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1
Content available remote An extragradient inertial algorithm for solving split fixed-point problems of demicontractive mappings, with equilibrium and variational inequality problems
Okeke Chibueze C., Ugwunnadi Godwin C., Jolaoso Lateef O.
Demonstratio Mathematica
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2022
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Vol. 55, nr 1
506--527
EN
The purpose of this article is to study and analyse a new extragradient-type algorithm with an inertial extrapolation step for solving split fixed-point problems for demicontractive mapping, equilibrium problem, and pseudomonotone variational inequality problem in real Hilbert spaces. One of the advantages of the proposed algorithm is that a strong convergence result is achieved without a prior estimate of the Lipschitz constant of the cost operator, which is very difficult to find. In addition, the stepsize is generated at each iteration by some simple computations, which allows it to be easily implemented without the prior knowledge of the Lipschitz constant of the cost operator. Some numerical experiments are reported to show the performance and behaviour of the sequence generated by our algorithm. The obtained results in this article extend and improve many related recent results in this direction in the literature.
2
Content available remote Two strongly convergent self-adaptive iterative schemes for solving pseudo-monotone equilibrium problems with applications
Pakkaranang Nuttapol, Rehman Habib ur, Kumam Wiyada
Demonstratio Mathematica
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2021
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Vol. 54, nr 1
280--298
EN
The aim of this paper is to propose two new modified extragradient methods to solve the pseudo-monotone equilibrium problem in a real Hilbert space with the Lipschitz-type condition. The iterative schemes use a new step size rule that is updated on each iteration based on the value of previous iterations. By using mild conditions on a bi-function, two strong convergence theorems are established. The applications of proposed results are studied to solve variational inequalities and fixed point problems in the setting of real Hilbert spaces. Many numerical experiments have been provided in order to show the algorithmic performance of the proposed methods and compare them with the existing ones.
3
Content available remote Strong convergence inertial projection algorithm with self-adaptive step size rule for pseudomonotone variational inequalities in Hilbert spaces
Wairojjana Nopparat, Pakkaranang Nuttapol, Pholasa Nattawut
Demonstratio Mathematica
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2021
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Vol. 54, nr 1
110--128
EN
In this paper, we introduce a new algorithm for solving pseudomonotone variational inequalities with a Lipschitz-type condition in a real Hilbert space. The algorithm is constructed around two algorithms: the subgradient extragradient algorithm and the inertial algorithm. The proposed algorithm uses a new step size rule based on local operator information rather than its Lipschitz constant or any other line search scheme and functions without any knowledge of the Lipschitz constant of an operator. The strong convergence of the algorithm is provided. To determine the computational performance of our algorithm, some numerical results are presented.
4
Content available remote A self-adaptive Tseng extragradient method for solving monotone variational inequality and fixed point problems in Banach spaces
Jolaoso Lateef Olakunle
Demonstratio Mathematica
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2021
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Vol. 54, nr 1
527--547
EN
In this paper, we introduce a self-adaptive projection method for finding a common element in the solution set of variational inequalities (VIs) and fixed point set for relatively nonexpansive mappings in 2-uniformly convex and uniformly smooth real Banach spaces. We prove a strong convergence result for the sequence generated by our algorithm without imposing a Lipschitz condition on the cost operator of the VIs. We also provide some numerical examples to illustrate the performance of the proposed algorithm by comparing with related methods in the literature. This result extends and improves some recent results in the literature in this direction.
5
Content available remote Two modifications of the inertial Tseng extragradient method with self-adaptive step size for solving monotone variational inequality problems
Alakoya Timilehin Opeyemi, Jolaoso Lateef Olakunle, Mewomo Oluwatosin Temitope
Demonstratio Mathematica
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2020
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Vol. 53, nr 1
208--224
EN
In this work, we introduce two new inertial-type algorithms for solving variational inequality problems (VIPs) with monotone and Lipschitz continuous mappings in real Hilbert spaces. The first algorithm requires the computation of only one projection onto the feasible set per iteration while the second algorithm needs the computation of only one projection onto a half-space, and prior knowledge o fthe Lipschitz constant of the monotone mapping is not required in proving the strong convergence theorems for the two algorithms. Under some mild assumptions, we prove strong convergence results for the proposed algorithms to a solution of a VIP. Finally, we provide some numerical experiments to illustrate the efficiency and advantages of the proposed algorithms.
6
Content available remote Generalized implicit viscosity approximation method for multivalued mappings in CAT(0) spaces
Abbas Mujahid, Iqbal Hira, de la Sen Manuel
Demonstratio Mathematica
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2019
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Vol. 52, nr 1
347--360
EN
We prove strong convergence of the sequence generated by implicit viscosity approximation method involving a multivalued nonexpansive mapping in framework of CAT(0) space. Under certain appropriate conditions on parameters, we show that such a sequence converges strongly to a fixed point of the mapping which solves a variational inequality. We also present some convergence results for the implicit viscosity approximation method in complete R-trees without the endpoint condition.
