418 results on '"Variational Inequalities"'
Search Results
2. Modelling, Analysis and Computation in Plasticity.
- Author
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Reddy, B. Daya
- Subjects
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NUMERICAL solutions to partial differential equations , *ALGORITHMS , *MATHEMATICAL inequalities , *MATERIAL plasticity , *APPROXIMATION theory , *STOCHASTIC convergence - Abstract
The typical problem in the mechanics of deformable solids comprises a mathematical model in the form of systems of partial differential equations or inequalities. Subsequent investigations are then concerned with analysis of the model to determine its well-posedness, followed by the development and implementation of algorithms to obtain approximate solutions to problems that are generally intractable in closed form. These processes of modelling, analysis, and computation are discussed with a focus on the behaviour of elastic-plastic bodies; these are materials which exhibit path-dependence and irreversibility in their behaviour. The resulting variational problem is an inequality that is not of standard elliptic or parabolic type. Properties of this formulation are reviewed, as are the conditions under which fully discrete approximations converge. A solution algorithm, motivated by the predictor-corrector algorithms that are common in elastoplastic problems, is presented and its convergence properties summarized. An important extension of the conventional theory is that of straingradient plasticity, in which gradients of the plastic strain appear in the formulation, and which includes a length scale not present in the conventional theory. Some recent results for strain-gradient plasticity are presented, and the work concludes with a brief description of current investigations. [ABSTRACT FROM AUTHOR] more...
- Published
- 2019
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3. A new inexact SQP algorithm for nonlinear systems of mixed equalities and inequalities.
- Author
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Gu, Chao, Zhu, Detong, and Pei, Yonggang
- Subjects
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QUADRATIC programming , *NONLINEAR systems , *MATHEMATICAL inequalities , *COMPLEMENTARITY constraints (Mathematics) , *STOCHASTIC convergence - Abstract
Traditional inexact SQP algorithm can only solve equality constrained optimization (Byrd et al. Math. Program. 122, 273-299 2010). In this paper, we propose a new inexact SQP algorithm with affine scaling technique for nonlinear systems of mixed equalities and inequalities, which arise in complementarity and variational inequalities. The nonlinear systems are transformed into a special nonlinear optimization with equality and bound constraints, and then we give a new inexact SQP algorithm for solving it. The new algorithm equipped with affine scaling technique does not require a quadratic programming subproblem with inequality constraints. The search direction is computed by solving one linear system approximately using iterative linear algebra techniques. Under mild assumptions, we discuss the global convergence. The preliminary numerical results show the effectiveness of the proposed algorithm. [ABSTRACT FROM AUTHOR] more...
- Published
- 2018
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4. Existence theorems via duality for equilibrium problems with trifunctions.
- Author
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Inoan, D. and Kolumbán, J.
- Subjects
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EXISTENCE theorems , *MATHEMATICAL inequalities , *SADDLEPOINT approximations , *VECTOR topology , *NASH equilibrium - Abstract
We present an existence result for an equilibrium problem formulated with trifunctions, which is motivated by variational inequalities governed by quasimonotone operators. To prove the existence result, we define the dual problem, and some monotonicity notions for trifunctions. From the main result follow, among others, the Browder-Minty theorem for variational inequalities and Ky Fan’s Minimax theorem. Some applications for mixed equilibrium problems and variational inequalities are given. [ABSTRACT FROM AUTHOR] more...
- Published
- 2018
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5. A scalable variational inequality approach for flow through porous media models with pressure-dependent viscosity.
- Author
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Mapakshi, N.K., Chang, J., and Nakshatrala, K.B.
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POROUS materials , *PRESSURE measurement , *MATHEMATICAL inequalities , *VISCOSITY , *ANISOTROPY - Abstract
Mathematical models for flow through porous media typically enjoy the so-called maximum principles, which place bounds on the pressure field. It is highly desirable to preserve these bounds on the pressure field in predictive numerical simulations, that is, one needs to satisfy discrete maximum principles (DMP). Unfortunately, many of the existing formulations for flow through porous media models do not satisfy DMP. This paper presents a robust, scalable numerical formulation based on variational inequalities (VI), to model non-linear flows through heterogeneous, anisotropic porous media without violating DMP. VI is an optimization technique that places bounds on the numerical solutions of partial differential equations. To crystallize the ideas, a modification to Darcy equations by taking into account pressure-dependent viscosity will be discretized using the lowest-order Raviart–Thomas (RT0) and Variational Multi-scale (VMS) finite element formulations. It will be shown that these formulations violate DMP, and, in fact, these violations increase with an increase in anisotropy. It will be shown that the proposed VI-based formulation provides a viable route to enforce DMP. Moreover, it will be shown that the proposed formulation is scalable, and can work with any numerical discretization and weak form. A series of numerical benchmark problems are solved to demonstrate the effects of heterogeneity, anisotropy and non-linearity on DMP violations under the two chosen formulations (RT0 and VMS), and that of non-linearity on solver convergence for the proposed VI-based formulation. Parallel scalability on modern computational platforms will be illustrated through strong-scaling studies, which will prove the efficiency of the proposed formulation in a parallel setting. Algorithmic scalability as the problem size is scaled up will be demonstrated through novel static-scaling studies. The performed static-scaling studies can serve as a guide for users to be able to select an appropriate discretization for a given problem size. [ABSTRACT FROM AUTHOR] more...
