# Combined methods for solving degenerate unconstrained optimization problems

### Abstract

UDC 519.853.6 : 519.613.2

We present constructive second- and fourth-order methods for solving degenerate unconstrained optimization problems. The fourth-order method applied in the present work is a combination of the Newton method and the method that uses fourth-order derivatives. Our approach is based on the decomposition of $\mathbb{R}^n$ into the direct sum of the kernel of a Hessian matrix and its orthogonal complement. The fourth-order method is applied to the kernel of the Hessian matrix, whereas the Newton method is applied to its orthogonal complement. This method proves to be efficient in the case of a one-dimensional kernel of the Hessian matrix. In order to get the second-order method, Newton's method is combined with the steepest-descent method. We study the efficiency of these methods and analyze their convergence rates. We also propose a new adaptive combined quasi-Newton-type method (ACQNM) based on the use of the second- and fourth-order methods in the degenerate case. The efficiency of ACQNM is demonstrated by analyzing an example of some most common test functions.

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DOI: https://doi.org/10.15421/142204

V. M. Zadachyn, *Modified Newton and quasi-Newtonian-type methods with pseudo-inversions for solving degenerate problems*, Ph. D. Thesis, Lomonosov Moscow State University, Moscow, CA (1988) (in Russian); https://search.rsl.ru/ru/record/01000049990.

V. I. Meleshko, V. M. Zadachin, *Factorizations and pseudo-inversions of singular perturbed matrices with nonfixed signs*, Izv. Vyss. Uchebn. Zaved. Mat., **11**, 42–50 (1987).

E. G. Birgin, J. M. Martinez, *The use of quadratic regularization with a cubic descent condition for unconstrained optimization*, SIAM J. Optim., **27**, № 2, 1049–1074 (2017).

E. G. Birgin, J. M. Martinez, *Newton-like method with mixed factorizations and cubic regularization for unconstrained minimization*, Comput. Optim. and Appl., **73**, 707–753 (2019).

S. Javed, A. Khan, *Efficient regularized Newton-type algorithm for solving convex optimization problem*, J. Appl. Math. and Comput., **68**, № 4, 2343–2363 (2022).

H. Zhang, Q. Ni, *A new regularized quasi-Newton method for unconstrained optimization*, Optim. Lett., **12**, № 7, 1639–1658 (2018).

*Ukrains’kyi Matematychnyi Zhurnal*, Vol. 76, no. 5, June 2024, pp. 695 -18, doi:10.3842/umzh.v76i5.7395.

Copyright (c) 2024 Задачин Віктор, Максим Бебія

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