nash equilibrium problems with extrapolation technique

The generalized Nash equilibrium problem is a kind of noncooperative game, whose strategy set of each player depends on the other rivals' decisions. Projection-type methods can be used to solve the corresponding quasi-variational inequalities problem. In this paper, we propose an accelerated method based on the half-space projection algorithm. At each iteration, we get a prediction point firstly by the half-space projection method. Then we use the extrapolation technique to obtain the next iterate. Numerical results show that our algorithm is effective.

An Accelerated Conjugate Gradient Algorithm for Solving Nonlinear Monotone Equations and Image Restoration Problems

Combining the three-term conjugate gradient method of Yuan and Zhang and the acceleration step length of Andrei with the hyperplane projection method of Solodov and Svaiter, we propose an accelerated conjugate gradient algorithm for solving nonlinear monotone equations in this paper. The presented algorithm has the following properties: (i) All search directions generated by the algorithm satisfy the sufficient descent and trust region properties independent of the line search technique. (ii) A derivative-free search technique is proposed along the direction to obtain the step length αk. (iii) If ϕk=−αkhk−hwkTdk>0, then an acceleration scheme is used to modify the step length in a multiplicative manner and create a point. (iv) If the point satisfies the given condition, then it is the next point; otherwise, the hyperplane projection technique is used to obtain the next point. (v) The global convergence of the proposed algorithm is established under some suitable conditions. Numerical comparisons with other conjugate gradient algorithms show that the accelerated computing scheme is more competitive. In addition, the presented algorithm can also be applied to image restoration.

Enhanced Dai–Liao conjugate gradient methods for systems of monotone nonlinear equations

In this paper, we propose two conjugate gradient methods for solving large-scale monotone nonlinear equations. The methods are developed by combining the hyperplane projection method by Solodov and Svaiter (Reformulation: nonsmooth, piecewise smooth, semismooth and smoothing methods. Springer, pp 355–369, 1998) and two modified search directions of the famous Dai and Liao (Appl Math Optim 43(1): 87–101, 2001) method. It is shown that the proposed schemes satisfy the sufficient descent condition. The global convergence of the methods are established under mild conditions, and computational experiments on some benchmark test problems show that the methods are promising.

A Modified Hestenes-Stiefel-Type Derivative-Free Method for Large-Scale Nonlinear Monotone Equations

The goal of this paper is to extend the modified Hestenes-Stiefel method to solve large-scale nonlinear monotone equations. The method is presented by combining the hyperplane projection method (Solodov, M.V.; Svaiter, B.F. A globally convergent inexact Newton method for systems of monotone equations, in: M. Fukushima, L. Qi (Eds.)Reformulation: Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods, Kluwer Academic Publishers. 1998, 355-369) and the modified Hestenes-Stiefel method in Dai and Wen (Dai, Z.; Wen, F. Global convergence of a modified Hestenes-Stiefel nonlinear conjugate gradient method with Armijo line search. Numer Algor. 2012, 59, 79-93). In addition, we propose a new line search for the derivative-free method. Global convergence of the proposed method is established if the system of nonlinear equations are Lipschitz continuous and monotone. Preliminary numerical results are given to test the effectiveness of the proposed method.

A new projection algorithm for solving constrained equilibrium problems in Hilbert spaces

ABSTRACTIn this paper, we introduce a new iterative scheme by combining the hyperplane projection method and the inertial technique for constrained equilibrium problems in real Hilbert spaces. The convergence of the proposed algorithm is established without requiring strict paramonotonicity property. The results presented in the paper extend and improve some recent results in the literature. In addition, a numerical example is given to illustrate the efficiency and performance of the proposed method.

A descent Dai-Liao conjugate gradient method for nonlinear equations

In this work, we propose an algorithm for solving system of nonlinear equations. The idea is a combination of the descent Dai-Liao method by Babaie-Kafaki and Gambari (Optim. Meth. Soft. 29(3), 583–591, 2014) and the hyperplane projection method. Using the monotonicity and Lipschitz continuity assumptions, we prove that the proposed method is globally convergent. Examples of numerical experiment show that the method is promising and efficient compared to the method proposed by Sun et al. (Journal of Inequalities and Applications 236, 1–8, 2017).

A globally convergent BFGS method for pseudo-monotone variational inequality problems

In this paper, we propose a globally convergent BFGS method to solve Variational Inequality Problems (VIPs). In fact, a globalization technique on the basis of the hyperplane projection method is applied to the BFGS method. The technique, which is independent of any merit function, is applicable for pseudo-monotone problems. The proposed method applies the BFGS direction and tries to reduce the distance of iterates to the solution set. This property, called Fejer monotonicity of iterates with respect to the solution set, is the basis of the convergence analysis. The method applied to pseudo-monotone VIP is globally convergent in the sense that subproblems always have unique solutions, and the sequence of iterates converges to a solution to the problem without any regularity assumption. Finally, some numerical simulations are included to evaluate the efficiency of the proposed algorithm.