- Published
- 2018
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6. On Hölder continuity of approximate solution maps to vector equilibrium problems.
- Author
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Lam QUOC ANH, Kien TRUNG NGUYEN, and Tran NGOC TAM
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MAPS , *EQUILIBRIUM , *VECTOR algebra , *APPROXIMATE solutions (Logic) , *MATHEMATICAL inequalities - Abstract
In this article, we consider parametric vector equilibrium problems in normed spaces. Sufficient conditions for Hölder continuity of approximate solution mappings where they are set-valued are established. As applications of these results, the Hölder continuity of the approximate solution mappings for vector optimization problems and vector variational inequalities are derived at the end of the paper. Our results are new and include the existing ones in the literature. [ABSTRACT FROM AUTHOR] more...
- Published
- 2017
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7. Viscosity solutions of obstacle problems for fully nonlinear path-dependent PDEs.
- Author
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Ekren, Ibrahim
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VISCOSITY solutions , *PARTIAL differential equations , *LIPSCHITZ spaces , *STOCHASTIC differential equations , *MATHEMATICAL inequalities - Abstract
In this article, we adapt the definition of viscosity solutions to the obstacle problem for fully nonlinear path-dependent PDEs with data uniformly continuous in ( t , ω ) , and generator Lipschitz continuous in ( y , z , γ ) . We prove that our definition of viscosity solutions is consistent with the classical solutions, and satisfy a stability result. We show that the value functional defined via the second order reflected backward stochastic differential equation is the unique viscosity solution of the variational inequalities. [ABSTRACT FROM AUTHOR] more...
- Published
- 2017
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8. A migration equilibrium model with uncertain data and movement costs.
- Author
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Causa, A., Jadamba, B., and Raciti, F.
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UNCERTAINTY (Information theory) ,INFORMATION theory in economics ,DISTRIBUTION (Probability theory) ,MATHEMATICAL inequalities ,UTILITY functions - Abstract
In this paper, we consider a variational inequality model of migration in which populations move according to the utility functions corresponding to various locations and movement costs. In contrast to previously studied models based on variational inequalities, we take into account the possibility that some of the data in the model are not deterministic but, instead, are known through their probability distributions. Our theoretical framework is that of random variational inequalities in Lebesgue spaces. [ABSTRACT FROM AUTHOR] more...
- Published
- 2017
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9. A posteriori error control and adaptivity of [formula omitted]-finite elements for mixed and mixed-hybrid methods.
- Author
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Petsche, Jan and Schröder, Andreas
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HYBRID systems , *ERROR analysis in mathematics , *FINITE element method , *MATHEMATICAL inequalities , *PARAMETER estimation , *LAGRANGE multiplier - Abstract
In this paper, mixed and mixed-hybrid methods for h - and h p -adaptive finite elements on quadrilateral meshes are discussed for variational equations and, in particular, for variational inequalities. The main result is the derivation of reliable error estimates for mixed methods for the obstacle problem. The estimates rely on the use of a post-processing of the potential in H 1 and on the introduction of a certain Lagrange multiplier which is associated with the obstacle constraints. The error estimates consist of the dual norm of the residual, which is defined by an appropriate approximation of the Lagrange multiplier, plus some computable remainder terms. In numerical experiments, the applicability of the post-processing procedure on quadrilateral meshes with multilevel hanging-nodes is verified and the use of the estimates in h - and h p -adaptive schemes is demonstrated by means of convergence rates and effectivity indices. [ABSTRACT FROM AUTHOR] more...
- Published
- 2017
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10. Density of convex intersections and applications.
- Author
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Hintermüller, M., Rautenberg, C. N., and Rösel, S.
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SET theory , *FUNCTION spaces , *STOCHASTIC convergence , *MATHEMATICAL regularization , *DISCRETIZATION methods , *MATHEMATICAL inequalities - Abstract
In this paper, we address density properties of intersections of convex sets in several function spaces. Using the concept of Γ-convergence, it is shown in a general framework, how these density issues naturally arise from the regularization, discretization or dualization of constrained optimization problems and from perturbed variational inequalities. A variety of density results (and counterexamples) for pointwise constraints in Sobolev spaces are presented and the corresponding regularity requirements on the upper bound are identified. The results are further discussed in the context of finite-element discretizations of sets associated with convex constraints. Finally, two applications are provided, which include elasto-plasticity and image restoration problems. [ABSTRACT FROM AUTHOR] more...