7
Content available remote A self adaptive inertial subgradient extragradient algorithm for variational inequality and common fixed point of multivalued mappings in Hilbert spaces
Jolaoso Lateef Olakunle, Taiwo Adeolu, Alakoya Timilehin Opeyemi, Mewomo Oluwatosin Temitope
Demonstratio Mathematica
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2019
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Vol. 52, nr 1
183--203
EN
We consider a new subgradient extragradient iterative algorithm with inertial extrapolation for approximating a common solution of variational inequality problems and fixed point problems of a multivalued demicontractive mapping in a real Hilbert space. We established a strong convergence theorem for our proposed algorithm under some suitable conditions and without prior knowledge of the Lipschitz constant of the underlying operator. We present numerical examples to show that our proposed algorithm performs better than some recent existing algorithms in the literature.
8
Content available remote A non-interior-point algorithm predictor-corrector algorithm for Variational Inequality Problem with equality linear constraints
Hu X., Huang C., Luo A., Chen H.
Przegląd Elektrotechniczny
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2013
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R. 89, nr 3b
101--103
EN
We present a noninterior-point predictor-corrector algorithm for variational inequality Problem with equality linear constraints based on Chen-Kanzow-Smale smoothing techniques. This method is based upon a modified predictor-corrector interior-point algorithm. It is established the global linear convergence.
PL
W artykule przedstawiono metodę punktów nie-wewnętrznych predykator-korektor, uwzględniający nierówności wariancyjne z liniowym ograniczeniem liniowości. Jego działanie oparto na technice Chen-Kanzow-Smale oraz zmodyfikowanej metodzie punktów wewnętrznych predykator-korektor. Analizie poddano liniową zbieżność algorytmu.
9
Content available remote A sign preserving mixed finite element approximation for contact problems
Hild P.
International Journal of Applied Mathematics and Computer Science
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2011
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Vol. 21, no 3
487-498
EN
This paper is concerned with the frictionless unilateral contact problem (i.e., a Signorini problem with the elasticity operator). We consider a mixed finite element method in which the unknowns are the displacement field and the contact pressure. The particularity of the method is that it furnishes a normal displacement field and a contact pressure satisfying the sign conditions of the continuous problem. The a priori error analysis of the method is closely linked with the study of a specific positivity preserving operator of averaging type which differs from the one of Chen and Nochetto. We show that this method is convergent and satisfies the same a priori error estimates as the standard approach in which the approximated contact pressure satisfies only a weak sign condition. Finally we perform some computations to illustrate and compare the sign preserving method with the standard approach.
10
Content available remote Topological derivatives for semilinear elliptic equations
Iguernane M., Nazarov S. A., Roche J. R., Sokolowski J., Szulc K.
International Journal of Applied Mathematics and Computer Science
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2009
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Vol. 19, no 2
191-205
EN
The form of topological derivatives for an integral shape functional is derived for a class of semilinear elliptic equations. The convergence of finite element approximation for the topological derivatives is shown and the error estimates in the L [...] norm are obtained. The results of numerical experiments which confirm the theoretical convergence rate are presented.
11
Content available remote A level set method in shape and topology optimization for variational inequalities
Fulmański P., Laurain A., Scheid J. F., Sokołowski J.
International Journal of Applied Mathematics and Computer Science
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2007
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Vol. 17, no 3
413-430
EN
The level set method is used for shape optimization of the energy functional for the Signorini problem. The boundary variations technique is used in order to derive the shape gradients of the energy functional. The conical differentiability of solutions with respect to the boundary variations is exploited. The topology modifications during the optimization process are identified by means of an asymptotic analysis. The topological derivatives of the energy shape functional are employed for the topology variations in the form of small holes. The derivation of topological derivatives is performed within the framework proposed in (Sokołowski and Żochowski, 2003). Numerical results confirm that the method is efficient and gives better results compared with the classical shape optimization techniques.
12
Content available remote On a Regularization Method for Variational Inequalities with P0 Mappings
Konnov I., Mazurkevich E., Ali M.
International Journal of Applied Mathematics and Computer Science
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2005
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Vol. 15, no 1
35-44
EN
We consider partial Browder-Tikhonov regularization techniques for variational inequality problems with P0 cost mappings and box-constrained feasible sets. We present classes of economic equilibrium problems which satisfy such assumptions and propose a regularization method for these problems.
13
A variational inequality approach to an elastoplastic plate-foundation system
Kuczma M. S., Litewka P., Rakowski J., Whiteman J. R.
