Education, Science, Technology, Innovation and Life
Open Access

### A Semismooth and an Inexact Newton Method for Solving Nonsmooth Operator Equations

#### Author(s)

Huicheng Liu 1, Han Zhang 2

#### Affiliation(s)

1 School of Insurance, Guangdong University of Finance, Guangzhou, 510521, Guangdong, China
2 School of Business Administration, Guangdong University of Finance, Guangzhou, 510521, Guangdong, China

Han Zhang

#### ABSTRACT

After a series of changes, many application problems in the fields of transportation, finance, economy, engineering technology and resource allocation are equivalent to solve a class of nonsmooth operator equations. In order to solve the nonsmooth operator equations and study its application in Banach space, this paper develops a semismooth Newton method and inexact-Newton method via a generalized inverse. The global, local and superlinear convergence of a semismooth Newton method are shown, the superlinear and linear convergence properties of an inexact Newton method are also proved. The present methods can be easier to perform than the previous ones in some applications, and viewed as the extensions of existing methods to solve nonsmooth operator equations.

#### KEYWORDS

Nonsmooth Operator Equations, Semismooth Newton Method, Inexact-Newton Method, Superlinear Convergence, Bounded Outer Inverse

#### CITE THIS PAPER

Huicheng Liu, Han Zhang, A Semismooth and an Inexact Newton Method for Solving Nonsmooth Operator Equations. Transactions on Computational and Applied Mathematics (2022) Vol. 2: 1-7. DOI: http://dx.doi.org/10.23977/tracam.2022.020101.

#### REFERENCES

[1] Argyros, I.K.(2004)Weak sufficient convergence conditions and applications for Newton methods. Journal of Applied Mathematics and Computing, 16,1-17.
[2] Ulbrich, M. (2003)Semismooth Newton methods for operator equations in function spaces. SIAM Journal on Optimization,13,805-842.
[3] Liu,J. and Gao, Y. (2006) Inexact-Newton method for solving operator equations in infinite-dimensional space. Journal of Applied Mathmatics and Computing, 22,351-360.
[4] Mosi´c, D. and Marovt, J. (2021) Weighted weak core inverse of operators. Linear and Multiline Algebra, 1-23. https://doi.org/10.1080/03081087.2021.1902462.
[5] Benahmed, B., Nachi, K. and Yassine, A. (2016)Non-smooth singular Newton's method for positive semidefinite solution of nonlinear matrix equations. International Journal of Operational Research, 27,303-315.
[6] Cibulka, R., Dontchev, A. and Geoffroy, M.H.(2015) Inexact Newton Methods and Dennis-Moré Theorems for Nonsmooth Generalized Equations. SIAM Journal on Control and Optimization, 53, 1003-1019.
[7] Ghadimi, S. and Zhang, H.(2016)Mini-batch stochastic approximation methods for nonconvex stochastic composite optimization. Mathematical Programming, 155, 267-305.
[8] Hu,Y.H. and Liang, F. (2018)Two modifications of efficient newton-type iterative method and two variants of Super-Halley’s method for solving nonlinear equations. Journal of Computational Methods in Sciences and Engineering,1,1-10.
[9] Milzarek, A., Xiao, X., Cen, S., Wen, Z. and Ulbrich,M. (2019) A Stochastic Semismooth Newton Method for Nonsmooth Nonconvex Optimization. SIAM Journal on Optimization, 29,1-40.
[10] Deuflhard, P. and Heindl, G.(1979) Affine invariant convergence theorems for Newtons method and extensions to related methods. SIAM Journal of Numerical Analysis,16,1-10.
[11] Wang, Y.W.(2005) Operator generalized inverse theorem and application in Banach space (in Chinese), Beijing:Science Press.
[12] Zhang, N. and Wei, Y.(2008) A note on the perturbation of an outer inverse. Calcolo,45,263-273.
[13] Boichuk, A.A. and  Samoilenko, A.M. (2016) Generalized Inverse Operators. Berlin: De Gruyter.
[14] Wang, G., Wei,Y. and Qiao, S. (2018) Generalized Inverses: Theory and Computations. Beijing: Science Press.
[15] Ulbrich, M. (2002) Nonsmooth Newton-like methods for variational inequalities and contrained optimization problems in function spaces. München:Technische Universität München.