- Published
- 2017
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11. A smoothing inexact Newton method for variational inequalities with nonlinear constraints.
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Ge, Zhili, Ni, Qin, and Zhang, Xin
- Subjects
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NEWTON-Raphson method , *MATHEMATICAL inequalities , *LINEAR systems , *STOCHASTIC convergence , *CONCAVE functions - Abstract
In this paper, we propose a smoothing inexact Newton method for solving variational inequalities with nonlinear constraints. Based on the smoothed Fischer-Burmeister function, the variational inequality problem is reformulated as a system of parameterized smooth equations. The corresponding linear system of each iteration is solved approximately. Under some mild conditions, we establish the global and local quadratic convergence. Some numerical results show that the method is effective. [ABSTRACT FROM AUTHOR] more...
- Published
- 2017
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12. A note on strongly nonlinear parabolic variational inequalities.
- Author
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El-Dessouky, A. T.
- Subjects
NONLINEAR analysis ,PARABOLA ,MATHEMATICAL inequalities ,PARABOLIC operators ,POLYNOMIALS - Abstract
We prove the existence of weak solutions of variational inequalities for general quasilinear parabolic operators of order m = 2 with strongly nonlinear perturbation term. The result is based on a priori bound for the time derivatives of the solutions. [ABSTRACT FROM AUTHOR] more...
- Published
- 2017
- Full Text
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13. The pseudomonotone polar for multivalued operators.
- Author
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Bueno, Orestes and Cotrina, John
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MONOTONE operators , *MANY-valued logic , *OPERATOR theory , *MATHEMATICAL inequalities , *ORTHOGONAL polynomials - Abstract
In this work, we study the pseudomonotonicity of multivalued operators from the point of view of polarity, in an analogous way as the well-known monotone polar due to Martínez-Legaz and Svaiter, and the quasimonotone polar recently introduced by Bueno and Cotrina. We show that this new polar, adapted for pseudomonotonicity, possesses analogous properties to the monotone and quasimonotone polar, among which are a characterization of pseudomonotonicity, maximality and pre-maximality. Furthermore, we characterize the notion ofD-maximal pseudomonotonicity introduced by Hadjisavvas. We conclude this work studying the connections between pseudomonotonicity, variational inequality problems and upper sign-continuity. [ABSTRACT FROM PUBLISHER] more...
- Published
- 2017
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14. A new LQP alternating direction method for solving variational inequality problems with separable structure.
- Author
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Bnouhachem, Abdellah, Hamdi, Abdelouahed, and Xu, M. H.
- Subjects
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MATHEMATICAL inequalities , *LOGARITHMIC functions , *QUADRATIC forms , *PROBLEM solving , *CONVEX programming - Abstract
We presented a new logarithmic-quadratic proximal alternating direction scheme for the separable constrained convex programming problem. The predictor is obtained by solving series of related systems of non-linear equations in a parallel wise. The new iterate is obtained by searching the optimal step size along a new descent direction. The new direction is obtained by the linear combination of two descent directions. Global convergence of the proposed method is proved under certain assumptions. We show theO(1 / t) convergence rate for the parallel LQP alternating direction method. [ABSTRACT FROM PUBLISHER] more...
- Published
- 2016
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15. Vector valued q-variation for ergodic averages and analytic semigroups.
- Author
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Hong, Guixiang and Ma, Tao
- Subjects
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VECTOR valued groups , *ERGODIC theory , *SEMIGROUPS (Algebra) , *LATTICE theory , *MATHEMATICAL inequalities , *STOCHASTIC convergence , *EXTRAPOLATION - Abstract
In this paper, we establish UMD lattice-valued variational inequalities for ergodic averages of contractively regular operators and analytic semigroups. These results generalize, on the one hand some scalar-valued variational inequalities in ergodic theory, on the other hand Xu's very recent result on UMD lattice-valued maximal inequality. As an application, we deduce the jump estimates and obtain quantitative information on the rate of the pointwise convergence for semigroups acting on UMD lattice-valued functions. [ABSTRACT FROM AUTHOR] more...
- Published
- 2016
- Full Text
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16. Merit functions: a bridge between optimization and equilibria.
- Author
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Pappalardo, Massimo, Mastroeni, Giandomenico, and Passacantando, Mauro
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MATHEMATICAL inequalities , *SMOOTHNESS of functions , *MATHEMATICAL optimization , *MATHEMATICAL reformulation , *VARIATIONAL inequalities (Mathematics) , *CONVEX domains - Abstract
In the last decades, many problems involving equilibria, arising from engineering, physics and economics, have been formulated as variational mathematical models. In turn, these models can be reformulated as optimization problems through merit functions. This paper aims at reviewing the literature about merit functions for variational inequalities, quasi-variational inequalities and abstract equilibrium problems. Smoothness and convexity properties of merit functions and solution methods based on them will be presented. [ABSTRACT FROM AUTHOR] more...
