Abstract
Application of penalty function methods for nonlinear constrained extremum problems allows using unconstrained optimization methods. In this direction, the works of such authors as A.V. Fiacco, G.P. McCormick, J. Cea, E. Polak, I.I. Eremin, B.T. Polyak and others are well known. Investigation of the penalty function method convergence for convex programming problems with a finite number of constraints is the most complete in the literature. In this case, a completely defined range of functions is considered as a penalty.We consider the minimization problem of non-linear convex functional on the convex closed set of Sobolev spaces. To solve this problem, one class of integral penalty functions introduced in papers by A.A. Kaplan is used. This leads to extremum problem on the whole Sobolev space. The estimation of convergence rate of the penalty method with integral penalty functions is obtained by generalizing the investigation methods for the case of a finite number of restrictions on the case of integral penalty functions. The obtained results can be used in numerical studies of similar problems.DOI 10.14258/izvasu(2018)1-22
Highlights
В данной работе проводится исследование интегральных штрафных функций в сравнении с результатами исследований для случая конечного числа ограничений
A completely defined range of functions is considered as a penalty
This leads to extremum problem on the whole Sobolev space
Summary
Применение методов штрафных функций при решении нелинейных экстремальных задач с ограничениями позволяет использовать методы безусловной оптимизации. Наиболее полно в литературе представлены исследования вопросов сходимости метода штрафных функций для задач выпуклого программирования с конечным числом ограничений. При этом рассматривается вполне определенный круг функций в качестве штрафных. Для сведения численного решения задачи минимизации нелинейного выпуклого функционала на выпуклом замкнутом множестве в пространстве Соболева к решению экстремальной задачи на всем пространстве предлагается использовать класс интегральных штрафных функций, введенный в работах А.А. Application of penalty function methods for nonlinear constrained extremum problems allows using unconstrained optimization methods. Investigation of the penalty function method convergence for convex programming problems with a finite number of constraints is the most complete in the literature. We consider the minimization problem of non-linear convex functional on the convex closed set of Sobolev spaces.
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