ISSN 2079-6900 (Print) 
ISSN 2587-7496 (Online)

Middle Volga Mathematical Society Journal

DOI 10.15507/2079-6900.26.202402.123-142

Original article

ISSN 2079-6900 (Print)

ISSN 2587-7496 (Online)

MSC2020 49N4

Modified projection generalized two-point two-stage extragradient quasinewton method for saddle point problems

V. G. Malinov

Independent researcher

Abstract. The purpose of this work is to investigate a new method mentioned in the article’s name. This method is designed for solving saddle problems with convexo-concave differentiable function that is defined on a convex closed subset of some finite-dimensional euclidean space and has «ravine» level hypersurfaces. The paper contains a brief survey of native publications devoted to new projection gradient methods for solving saddle problems. A mathematical statement of a saddle problem, information about solution method, some auxiliary inequalities, and method’s convergence are discussed in the article as well. Moreover, iterative formulas are exemplified for another perspective saddle method for convexo-concave differentiable saddle functions, which may be validated as well as formulas proved in this work. New auxiliary inequalities complete mathematical apparatus of convex analysis for justification of convergence and rate of convergence and have value also for justification of another methods of operations research. By using obtained inequalities, convex analysis and numerical mathematics, convergence of the saddle method for convexo-concave smooth saddle functions with Lipschitz partial gradients is proved. Under supplementary conditions, for twice continuously differentiable saddle functions, superlinear and quadratic rate of convergence of saddle method are proved, too.

Key Words: convexo-concave saddle function, saddle point problem, projection generalized two-point two-stage extragradient quasinewton saddle method

For citation: V. G. Malinov. Modified projection generalized two-point two-stage extragradient quasinewton method for saddle point problems. Zhurnal Srednevolzhskogo matematicheskogo obshchestva. 26:2(2024), 123–142. DOI:

Submitted: 14.11.2023; Revised: 12.04.2024; Accepted: 29.05.2024

Information about the author:

Valerian G. Malinov, Ph.D. (Phys.-Math.), Independent researcher, ORCID:,

The author have read and approved the final manuscript.

Conflict of interest: The author declare no conflict of interest.

Creative Commons Attribution 4.0 International License This is an open access article distributed under the terms of the Creative Commons Attribution 4.0 International License.