- Published
- 2016
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17. Finite volume method for the variational inequalities of first and second kinds.
- Author
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Zhang, Tie and Tang, Lixin
- Subjects
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VARIATIONAL inequalities (Mathematics) , *NUMERICAL analysis , *PARTIAL differential equations , *MATHEMATICAL inequalities , *INFINITE processes - Abstract
We propose and analyze the finite volume method for solving the variational inequalities of first and second kinds. The stability and convergence analysis are given for this method. For the elliptic obstacle problem,we derive the optimal error estimate in the H1-norm. For the simplified friction problem,we establish an abstract H1-error estimate, which implies the convergence if the exact solution u ϵ H1(Ω) and the optimal error estimate if u ϵ H11+α(Ω)0 < ⩽ 2. [ABSTRACT FROM AUTHOR] more...
- Published
- 2015
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18. Variational Inequalities Characterizing Weak Minimality in Set Optimization.
- Author
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Crespi, Giovanni, Rocca, Matteo, and Schrage, Carola
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MATHEMATICAL optimization , *MATHEMATICAL inequalities , *EQUILIBRIUM , *DERIVATIVES (Mathematics) , *PARETO distribution - Abstract
We introduce the notion of weak minimizer in set optimization. Necessary and sufficient conditions in terms of scalarized variational inequalities of Stampacchia and Minty type, respectively, are proved. As an application, we obtain necessary and sufficient optimality conditions for weak efficiency of vector optimization in infinite-dimensional spaces. A Minty variational principle in this framework is proved as a corollary of our main result. [ABSTRACT FROM AUTHOR] more...
- Published
- 2015
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19. The abstract Lewy–Stampacchia inequality and applications.
- Author
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Gigli, Nicola and Mosconi, Sunra
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MATHEMATICAL inequalities , *ESTIMATION theory , *MATHEMATICS theorems , *MATHEMATICAL proofs , *LAPLACIAN operator - Abstract
We identify submodularity as the key ingredient needed to get the Lewy–Stampacchia inequality in the potential case, by showing how it can be used in a simple and effective way to produce a very abstract and general version of such estimate. We then discuss how to reproduce more classical versions of it and, more importantly, how it can be used in conjunction with Laplacian comparison estimates to produce large class of functions with bounded Laplacian on spaces with a lower bound on the Ricci curvature. [ABSTRACT FROM AUTHOR] more...
- Published
- 2015
- Full Text
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20. Incremental constraint projection methods for variational inequalities.
- Author
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Wang, Mengdi and Bertsekas, Dimitri
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MATHEMATICAL inequalities , *RANDOM projection method , *VARIATIONAL inequalities (Mathematics) , *STOCHASTIC analysis , *STOCHASTIC approximation - Abstract
We consider the solution of strongly monotone variational inequalities of the form $$F(x^*)'(x-x^*)\ge 0$$ , for all $$x\in X$$ . We focus on special structures that lend themselves to sampling, such as when $$X$$ is the intersection of a large number of sets, and/or $$F$$ is an expected value or is the sum of a large number of component functions. We propose new methods that combine elements of incremental constraint projection and stochastic gradient. These methods are suitable for problems involving large-scale data, as well as problems with certain online or distributed structures. We analyze the convergence and the rate of convergence of these methods with various types of sampling schemes, and we establish a substantial rate of convergence advantage for random sampling over cyclic sampling. [ABSTRACT FROM AUTHOR] more...
- Published
- 2015
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21. VARIATIONAL CONVERGENCE OF BIFUNCTIONS: MOTIVATING APPLICATIONS.
- Author
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JOFRÉ, ALEJANDRO and WETSÍ, ROGER J.-B.
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STABILITY (Mechanics) , *POSTURAL balance , *FIXED point theory , *NONLINEAR operators , *MATHEMATICAL inequalities - Abstract
It is shown that a number of variational and equilibrium problems can be cast as finding the maxinf-points or minsup-points of bivariate functions, for short, bifunctions. These problems include linear and nonlinear complementarity problems, fixed points, variational inequalities, inclusions, noncooperative games, and Walras and Nash equilibrium problems. One appeals to the theory of lopsided convergence for bifunctions to derive stability results for each one of these problems. [ABSTRACT FROM AUTHOR] more...
- Published
- 2014
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22. An approach to the numerical verification of solutions for variational inequalities using Schauder fixed point theory.
