Applications of the PL homotopy algorithm for the computation of fixed points to unconstrained optimization problems

This paper describes the main aspects of the ”piecewise-linear homotopy method” for fixed point approximation proposed by Eaves and Saigal [Eaves, C. B. and Saigal, R., Homotopies for computation of fixed points on unbounded regions, Mathematical Programming, 3 (1972), No. 1, 225–237]. The implementation of the method is developed using the modern programming language C# and then is used for solving some unconstrained optimization problems. The PL homotopy algorithm appears to be more reliable than the classical Newton method in the case of the problem of finding a local minima for Schwefel’s function and other optimization problems.

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A comparative study of the PL homotopy and BFGS methods for some nonsmooth optimization problems

Creative Mathematics and Informatics ◽

10.37193/cmi.2019.02.01 ◽

2019 ◽

Vol 28 (2) ◽

pp. 97-104

Author(s):

ANDREI BOZANTAN ◽

VASILE BERINDE

Keyword(s):

Fixed Points ◽

Nonsmooth Optimization ◽

Optimization Problems ◽

Piecewise Linear ◽

Smooth Functions ◽

Inexact Line Search ◽

Homotopy Algorithm ◽

Unconstrained Optimization Problems ◽

Piecewise Linear Homotopy ◽

Computation Of Fixed Points

We consider some non-smooth functions and investigate the numerical behavior of the Piecewise Linear Hompotopy (PLH) method ([Bozântan, A., An implementation of the piecewise-linear homotopy algorithm for the computation of fixed points, Creat. Math. Inform., 19 (2010), No.~2, 140–148] and [Bozântan, A. and Berinde, V., Applications of the PL homotopy algorithm for the computation of fixed points to unconstrained optimization problems, Creat. Math. Inform., 22 (2013), No. 1, 41–46]). We compare the PLH method with the BFGS with inexact line search, a quasi-Newton method, having some results reported in [Lewis, A. S. and Overton, M. L., Nonsmooth optimization via BFGS, submitted to SIAM J. Optimiz, (2009)]. For most of the considered cases, the characteristics of the PLH method are quite similar to the BFGS method, that is, the PLH method converges to local minimum values and the convergence rate seems to be linear with respect to the number of function evaluations, but we also identify some issues with the PLH method.

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A MODIFIED BFGS METHOD VIA NEW RATIONAL APPROXIMATION MODEL FOR SOLVING UNCONSTRAINED OPTIMIZATION PROBLEMS AND ITS APPLICATION

Advances in Mathematics: Scientific Journal ◽

10.37418/amsj.9.12.61 ◽

2020 ◽

Vol 9 (12) ◽

pp. 10771-10786

Author(s):

K. Kamfa ◽

S. Ibrahim ◽

S. F. Sufahani ◽

R. B. Yunus ◽

M. Mamat

Keyword(s):

Unconstrained Optimization ◽

Rational Approximation ◽

Optimization Problems ◽

Bfgs Method ◽

Approximation Model ◽

Unconstrained Optimization Problems

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A Non-Monotone Line Search Combination Technique for Unconstrained Optimization

The Open Electrical & Electronic Engineering Journal ◽

10.2174/1874129001408010218 ◽

2014 ◽

Vol 8 (1) ◽

pp. 218-221 ◽

Cited By ~ 2

Author(s):

Ping Hu ◽

Zong-yao Wang

Keyword(s):

Global Convergence ◽

Unconstrained Optimization ◽

Numerical Experiments ◽

Line Search ◽

Optimization Problems ◽

Search Algorithm ◽

New Combination ◽

Combination Rule ◽

Combination Technique ◽

Unconstrained Optimization Problems

We propose a non-monotone line search combination rule for unconstrained optimization problems, the corresponding non-monotone search algorithm is established and its global convergence can be proved. Finally, we use some numerical experiments to illustrate the new combination of non-monotone search algorithm’s effectiveness.

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