Some Results About Proximal-Like Methods

Kaplan, A.; Tichatschke, R.

doi:10.1007/3-540-28258-0_5

Cited by 4 publications

(9 citation statements)

References 36 publications

(58 reference statements)

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“…Considering the application of the GPP-method to VI(Q, K ) (with Q in (31) and K in (32) or (26)), we suppose, as already mentioned, that Ω is a convex polygonal domain, VI(Q, K ) is solvable and its solution u * belongs to H 2 (Ω).…”

Section: Application Of Gpp-methods To Minimal Surface Problemsmentioning

confidence: 99%

“…The uniqueness of a solution (if it exists) in case K is given by (32) or (26) is a rather evident corollary of the strict convexity of the functional…”

Section: Remarkmentioning

confidence: 97%

“…If K is given by (32), the relation (40) is well-known. The application of Proposition 1 and (40) yields…”

Section: One Can Easily Show That the Operatorsmentioning

confidence: 98%

“…Theorem 3. (i) The problems {(P k )}, corresponding to VI(Q, K ) (with Q defined by (31) and K by (32) or (26)) are solvable. (1) and (2) and…”

Section: One Can Easily Show That the Operatorsmentioning

confidence: 99%

“…Suppose that a solution u of VI(Q, K ), with Q in (31) and K in (32) or (26), belongs to W 1,∞ (Ω). Then Assumption 3 is valid with Y = W 1,1 (Ω), arbitrary ρ > u and τ (s) := c(ρ)s 2 .…”

Section: Propositionmentioning

confidence: 99%

See 4 more Smart Citations

Proximal point method and elliptic regularization

Kaplan

Tichatschke

2009

Nonlinear Analysis: Theory, Methods & Applications

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Section: Application Of Gpp-methods To Minimal Surface Problemsmentioning

confidence: 99%

“…The uniqueness of a solution (if it exists) in case K is given by (32) or (26) is a rather evident corollary of the strict convexity of the functional…”

Section: Remarkmentioning

confidence: 97%

“…If K is given by (32), the relation (40) is well-known. The application of Proposition 1 and (40) yields…”

Section: One Can Easily Show That the Operatorsmentioning

confidence: 98%

“…Theorem 3. (i) The problems {(P k )}, corresponding to VI(Q, K ) (with Q defined by (31) and K by (32) or (26)) are solvable. (1) and (2) and…”

Section: One Can Easily Show That the Operatorsmentioning

confidence: 99%

Section: Propositionmentioning

confidence: 99%

See 3 more Smart Citations

Proximal point method and elliptic regularization

Kaplan

Tichatschke

2009

Nonlinear Analysis: Theory, Methods & Applications

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Variational Inequalities

Konnov¹

2011

Wiley Encyclopedia of Operations Research and Management Science

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We consider variational inequality problems in a finite‐dimensional space setting. We first describe existence and uniqueness results under norm monotonicity assumptions. Relations to some other general nonlinear problems are discussed. Then we describe basic solution methods utilizing the above monotonicity concepts and their convergence properties. Some extensions for the multivalued case are also discussed.

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On Applicability of a Class of Parametric Functions as Exterior Penalty Operators to Solve Nonlinear Problems with Constraints

Саженкова¹,

Саженков²,

Плотникова³

et al. 2019

izvasu

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Проведено исследование однопараметрического класса функций, введенного А.А. Капланом, для решения задачи условной минимизации нелинейной выпуклой функции на множестве, заданном с помощью ограничений-неравенств. Предполагается при этом, что множество, задаваемое ограничениями, не пусто и имеет внутренние точки. В монографиях А. Фиакко, Г. МакКормика, Э. Полака, А.А. Каплана представлено систематическое изложение теории методов штрафов и классификации штрафных функций. Опираясь на приемы и методы, изложенные в них, в данной работе устанавливается принадлежность исследуемого класса функций к внешним штрафам для задач выпуклого программирования. Применение методов штрафных функций при решении нелинейных экстремальных задач с ограничениями позволяет использовать методы безусловной нелинейной оптимизации, в том числе градиентные методы. Штрафные функции А.А. Каплана обладают хорошими дифференциальными свойствами и тем самым удобны при использовании итерационных градиентных методов приближенного решения задач на безусловный экстремум. Далее в работе доказывается теорема сходимости последовательности получающихся приближенных решений задач на безусловный экстремум к точному решению исходной задачи. Получена оценка скорости сходимости метода штрафов с использованием рассматриваемого однопараметрического класса функций в качестве штрафных функций. Представляемая оценка скорости сходимости метода получена в предположении, что осуществляется точное решение последовательности задач безусловной оптимизации. Полученные результаты могут быть применены при численном исследовании задач рассматриваемого вида.Ключевые слова: методы штрафных функций, выпуклое программирование, оценки скорости сходимости.Kaplan's one-parametric class of functions for solving nonlinear convex minimization problems with one-sided constraints is studied. The set defined by constraints is assumed to have a nonempty interior. In the monographs written the penalty methods theory and classification of penalty functions are presented quite systematically. With these methods and approaches as backbones, in the present article, we establish that the class of functions under study belongs to the class of exterior penalty functions for problems of convex programming. Application of penalty methods to the solution of nonlinear extremal problems with constraints allows using the toolbox of unconstrained nonlinear optimization, including gradient methods. Kaplan's penalty functions have fine differential properties. Therefore, they are suitable for use in iterative gradient methods of approximate solution of unconstrained extremal problems. After this, we prove the theorem on convergence of the sequence of approximate solutions of penalized unconstrained extremal problems to the exact solution of the original problem with constraints. As well, we establish a bound on the convergence rate for the penalty method with the one-parametric class of functions serving as penalty functions. This bound is derived provided that the exact resolving of the sequence of unconstrai...

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Some Results About Proximal-Like Methods

Cited by 4 publications

References 36 publications

Proximal point method and elliptic regularization

Proximal point method and elliptic regularization

Variational Inequalities

On Applicability of a Class of Parametric Functions as Exterior Penalty Operators to Solve Nonlinear Problems with Constraints

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