- Author
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Cheon Seoung Ryoo
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NUMERICAL analysis , *MATHEMATICAL inequalities , *SCHAUDER bases , *FIXED point theory , *PROBLEM solving - Abstract
In this paper, we describe a numerical method to verify the existence of solutions for a unilateral boundary value problems for second order equation governed by the variational inequalities. It is based on Nakao's method by using finite element approximation and its explicit error estimates for the problem. Using the Riesz representation theory in Hilbert space, we first transform the iterative procedure of variational inequalities into a fixed point form. Then, using Schauder fixed point theory, we construct a high efficiency numerical verification method that through numerical computation generates a bounded, closed, convex set which includes the approximate solution. Finally, a numerical example is illustrated. [ABSTRACT FROM AUTHOR] more...
- Published
- 2014
- Full Text
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23. Global convergence rate of a standard multigrid method for variational inequalities.
- Author
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Badea, Lori
- Subjects
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STOCHASTIC convergence , *MATHEMATICAL inequalities , *MONOTONE operators , *MATHEMATICAL bounds , *BANACH spaces , *FINITE element method - Abstract
The main purpose of this paper is to give an estimation of the global convergence rate for the standard monotone multigrid method applied to variational inequalities whose constraints are of the two-obstacle type. The numerical experiments using this method have highlighted global upper bounds of the convergence rate, but, to our knowledge, no theoretical justification exists so far. The method was introduced by Mandel in 1984 for complementarity problems and named later by Kornhuber as the standard monotone multigrid method. First, we introduce the method as a subspace correction algorithm in a reflexive Banach space, prove its global convergence and estimate the error after making some assumptions. By introducing finite element spaces, this algorithm becomes a multilevel or multigrid method. In this case, we prove that the assumptions that we made in the general theory are satisfied and write the convergence rate as a function of the number of levels. Finally, we compare our results with the estimations of the asymptotic convergence rate existing in the literature for complementarity problems. [ABSTRACT FROM PUBLISHER] more...
- Published
- 2014
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24. Iterative method for solving geometrically nonlinear inverse problems of structural element shaping under creep conditions.
- Author
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Bormotin, K.
- Subjects
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PROBLEM solving , *ITERATIVE methods (Mathematics) , *NONLINEAR analysis , *COMPUTER software , *FINITE element method , *UNIQUENESS (Mathematics) , *MATHEMATICAL inequalities - Abstract
An iterative method is proposed for solving geometrically nonlinear inverse problems of shaping structural elements under creep conditions. The method is implemented using a software package based on finite element analysis. [ABSTRACT FROM AUTHOR] more...
- Published
- 2013
- Full Text
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25. Viscosity approximation methods based on generalized contraction mappings for a countable family of strict pseudo-contractions, a general system of variational inequalities and a generalized mixed equilibrium problem in Banach spaces.
- Author
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Sunthrayuth, Pongsakorn and Kumam, Poom
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VISCOSITY , *APPROXIMATION theory , *MATHEMATICAL mappings , *MATHEMATICAL inequalities , *SET theory , *FIXED point theory - Abstract
Abstract: In this paper, we introduce viscosity approximation methods based on generalized contraction mappings for finding a set of common fixed points of a countable family of strict pseudo-contraction mappings, the common element of the set solutions of a general system of variational inequalities with Lipschitzian and relaxed cocoercive mappings and the set of solutions of a generalized mixed equilibrium problem. Furthermore, strong convergence theorems of the purposed iterative process are established in the framework of Banach spaces. The results presented in this paper improve and extend many recent important results. [Copyright &y& Elsevier] more...
- Published
- 2013
- Full Text
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26. SENSITIVITY ANALYSIS OF SOME QUASI VARIATIONAL INEQUALITIES.
- Author
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NOOR, MUHAMMAD ASLAM and NOOR, KHALIDA INAYAT
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SENSITIVITY analysis , *WIENER-Hopf equations , *MATHEMATICAL inequalities , *MATHEMATICAL optimization , *FIXED point theory , *MATHEMATICAL programming - Abstract
In this paper, we develop a unified and general framework of the sensitivity analysis for a class of quasi variational inequalities involving three operators using the Wiener-Hopf equations technique. Our analysis does not involve the differentiability of the given data. Since the extended general quasi variational inequalities include classical variational inequalities, quasi (mixed) variational inequalities and complementarity problems as special cases, results obtained in this paper continue to hold for these problems. In fact, our results can be considered as a significant extension of previously known results. [ABSTRACT FROM AUTHOR] more...
