TY - JOUR

T1 - Метод точных штрафов в задаче поиска оптимальной траектории

AU - Аббасов, Меджид Эльхан Оглы

AU - Горбунова, Анна Андреевна

AU - Мирошниченко, Даниил Викторович

PY - 2026/8/3

Y1 - 2026/8/3

N2 - The application of the exact penalty method to the problem of finding a cost-optimal tra¬jectory is considered. To minimize the exact penalty function, the apparatus of constructive nonsmooth analysis is employed. It is shown that the application of the exact penalty method in the original functional formulation leads to the generation of successive approximations in the form of non-elementary functions. Although this fact is theoretically expected, this work is the first to investigate its practical consequences: it is demonstrated that as early as the second iteration, the computational complexity increases to such an extent that the analytical implementation of the algorithm becomes infeasible (a slowdown of more than 10. 000 times compared to the Ritz method). To overcome these difficulties, a grid-based adaptation of the method is proposed, which allows obtaining a solution in a reasonable time. The obtained results may be useful for researchers planning to apply the exact penalty method to problems in the fields of calculus of variations and control theory, as well as for solving trajectory optimization and route planning problems.

AB - The application of the exact penalty method to the problem of finding a cost-optimal tra¬jectory is considered. To minimize the exact penalty function, the apparatus of constructive nonsmooth analysis is employed. It is shown that the application of the exact penalty method in the original functional formulation leads to the generation of successive approximations in the form of non-elementary functions. Although this fact is theoretically expected, this work is the first to investigate its practical consequences: it is demonstrated that as early as the second iteration, the computational complexity increases to such an extent that the analytical implementation of the algorithm becomes infeasible (a slowdown of more than 10. 000 times compared to the Ritz method). To overcome these difficulties, a grid-based adaptation of the method is proposed, which allows obtaining a solution in a reasonable time. The obtained results may be useful for researchers planning to apply the exact penalty method to problems in the fields of calculus of variations and control theory, as well as for solving trajectory optimization and route planning problems.

KW - calculus of variations

KW - constructive nonsmooth analysis

KW - cost functional

KW - exact penalty method

KW - nondifferentiable optimization

KW - optimal trajectory

KW - route planning

UR - https://www.mendeley.com/catalogue/3e152293-6f55-300d-ad70-7f085d353451/

U2 - 10.21638/spbu10.2026.201

DO - 10.21638/spbu10.2026.201

M3 - статья

VL - 22

SP - 184

EP - 195

JO - ВЕСТНИК САНКТ-ПЕТЕРБУРГСКОГО УНИВЕРСИТЕТА. ПРИКЛАДНАЯ МАТЕМАТИКА. ИНФОРМАТИКА. ПРОЦЕССЫ УПРАВЛЕНИЯ

JF - ВЕСТНИК САНКТ-ПЕТЕРБУРГСКОГО УНИВЕРСИТЕТА. ПРИКЛАДНАЯ МАТЕМАТИКА. ИНФОРМАТИКА. ПРОЦЕССЫ УПРАВЛЕНИЯ

SN - 1811-9905

IS - 2

ER -