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.
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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
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DOI: https://doi.org/10.1007/978-3-030-91059-4_1
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