- Published
- 2013
27. Stochastic optimal multi-modes switching with a viscosity solution approach
- Author
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El Asri, Brahim
- Subjects
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STOCHASTIC analysis , *SWITCHING theory , *VISCOSITY solutions , *COST functions , *EXISTENCE theorems , *PARTIAL differential equations , *MATHEMATICAL inequalities - Abstract
Abstract: We consider the problem of optimal multi-modes switching in finite horizon, when the state of the system, including the switching cost functions are arbitrary (). We show existence of the optimal strategy, via a verification theorem. Finally, when the state of the system is a Markov process, we show that the vector of value functions of the optimal problem is the unique viscosity solution to the system of variational partial differential inequalities with inter-connected obstacles. [Copyright &y& Elsevier] more...
- Published
- 2013
- Full Text
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28. Vibro-impact of a plate on rigid obstacles: existence theorem, convergence of a scheme and numerical simulations.
- Author
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Pozzolini, C., Renard, Y., and Salaun, M.
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EXISTENCE theorems , *STOCHASTIC convergence , *SCHEMES (Algebraic geometry) , *COMPUTER simulation , *DISCRETE systems , *APPROXIMATION theory , *MATHEMATICAL inequalities - Abstract
The purpose of this paper is to describe a fully discrete approximation and its convergence to the continuum dynamical impact problem for the fourth-order Kirchhoff–Love plate model with nonpenetration Signorini contact condition. We extend to the case of plates the theoretical results of weak convergence due to Y. Dumont and L. Paoli, which was stated for Euler–Bernouilli beams. In particular, this provides an existence result for the solution of this problem. Finally, we discuss the numerical results we obtain. [ABSTRACT FROM AUTHOR] more...
- Published
- 2013
- Full Text
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29. Solvability of Variational Inequalities on Hilbert Lattices.
- Author
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Nishimura, Hiroki and Ok, Efe A.
- Subjects
MATHEMATICAL inequalities ,HILBERT'S problems ,LATTICE theory ,MATHEMATICAL mappings ,GAME theory - Abstract
This paper provides a systematic solvability analysis for (generalized) variational inequalities on separable Hilbert lattices. By contrast to a large part of the existing literature, our approach is lattice-theoretic, and is not based on topological fixed point theory. This allows us to establish the solvability of certain types of (generalized) variational inequalities without requiring the involved (set-valued) maps be hemicontinuous or monotonic. Some of our results generalize those obtained in the context of nonlinear complementarity problems in earlier work, and appear to have scope for applications. This is illustrated by means of several applications to fixed point theory, optimization, and game theory. [ABSTRACT FROM AUTHOR] more...
- Published
- 2012
- Full Text
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30. Higher integrability in parabolic obstacle problems.
- Author
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Bögelein, Verena and Scheven, Christoph
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VARIATIONAL inequalities (Mathematics) , *DEGENERATE parabolic equations , *MATHEMATICAL singularities , *OPERATOR theory , *MATHEMATICAL inequalities , *MATHEMATICAL models , *MATHEMATICAL forms - Abstract
In this paper we establish the self-improving property of integrability for parabolic variational inequalities satisfying an obstacle constraint and involving possibly degenerate respectively singular operators in divergence form. In particular, our results apply to the model case of the variational inequality associated to the parabolic p-Laplacean operator. Thereby we do not impose any monotonicity assumption in time on the obstacle function. [ABSTRACT FROM AUTHOR] more...
- Published
- 2012
- Full Text
- View/download PDF
31. THREE-STEP PROJECTION METHOD FOR GENERAL VARIATIONAL INEQUALITIES.
- Author
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BNOUHACHEM, ABDELLAH and NOOR, MUHAMMAD ASLAM
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VARIATIONAL inequalities (Mathematics) , *MONOTONE operators , *STOCHASTIC convergence , *MATHEMATICAL inequalities , *MATHEMATICAL analysis , *OPERATOR theory - Abstract
In this paper, we suggest and analyze a new three-step iterative projection method for solving general variational inequalities in conjunction with a descent direction. We prove that the new method is globally convergent under suitable mild conditions. An example is given to illustrate the advantage and efficiency of the proposed method. [ABSTRACT FROM AUTHOR] more...
- Published
- 2012
- Full Text
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32. A Cooperative Coalitional Game in Duopolistic Supply-Chain Competition.
- Author
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Lin, Cheng-Chang and Hsieh, Chao-Chen
- Subjects
COOPERATIVE game theory ,SUPPLY chains ,ECONOMIC competition ,MATHEMATICAL models ,NASH equilibrium ,ALGORITHMS ,MATHEMATICAL inequalities - Abstract
Cooperative coalitional games study the selection of chain partners, the formation of supply chains and outcome allocations. The chain value of a coalition depends on the outcome of inter-chain competition. Subsequently, chain partners may accept their payoffs or decide to defect to another coalition that has made a higher tender offer. The formation and defection continues until a stable Cournot-Nash equilibrium is reached. This is the state where no player may unilaterally defect to another coalition and earn a higher profit. We formulate the cooperative coalitional game as a variational inequality problem and propose an iterative diagonalization algorithm to determine the steady state for the game. The computational results illustrated that (1) supply-chain competition may not necessarily preserve the same level of social welfare; (2) internalization of resources and costs may distort the general competition economy; and (3) wielding the power in a supply chain does not necessarily translate into higher profits. [ABSTRACT FROM AUTHOR] more...
