Hemivariational inequality approach to constrained problems for star-shaped admissible sets

Naniewicz, Z.

doi:10.1007/bf02191764

Cited by 23 publications

(9 citation statements)

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“…In the current paper, we treat the counterpart of problem (1) where the set K represents a set of admissible constraints which is star-shaped with respect to a certain ball in V. Note that for this class of nonconvex sets, some particular versions have been studied earlier in Section 7.4 of [8] if j = 0, in Section 7.3 of [8,12] if ϕ = 0, and Section 7.2 of [8] when ϕ = j = 0. The first novelty of our contribution is to study the class of fully variational-hemivariational inequalities involving nonconvex constraints.…”

Section: Introductionmentioning

confidence: 99%

Constrained Variational-Hemivariational Inequalities on Nonconvex Star-Shaped Sets

Migórski

Fengzhen

2020

Mathematics

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In this paper, we study a class of constrained variational–hemivariational inequality problems with nonconvex sets which are star-shaped with respect to a certain ball in a reflexive Banach space. The inequality is a fully nonconvex counterpart of the variational–hemivariational inequality of elliptic type since it contains both, a convex potential and a locally Lipschitz one. Two new results on the existence of a solution are proved by a penalty method applied to a variational–hemivariational inequality penalized by the generalized directional derivative of the distance function of the constraint set. In the first existence theorem, the strong monotonicity of the governing operator and a relaxed monotonicity condition of the Clarke subgradient are assumed. In the second existence result, these two hypotheses are relaxed and a suitable hypothesis on the upper semicontinuity of the operator is adopted. In both results, the penalized problems are solved by using the Knaster, Kuratowski, and Mazurkiewicz (KKM) lemma. For a suffciently small penalty parameter, the solution to the penalized problem solves also the original one. Finally, we work out an example on the interior and boundary semipermeability problem that ilustrate the applicability of our results.

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Section: Introductionmentioning

confidence: 99%

Constrained Variational-Hemivariational Inequalities on Nonconvex Star-Shaped Sets

Migórski

Fengzhen

2020

Mathematics

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“…For more details about the algorithmic treatment the reader is referred to Chapter 6 and to Bibliographical Remarks The application of nonconvex potentials, the formulation of nonconvex variational inequality problems and the hemivariational inequalities (after Panagiotopoulos) for the treatment of the corresponding problems, have been studied in several recent publications. The reader is referred to the references given in Chapter 1, the monographies , Naniewicz and Panagiotopoulos, 1995 and, among others, to the following publications: Panagiotopoulos, 1983, Baniotopoulos and Panagiotopoulos, 1987, Naniewicz, 1989, Motreanu and Panagiotopoulos, 1993, Miettinen, 1993, Naniewicz, 1994a, Naniewicz, 1994b Miettinen, 1995, Motreanu and Panagiotopoulos, 1995, Naniewicz, 1995, Goeleven, 1997.…”

Section: Heuristic Nonconvex Optimization Approachmentioning

confidence: 99%

Nonconvex Optimization in Mechanics

Mistakidis

Stavroulakis

1998

Nonconvex Optimization and Its Applications

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This book is dedicated to our teacher, Professor P. D. Pa nagiotopoulos 206 xiii XV xvi NONCONVEX OPTIMIZATION IN MECHANICSThe book is divided into four parts. The first part presents a number of engineering problems involving nonmonotone stress-strain or reaction-displacement laws, leading to nonconvex energy functions. The difficulties introduced by the nonconvexity and nonsmoothness is demonstrated by very simple examples involving one or two variables.The second part gives the mathematical background of applied convex and nonconvex optimization. After a short review of the literature, new algorithms inspired from engineering applications are proposed for the solution of the mathematical problem.The third part deals with the mathematical formulation of convex and nonconvex, smooth and nonsmooth problems in mechanics and engineering. Chapter 3 deals with convex problems, while in Chapter 4 nonconvex problems are presented. The presentation of the problems follows the same sequence in both Chapters in order to facilitate the comparison of nonconvex problems with respect to the corresponding convex problems. Starting from the basic relations of mechanics, a detailed formulation is presented for a large class of engineering problems. First, interface problems are treated and the corresponding variational relations are obtained. Then the treatment is extended in order to include material nonlinearities.In the last Chapter of this part {Chapter 5), some problems of optimal design of structures are briefly treated. After the theoretical formulation of the problem, certain algorithms are presented for its numerical treatment.The fourth part presents algorithms specialized for the treatment of a large class of nonconvex, nonsmooth optimization problems appearing in mechanics and engineering. The algorithms are presented in Chapter 6 and described in all possible detail; certain hints on their computer implementation are given. In Chapter 7 a large number of engineering applications is presented. After the detailed examination of the behaviour of the algorithms on very simple examples, realistic engineering applications are presented with large number of unknowns, exhibiting the potential of the optimization approach.The authors are deeply greatful to Professor P.O. Panagiotopoulos {Aristotle University of Thessaloniki, Greece and RWTH Aachen, Germany) for his encouragement to publish this book, for his helpful suggestions and his continuous support. We would also like to thank Professor K.T. Thomopoulos and Professor C.C. Baniotopoulos for the helpful discussions and their advices concerning the engineering oriented topics and Dr. O.K. Panagouli for her help on the topics concerning the application of fractal geometry to mechanics.

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“…We can also find a sequence {un ( n E Bv} such that t, :=I[ u, II-+ co, W n := unl II un II-w and Let us now come back to problem P. As in the article of Naniewicz [15] we will use a penalization method, but to prove the existence of a solution for the penalized problem we need another approach than the one used by Naniewicz [15] which is only valid for coercive problems.…”

Section: Preliminaries Notations and Basic Factsmentioning

confidence: 99%

“…neither convex nor differentiable. As a consequence of the contributions of Panagiotopoulos, the study of hemivariational inequalities has emerged as an interesting branch of applied mathematics and this topic is now the subject of the attention of several engineers and mathematicians (see Goeleven and Thera [13], Naniewicz [ 15]-[ 171 and Panagiotopoulos [ 18]- [22]). …”

Section: Introductionmentioning

confidence: 99%

Noncoercive hemivariational inequality approach to constrained problems for star-shaped admissible sets

Goeleven

1996

J Glob Optim

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Hemivariational inequality approach to constrained problems for star-shaped admissible sets

Cited by 23 publications

References 24 publications

Constrained Variational-Hemivariational Inequalities on Nonconvex Star-Shaped Sets

Constrained Variational-Hemivariational Inequalities on Nonconvex Star-Shaped Sets

Nonconvex Optimization in Mechanics

Noncoercive hemivariational inequality approach to constrained problems for star-shaped admissible sets

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