scholarly journals A new class of self-scaling for quasi-newton method based on the quadratic model

2021 ◽  
Vol 21 (3) ◽  
pp. 1830
Author(s):  
Basim A. Hassan ◽  
Ranen M. Sulaiman
Keyword(s):  
Newton Method ◽  
Optimization Problems ◽  
Quadratic Function ◽  
Quadratic Model ◽  
Newton Methods ◽  
New Class ◽  
Unconstrained Optimization Problems ◽  
The Common ◽  
Numerical Performance ◽  
Quasi Newton

<span id="docs-internal-guid-a04d8b24-7fff-eaad-9449-fe4b2527904b"><span>Quasi-Newton method is an efficient method for solving unconstrained optimization problems. Self-scaling is one of the common approaches in the modification of the quasi-Newton method. A large variety of self-scaling of quasi-Newton methods is very well known. In this paper, based on quadratic function we derive the new self-scaling of quasi-Newton method and study the convergence property. Numerical results on the collection of problems showed the self-scaling of quasi-Newton methods which improves overall numerical performance for BFGS method.</span></span>

scholarly journals A Sparse Quasi-Newton Method Based on Automatic Differentiation for Solving Unconstrained Optimization Problems

Symmetry ◽  
2021 ◽  
Vol 13 (11) ◽  
pp. 2093
Author(s):  
Huiping Cao ◽  
Xiaomin An
Keyword(s):  
Unconstrained Optimization ◽  
Newton Method ◽  
Automatic Differentiation ◽  
Optimization Problems ◽  
Symmetric Matrix ◽  
Matrix Completion ◽  
Bfgs Method ◽  
Limited Memory ◽  
Unconstrained Optimization Problems ◽  
Quasi Newton

In our paper, we introduce a sparse and symmetric matrix completion quasi-Newton model using automatic differentiation, for solving unconstrained optimization problems where the sparse structure of the Hessian is available. The proposed method is a kind of matrix completion quasi-Newton method and has some nice properties. Moreover, the presented method keeps the sparsity of the Hessian exactly and satisfies the quasi-Newton equation approximately. Under the usual assumptions, local and superlinear convergence are established. We tested the performance of the method, showing that the new method is effective and superior to matrix completion quasi-Newton updating with the Broyden–Fletcher–Goldfarb–Shanno (BFGS) method and the limited-memory BFGS method.

scholarly journals Exact linesearch limited-memory quasi-Newton methods for minimizing a quadratic function

2021 ◽  
Author(s):  
David Ek ◽  
Anders Forsgren
Keyword(s):  
Optimization Problems ◽  
Linear Equations ◽  
Quadratic Function ◽  
Compact Representation ◽  
Test Collection ◽  
Newton Methods ◽  
Limited Memory ◽  
Broyden Class ◽  
Systems Of Linear Equations ◽  
Quasi Newton

AbstractThe main focus in this paper is exact linesearch methods for minimizing a quadratic function whose Hessian is positive definite. We give a class of limited-memory quasi-Newton Hessian approximations which generate search directions parallel to those of the BFGS method, or equivalently, to those of the method of preconditioned conjugate gradients. In the setting of reduced Hessians, the class provides a dynamical framework for the construction of limited-memory quasi-Newton methods. These methods attain finite termination on quadratic optimization problems in exact arithmetic. We show performance of the methods within this framework in finite precision arithmetic by numerical simulations on sequences of related systems of linear equations, which originate from the CUTEst test collection. In addition, we give a compact representation of the Hessian approximations in the full Broyden class for the general unconstrained optimization problem. This representation consists of explicit matrices and gradients only as vector components.

A new three-term spectral conjugate gradient algorithm with higher numerical performance for solving large scale optimization problems based on Quasi-Newton equation

2021 ◽  
pp. 2150053
Author(s):  
Jie Guo ◽  
Zhong Wan
Keyword(s):  
Conjugate Gradient ◽  
Large Scale ◽  
Line Search ◽  
Optimization Problems ◽  
Gradient Algorithm ◽  
Large Scale Optimization ◽  
Newton Equation ◽  
Numerical Performance ◽  
Scale Optimization ◽  
Quasi Newton

A new spectral three-term conjugate gradient algorithm in virtue of the Quasi-Newton equation is developed for solving large-scale unconstrained optimization problems. It is proved that the search directions in this algorithm always satisfy a sufficiently descent condition independent of any line search. Global convergence is established for general objective functions if the strong Wolfe line search is used. Numerical experiments are employed to show its high numerical performance in solving large-scale optimization problems. Particularly, the developed algorithm is implemented to solve the 100 benchmark test problems from CUTE with different sizes from 1000 to 10,000, in comparison with some similar ones in the literature. The numerical results demonstrate that our algorithm outperforms the state-of-the-art ones in terms of less CPU time, less number of iteration or less number of function evaluation.

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