- Published
- 2012
- Full Text
- View/download PDF
33. A Regularized Hybrid Steepest Descent Method for Variational Inclusions.
- Author
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Moudafi, Abdellatif
- Subjects
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MATHEMATICAL regularization , *GENERALIZATION , *MONOTONE operators , *MATHEMATICAL inequalities , *LIPSCHITZ spaces , *MATHEMATICAL mappings , *MIMO systems , *ALGORITHMS - Abstract
This article is concerned with a generalization of the hybrid steepest descent method from variational inequalities to the multivalued case. This will be reached by replacing the multivalued operator by its Yosida approximate, which is always Lipschitz continuous. It is worth mentioning that the hybrid steepest descent method is an algorithmic solution to variational inequality problems over the fixed point set of certain nonexpansive mappings and has remarkable applicability to the constrained nonlinear inverse problems like image recovery and MIMO communication systems (see, e.g., [9, 10]). [ABSTRACT FROM PUBLISHER] more...
- Published
- 2012
- Full Text
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34. COUPLING FORWARD-BACKWARD WITH PENALTY SCHEMES AND PARALLEL SPLITTING FOR CONSTRAINED VARIATIONAL INEQUALITIES.
- Author
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Attouch, Hédy, Czarnecki, Marc-Olivier, and Peypouquet, Juan
- Subjects
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MATHEMATICAL inequalities , *HILBERT space , *CONVEX functions , *APPROXIMATION theory , *MATHEMATICS - Abstract
We are concerned with the study of a class of forward-backward penalty schemes for solving variation inequalities 0 ∈ Ax + NC(x) where … is a real Hilbert space, A : H ⇉ H is a maximal monotone operator, and NC is the outward normal cone to a closed convex set C ⊂ H. Let Ψ : H → ℝ be a convex differentiable function whose gradient is Lipschitz continuous and which acts ,as a penalization function with respect to the constraint x ∈ C. Given a sequence (βn) of penalization parameters which tends to infinity, and a sequence of positive time steps (λn) ∈ ℓ² \ ℓ¹, we consider the diagonal forward-backward algorithm xn+ 1 = (I + λnA)-1 (xn - λnβn∇Ψ(xn)). Assuming that (βn) satisfies the growth condition lim supn→∞ λnβn < 2/ϑ (where ϑ is the Lipschitz constant of ∇Ψ), we obtain weak ergodic convergence of the sequence (xn) to an equilibrium for a general maximal monotone operator A. We also obtain weak convergence of the whole sequence (xn) when A is the sub differential of a proper lower-semi continuous convex function. As a key ingredient of our analysis, we use the co coerciveness of the operator ∇Ψ. When specializing our results to coupled systems, we bring new light to Pasty's theorem and obtain convergence results of new parallel splitting algorithms for variation inequalities involving coupling in the constraint. We also establish robustness and stability results that account for numerical approximation errors. An illustration of compressive sensing is given. [ABSTRACT FROM AUTHOR] more...
- Published
- 2011
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35. Equilibrium in a vector supply-demand network with capacity constraints.
- Author
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Luc, Dinh The, Rocca, Mateo, and Papalia, Melanie
- Subjects
- *
ECONOMIC equilibrium , *VECTOR analysis , *SUPPLY & demand , *MATHEMATICAL inequalities , *PARETO optimum , *INFINITE processes , *MATHEMATICAL analysis - Abstract
In this article, we study a multi-class, multi-criteria supply-demand network with elasticity demand and capacity constraints. We introduce a concept of elementary flows and construct a vector variational inequality problem which is equivalent to the vector network problem. Then we present a general scheme for scalarizing vector networks which allows us to obtain and strengthen a number of existing results on weakly vector equilibrium. A particular scalarization by augmented smallest monotone functions leads to a complete characterization of vector equilibrium. Necessary and sufficient conditions of vector equilibrium in terms of solutions to scalarized variational inequalities and efficient solutions to vector optimization problems are also established. [ABSTRACT FROM AUTHOR] more...
- Published
- 2011
- Full Text
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36. Fejer algorithms with an adaptive step.
- Author
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Nurminski, E.
- Subjects
- *
CONVEX functions , *MATHEMATICAL inequalities , *ALGORITHMS , *STOCHASTIC convergence , *MATHEMATICS , *EUCLIDEAN algorithm - Abstract
For Fejer processes with attractants, a general adaptive scheme for step multiplier control is proposed and the convergence of this class of algorithms to stationary points is proved. Numerical results demonstrating that the convergence rate is generally linear are presented. [ABSTRACT FROM AUTHOR] more...
