Because of their computing efficiency and minimal memory requirements, conjugate gradient techniques are a fundamental family of algorithms for handling large-scale unconstrained nonlinear optimization problems. A new version of the Hestenes-Stiefel (HS) technique is presented in this study with the goal of improving convergence properties without compromising ease of use. We rigorously prove the global convergence qualities of the proposed approach under standard assumptions and show that it meets the conjugacy, descent, and adequate descent constraints. Numerous numerical tests, covering a wide range of benchmark issues, show that the suggested strategy routinely performs better than the traditional HS approach in terms of function evaluations and iteration count.
2025-08
Journal of Intelligent & Fuzzy Systems
(Issue : 46)
(Volume : 2)
Two new limited-memory preconditioned conjugate gradient algorithms for nonlinear optimization problems
The conjugate gradient (CG) techniques are a class of unconstrained optimization algorithms with strong local and
global convergence qualities and minimal memory needs. While the quasi-Newton methods are reliable and efficient on a
wide range of problems and thesemethods are converge faster than the conjugate gradientmethods and require fewer function
evaluations, however, they are request substantially more storage, and if the problem is ill-conditioned, they may require
several iterations. There is another class, termed preconditioned conjugate gradient method, it is a technique that combines
two methods conjugate gradient with quasi-Newton. In this work, we proposed a new two limited memory preconditioned
conjugate gradient methods (New1 and New2), to solve nonlinear unconstrained minimization problems, by using new
modified symmetric rank one (NMSR1) and new modified Davidon, Fletcher, Powell (NMDFP), and also using projected
vectors. We proved that these modifications fulfill some conditions. Also, the descent condition of the new technique has
been proved. The numerical results showed the efficiency of the proposed new algorithms compared with some standard
nonlinear, unconstrained problems.
2024-02
Mathematics and Statistics
(Issue : 1)
(Volume : 11)
Two New Preconditioned Conjugate Gradient Methods for Minimization Problems
In application to general function, each of the conjugate gradient and Quasi-Newton methods has particular
advantages and disadvantages. Conjugate gradient (CG) techniques are a class of unconstrained optimization algorithms
with strong local and global convergence qualities and minimal memory needs. Quasi-Newton methods are reliable and
efficient on a wide range of problems and they converge faster than the conjugate gradient method and require fewer
function evaluations but they have the disadvantage of requiring substantially more storage and if the problem is
ill-conditioned, they may take several iterations. A new class has been developed, termed preconditioned conjugate
gradient (PCG) method. It is a method that combines two methods, conjugate gradient and Quasi-Newton. In this work,
two new preconditioned conjugate gradient algorithms are proposed namely New PCG1 and New PCG2 to solve
nonlinear unconstrained optimization problems. A new PCG1 combines conjugate gradient method Hestenes-Stiefel (HS)
with new self-scaling symmetric Rank one (SR1), and a new PCG2 combines conjugate gradient method Hestenes-Stiefel
(HS) with new self-scaling Davidon, Flecher and Powell (DFP). The algorithm uses the strong Wolfe line search
condition. Numerical comparisons with standard preconditioned conjugate gradient algorithms show that for these new
algorithms, computational scheme outperforms the preconditioned conjugate gradient.
2023-01
Science Journal of University of Zakho
(Issue : 4)
(Volume : 10)
A NEW MODIFIED CONJUGATE GRADIENT FOR NONLINEAR MINIMIZATION PROBLEMS
The conjugate gradient is a highly effective technique to solve the unconstrained nonlinear minimization problems and it is one of
the most well-known methods. It has a lot of applications. For large-scale and unconstrained minimization problems, conjugate
gradient techniques are widely applied. In this paper, we will suggest a new parameter of conjugate gradient to solve the nonlinear
unconstrained minimization problems, based on the parameter of Dai and Liao. We will study the property of the descent , the
property of the sufficient descent and property of the global convergence of the new method. We introduce some numerical data
to prove the efficacy of the our method.
2022-10
General Letters in Mathematics (GLM)
(Issue : 2)
(Volume : 9)
A New conjugate gradient method for unconstrained optimization problems with descent property
In this paper, we propose a new conjugate gradient method for solving nonlinear unconstrained optimization.
The new method consists of three parts, the first part of them is the parameter of Hestenes-Stiefel (HS). The
proposed method is satisfying the descent condition, sufficient descent condition and conjugacy condition. We
give some numerical results to show the efficiency of the suggested method.
2021-05
General Letters in Mathematics
(Issue : 1)
(Volume : 7)
A New Parameter Conjugate Gradient Method Based on Three Terms Unconstrained Optimization
In this paper, we suggest a new conjugate gradient method for solving nonlinear uncons trained
optimization problems by using three term conjugate gradient method , We give a descent condi tion and the sufficient
descent condition of the suggested method.
2019-11
Journal of University of Zakho
(Issue : 1)
(Volume : 4)
New conjugate gradient method for unconstrained optimization with logistic mapping
In this paper , we suggested a new conjugate gradient algorithm for unconstrained optimization based
on logistic mapping, descent condition and sufficient descent condition for our method are provided.
Numerical results show that our presented algorithm is more efficient for solving nonlinear unconstrained
optimization problems comparing with (DY) .
A new hybrid conjugate gradient method for unconstrained optimization by using homotopy formula
, we computed the parameter as aconvex combination of (Polak-Ribiere (PR))[9] and (Al-Bayati
and Al-Assady (BA))[1].
In this paper, we suggest a hybrid conjugate gradient method for unconstrained optimization by
using homotopy formula , We calculate the parameter �� as aconvex combination of �𝑆 (Hestenes Stiefel)[5]
and ��� (Conjugate descent)[3].
2014-05
Thesis
2014-12-29
Homotopy and Homotopy perturbation Methods for unconstrained Optimization