Foundations of Civil and Environmental Engineering
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2004
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No. 4
31-48
EN
A variational formulation of the inequality problem of a plate resting on a foundation is developed. The unilateral interaction conditions between the plate and its foundation are taken into account. For the plate the kinematical model of Reissner-Mindlin is applied and the elastoplastic material behaviour is assumed. The foundation is modelled as an elastoplastic medium of Winkler type. These fundamental assumptions impose inequality constraints on some quantities describing the plate-foundation system and, on the mathematical side, lead to an unilateral boundary value problem. We present a unified approach to this problem in which both the conditions of unilateral contact and those of elastoplastic behaviour are governed by variational inequalities. The proposed variational formulation consists of a variational equation and three variational inequalities. The finite element solution to the weak formulation is defined and numerical results for some test examples are presented.
PL
Artykuł dotyczy zagadnienia współpracy płyty z podłożem o więzach jednostronnych. Przyjęto, że zarówno materiał płyty, jak i materiał podłoża są spreżystopla-styczne, oraz dopuszczono możliwość odrywania się płyty od podłoża. Wykorzystując analogiczną strukturę matematyczną związków plastycznego płynięcia i więzów jednostronnych (bez tarcia), autorom udało się ująć analizowane zagadnienie w jednolitej postaci nierówności wariacyjnej. Po dyskretyzacji metoda elementów skończonych to nieliniowe ewolucyjne zagadnienie brzegowe sprowadzono do sekwencji zagadnień liniowego dopełnienia. Artykuł zawiera wyniki obliczeń numerycznych dla sprężystej płyty i spreżysto-plastycznego podłoża z więzami jednostronnymi.
14
Content available remote Robinson's implicit function theorem
Dontchev A. L.
Control and Cybernetics
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2003
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Vol. 32, no 3
529-541
EN
Robinson's implicit function theorem has played a mayor role in the analysis of stability of optimization problems in the last two decades. In this paper we take a new look at this theorem, and with an updated terminology go back to the roots and present some extensions.
15
Content available remote On the Solution of a Finite Element Approximation of a Linear Obstacle Plate Problem
Fernandes L. M., Figueiredo I. N., Judice J. J.
International Journal of Applied Mathematics and Computer Science
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2002
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Vol. 12, no 1
27-40
EN
In this paper the solution of a finite element approximation of a linear obstacle plate problem is investigated. A simple version of an interior point method and a block pivoting algorithm have been proposed for the solution of this problem. Special purpose implementations of these procedures are included and have been used in the solution of a set of test problems. The results of these experiences indicate that these procedures are quite efficient to deal with these instances and compare favourably with the path-following PATH and the active-set MINOS codes of the commercial GAMS collection
16
Content available Proximal-based regularization methods and successive approximation of variational inequalities in Hilbert spaces
Kaplan A., Tichatschke R.
Control and Cybernetics
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2002
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Vol. 31, no 3
521-544
EN
For variational inequalities with multi-valued, maximal monotone operators in Hilbert spaces we study proximal-based methods with an improvement of the data approximation after each (approximately performed) proximal iteration. The standard conditions on a distance functional of Bregman's type are weakened, depending on a "reserve of monotonicity" of the operator in the variational inequality, and the enlargement concept is used for approximating the operator. Weak convergence of the proxinnal iterates to a solution of tire original problem is proved. The construction of the [epsilon]-enlargement of monotone operators is analyzed for some particular cases.
17
Content available On some optimization problem related to economic equilibrium
Naniewicz Z.
Control and Cybernetics
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2002
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Vol. 31, no 1
141-165
EN
The paper considers an optimization problem in which the minima of a finite collection of objective functions satisfy some unilateral constraints and are linked together by a certain subdifferential relationship. The governing relations are stated as a variational inequality defined on a nonconvex feasible set. By the reduction to the variational inequality involving nonmonotone multivalued mapping, defined over nonnegative orthant, the existence of solutions is examined. The prototype is the general economic equilibrium problem. The exemplification of the theory for the quadratic multi-objective function is provided.
18
Content available remote Variational Analysis of a Frictional Contact Problem for the Bingham Fluid
Awbi B., Selmani L., Sofonea M.
International Journal of Applied Mathematics and Computer Science
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1999
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Vol. 9, no 2
371-385
EN
We consider a mathematical model which describes the flow of a Bingham fluid with friction. We assume a stationary flow and we model the contact with damped response and a local version of Coulomb's law of friction.The problem leads to a quasi-variational inequality for the velocity field. We establish the existence of a weak solution and, under additional assumptions, its uniqueness. The proofs are based on a new result obtained in (Motreanu and Sofonea, 1999). We also establish the continuous dependence of the solution with respect to the contact conditions.
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