Abstract
In the paper, we introduce a new nonsingular operator instead of a degenerate operator of the first derivative in a singular case for solving and describing nonregular optimization problems and some problems in calculus. Such operator is called p-factor-operator and its construction is based on the derivatives up to order p as well as on some element h, which we call the “exit from singularity”. The special variant of the method of the Modified Lagrangian Functions for optimization problems with inequality constraints is justified on the basis of the 2-factor transformation and constructions of p-regularity theory. These results are used in some classical branches of calculus: implicit function theorem is given for the singular case and is shown the existence of solutions to a boundary-valued problem for a nonlinear differential equation in the resonance case. New numerical methods are proposed including the p-factor method for solving ODEs with a small parameter and new formula is obtained for the solutions of such type equations.
This work was supported in part by the Russian Foundation for Basic Research, project No. 21-71-30005.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Brezhneva, O.A., Tretyakov, A.A.: Methods for solving essentially nonlinear problems. CCAS of RAS, Moscow (2000).(in Russian)
Tret’yakov, A.A., Marsden, J.E.: Factor-analysis of nonlinear mapping: \(p\)-regularity Theory. Commun. Pure Appl. Anal. 2, 425–445 (2003)
Izmailov, A.F., Tretyakov, A.A.: 2-regular solutions to nonlinear problems. Fizmatgiz, Moscow (1999). (in Russian)
Evtushenko, Yu.: Generalized Lagrange multiplier technique for nonlinear programming. J. Optim. Theory Appl. 21(2), 121–135 (1977)
Tret’yakov, A.A.: The implicit function theorem in degenerate problems. Russ. Math. Surv. 42, 179–180 (1987)
Belash, K.N., Tretyakov, A.A.: Methods for solving degenerate problems. USSR Comput. Math. Math. Phys. 28(4), 90–94 (1988)
Tretyakov, A.A.: Structures of nonlinear degenerate mappings and their application to construction numerical methods. Doct. diss, Moscow (1987). (in Russian)
Mangasarian, O.L.: A Newton method for linear programming. J. Optimiz. Theory Appl. 121, 1–18 (2004)
Golikov, A.I., Evtushenko, Y.G., Mollaverdi, N.: Application of Newton’s method for solving large linear programming problems. Comput. Math. Math. Phys. 44, 1484–1493 (2004)
Izmailov, A.F., Tretyakov, A.A.: Factor Analysis of Nonlinear Mappings. Nauka, Moscow (1994).(in Russian)
El’sgol’ts, L.E.: Ordinary Dierential Equations. Gostekhteorizdat, Moscow (1950). (in Russian)
Kamke, E.: Gewöhnliche Differentialgleichungen. Akademie-Verlag, Leipzig, Nauka, Moscow (1971)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 Springer Nature Switzerland AG
About this paper
Cite this paper
Evtushenko, Y., Malkova, V., Tret’yakov, A. (2021). Exit from Singularity. New Optimization Methods and the p-Regularity Theory Applications. In: Olenev, N.N., Evtushenko, Y.G., Jaćimović, M., Khachay, M., Malkova, V. (eds) Optimization and Applications. OPTIMA 2021. Lecture Notes in Computer Science(), vol 13078. Springer, Cham. https://doi.org/10.1007/978-3-030-91059-4_1
Download citation
DOI: https://doi.org/10.1007/978-3-030-91059-4_1
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-91058-7
Online ISBN: 978-3-030-91059-4
eBook Packages: Computer ScienceComputer Science (R0)