A New Hybrid PRPFR Conjugate Gradient Method for Solving Nonlinear Monotone Equations and Image Restoration Problems

Abstract

A new hybrid PRPFR conjugate gradient method is presented in this paper, which is designed such that it owns sufficient descent property and trust region property. This method can be considered as a convex combination of the PRP method and the FR method while using the hyperplane projection technique. Under accelerated step length, the global convergence property is gained with some appropriate assumptions. Comparing with other methods, the numerical experiments show that the PRPFR method is more competitive for solving nonlinear equations and image restoration problems.

Another hybrid approach for solving monotone operator equations and application to signal processing

“…The standard metrics used in this experiment are number of iterations (ITER), number of function evaluations (FVAL), and CPU time (TIME). To assess the strength of Algorithm 1, we compare it with following algorithms: The algorithm called PRPFR proposed by Zhou et al 37 The algorithm called HCGP proposed by Koorapetse and Kaelo 38 …”

“…• The algorithm called PRPFR proposed by Zhou et al 37 • The algorithm called HCGP proposed by Koorapetse and Kaelo. 38 For the experiments, we use the following:…”

“…• Five initial points: • Control parameters: We select 𝜌 = 0.8, 𝜎 = 10 −4 , 𝛾 = 1.2. As for PRPFR and HCGP, all parameters are as selected in Zhou et al 37 and Koorapetse and Kaelo, 38 respectively. • Stopping condition: Iterations are stopped when ||G (k) || ≤ 10 −5 and/or the number of iterations exceed 1,000 without reaching a solution.…”

“…For the experiments, we use the following: Number of problems: 10. Five dimensions: 1,000, 5,000, 10,000, 50,000, 100,000. Five initial points: $$$ {x}_1={\left(0.1,0.1,\cdots, 0.1\right)}^T,{x}_2={\left(0.2,0.2,\cdots, 0.2\right)}^T,{x}_3={\left(0.5,0.5,\cdots, 0.5\right)}^T,{x}_4={\left(1.5,1.5,\cdots 1.5\right)}^T,{x}_5={\left(2,2,\cdots, 2\right)}^T $$$. Control parameters: We select $$$ \rho =0.8,\sigma =1{0}^{-4},\gamma =1.2 $$$. As for PRPFR and HCGP, all parameters are as selected in Zhou et al 37 and Koorapetse and Kaelo, 38 respectively. Stopping condition: Iterations are stopped when $$$ \left\Vert {G}^{(k)}\right\Vert \le 1{0}^{-5} $$$ and/or the number of iterations exceed 1,000 without reaching a solution. The symbol − is used to indicate that an algorithm failed due to number of iterations exceeding 1,000 with reaching a solution. …”

A New Hybrid Three-Term Conjugate Gradient Algorithm for Large-Scale Unconstrained Problems

“…Conjugate gradient (CG) methods [4][5][6][7][8][9][10][11][12] are much more effective for unconstrained problems, especially for large-scale cases by low memory requirements and strong convergence properties [6,[8][9][10][11], etc. Meanwhile, CG methods have been applied to image restoration problems, optimal control problems and optimal problems in machine learning [13][14][15], etc. In this paper, we design a CG method for (1).…”