A Cutting Hyperplane Projection Method for Solving Generalized Quasi-Variational Inequalities

The generalized quasi-variational inequality is a generalization of the generalized variational inequality and the quasi-variational inequality. The study for the generalized quasi-variational inequality is mainly concerned with the solution existence theory. In this paper, we present a cutting hyperplane projection method for solving generalized quasi-variational inequalities. Our method is new even if it reduces to solve the generalized variational inequalities. The global convergence is proved under certain assumptions. Numerical experiments have shown that our method has less total number of iterative steps than the most recent projection-like methods of Zhang et al. (Comput Optim Appl 45:89–109, 2010) for solving quasi-variational inequality problems and outperforms the method of Li and He (J Comput Appl Math 228:212–218, 2009) for solving generalized variational inequality problems.

A modified Perry’s conjugate gradient method-based derivative-free method for solving large-scale nonlinear monotone equations

In this paper, we propose a derivative-free method for solving large-scale nonlinear monotone equations. It combines the modified Perry’s conjugate gradient method (I.E. Livieris, P. Pintelas, Globally convergent modified Perrys conjugate gradient method, Appl. Math. Comput., 218 (2012) 9197–9207) for unconstrained optimization problems and the hyperplane projection method (M.V. Solodov, B.F. Svaiter, A globally convergent inexact Newton method for systems of monotone equations, in: M. Fukushima, L. Qi (Eds.), Reformulation: Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods, Kluwer Academic Publishers, 1998, pp. 355–369). We prove that the proposed method converges globally if the equations are monotone and Lipschitz continuous without differentiability requirement on the equations, which makes it possible to solve some nonsmooth equations. Another good property of the proposed method is that it is suitable to solve large-scale nonlinear monotone equations due to its lower storage requirement. Preliminary numerical results show that the proposed method is promising.

Two spectral gradient projection methods for constrained equations and their linear convergence rate

Due to its simplicity and numerical efficiency for unconstrained optimization problems, the spectral gradient method has received more and more attention in recent years. In this paper, two spectral gradient projection methods for constrained equations are proposed, which are combinations of the well-known spectral gradient method and the hyperplane projection method. The new methods are not only derivative-free, but also completely matrix-free, and consequently they can be applied to solve large-scale constrained equations. Under the condition that the underlying mapping of the constrained equations is Lipschitz continuous or strongly monotone, we establish the global convergence of the new methods. Compared with the existing gradient methods for solving such problems, the new methods possess a linear convergence rate under some error bound conditions. Furthermore, a relax factor γ is attached in the update step to accelerate convergence. Preliminary numerical results show that they are efficient and promising in practice.

An Liu-Storey-Type Method for Solving Large-Scale Nonlinear Monotone Equations

An algorithm for solving nonlinear monotone equations is proposed, which combines a modified Liu-Storey conjugate gradient method with hyperplane projection method. Under mild conditions, the global convergence of the proposed method is established with a suitable line search method. The method can be applied to solve large-scale problems for its lower storage requirement. Numerical results indicate that our method is efficient.

A Globally Convergent Matrix-Free Method for Constrained Equations and Its Linear Convergence Rate

A matrix-free method for constrained equations is proposed, which is a combination of the well-known PRP (Polak-Ribière-Polyak) conjugate gradient method and the famous hyperplane projection method. The new method is not only derivative-free, but also completely matrix-free, and consequently, it can be applied to solve large-scale constrained equations. We obtain global convergence of the new method without any differentiability requirement on the constrained equations. Compared with the existing gradient methods for solving such problem, the new method possesses linear convergence rate under standard conditions, and a relax factorγis attached in the update step to accelerate convergence. Preliminary numerical results show that it is promising in practice.

Sufficient Descent Conjugate Gradient Methods for Solving Convex Constrained Nonlinear Monotone Equations

Two unified frameworks of some sufficient descent conjugate gradient methods are considered. Combined with the hyperplane projection method of Solodov and Svaiter, they are extended to solve convex constrained nonlinear monotone equations. Their global convergence is proven under some mild conditions. Numerical results illustrate that these methods are efficient and can be applied to solve large-scale nonsmooth equations.

A global convergent quasi-Newton method for systems of monotone equations

In this paper, we propose a quasi-Newton method for solving systems of monotone equations. The method is a combination of the Broyden method and the hyperplane projection method. Under appropriate conditions, we prove that the proposed method is globally convergent. Preliminary numerical results show that the proposed method is promising.

A PRP type method for systems of monotone equations

In this paper, we propose an algorithm for solving systems of monotone equations. The method is a combination of the well-known PRP method and the hyperplane projection method. We prove that the proposed method is globally convergent if the equation is monotone and Lipschitz continuous without any differentiability requirement on the equation. Preliminary numerical results show that the proposed method is promising.

A globally convergent BFGS method for nonlinear monotone equations without any merit functions

r. Since 1965, there has been significant progress in the theoretical study on quasi-Newton methods for solving nonlinear equations, especially in the local convergence analysis. However, the study on global convergence of quasi-Newton methods is relatively fewer, especially for the BFGS method. To ensure global convergence, some merit function such as the squared norm merit function is typically used. In this paper, we propose an algorithm for solving nonlinear monotone equations, which combines the BFGS method and the hyperplane projection method. We also prove that the proposed BFGS method converges globally if the equation is monotone and Lipschitz continuous without differentiability requirement on the equation, which makes it possible to solve some nonsmooth equations. An attractive property of the proposed method is that its global convergence is independent of any merit function.We also report some numerical results to show efficiency of the proposed method.