- Published
- 2011
- Full Text
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37. Barrier subgradient method.
- Author
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Nesterov, Yurii
- Subjects
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MATHEMATICAL optimization , *MAXIMA & minima , *MATHEMATICAL analysis , *STOCHASTIC analysis , *STOCHASTIC processes , *MATHEMATICAL inequalities - Abstract
In this paper we develop a new affine-invariant primal-dual subgradient method for nonsmooth convex optimization problems. This scheme is based on a self-concordant barrier for the basic feasible set. It is suitable for finding approximate solutions with certain relative accuracy. We discuss some applications of this technique including fractional covering problem, maximal concurrent flow problem, semidefinite relaxations and nonlinear online optimization. For all these problems, the rate of convergence of our method does not depend on the problem's data. [ABSTRACT FROM AUTHOR] more...
- Published
- 2011
- Full Text
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38. SOME EXTRAGRADIENT METHODS FOR NONCONVEX QUASI VARIATIONAL INEQUALITIES.
- Author
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NOOR, MUHAMMAD ASLAM, NOOR, KHALIDA INAYAT, AL-SAID, EISA, and MOUDAFI, ABDELLATIF
- Subjects
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VARIATIONAL inequalities (Mathematics) , *DIFFERENTIAL inequalities , *ITERATIVE methods (Mathematics) , *NUMERICAL analysis , *MATHEMATICAL inequalities - Abstract
In this paper, we introduce and consider a new class of variational inequalities involving two operators, which is called the general nonconvex quasi variational inequality. Several special cases are discussed. We use the projection technique to establish the equivalence between the general nonconvex quasi variational inequalities and the fixed point problems. This alternative equivalent formulation is used to study the existence of a solution of the general nonconvex quasi variational inequalities. Using these equivalent formulations, we suggest and analyze a wide class of new extragradient methods for solving the general nonconvex quasi variational inequalities. Convergence criteria of these new iterative methods is considered under some suitable conditions. [ABSTRACT FROM AUTHOR] more...
- Published
- 2011
39. Vanishing viscosity method for quasilinear variational inequalities.
- Author
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Szász, Tünde Zsuzsánna
- Subjects
VISCOSITY solutions ,QUASILINEARIZATION ,MATHEMATICAL inequalities ,DIFFERENTIAL inequalities ,VARIATIONAL inequalities (Mathematics) - Abstract
In this paper we first define the notion of viscosity solution for the following partial differential quasilinear variational inequalities involving a subdifferential operator:∂u(t, x)/∂t + F(t, x, u(t, x)) · Du(t, x) + f(t, x, u(t, x)) ϵ ∂ͬ(u(t, x)) in O t ϵ [0, T] , x ϵ R
d , where ∂ͬ is the subdifferential operator of the proper convex lower semicontinuous function ͬ : Rd → (-∞,+∞]. We prove the existence of a viscosity solution u : O → Rn , where O an open set in [0, T] × Rd . [ABSTRACT FROM AUTHOR] more...- Published
- 2011
40. Duality and optimality conditions for generalized equilibrium problems involving DC functions.
- Author
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Dinh, N., Strodiot, J. J., and Nguyen, V. H.
- Subjects
MATHEMATICAL optimization ,CONVEX programming ,NONLINEAR statistical models ,MATHEMATICAL inequalities ,MATHEMATICAL functions - Abstract
We consider a generalized equilibrium problem involving DC functions which is called (GEP). For this problem we establish two new dual formulations based on Toland-Fenchel-Lagrange duality for DC programming problems. The first one allows us to obtain a unified dual analysis for many interesting problems. So, this dual coincides with the dual problem proposed by Martinez-Legaz and Sosa (J Glob Optim 25:311-319, 2006) for equilibrium problems in the sense of Blum and Oettli. Furthermore it is equivalent to Mosco's dual problem (Mosco in J Math Anal Appl 40:202-206, 1972) when applied to a variational inequality problem. The second dual problem generalizes to our problem another dual scheme that has been recently introduced by Jacinto and Scheimberg (Optimization 57:795-805, 2008) for convex equilibrium problems. Through these schemes, as by products, we obtain new optimality conditions for (GEP) and also, gap functions for (GEP), which cover the ones in Antangerel et al. (J Oper Res 24:353-371, 2007, Pac J Optim 2:667-678, 2006) for variational inequalities and standard convex equilibrium problems. These results, in turn, when applied to DC and convex optimization problems with convex constraints (considered as special cases of (GEP)) lead to Toland-Fenchel-Lagrange duality for DC problems in Dinh et al. (Optimization 1-20, 2008, J Convex Anal 15:235-262, 2008), Fenchel-Lagrange and Lagrange dualities for convex problems as in Antangerel et al. (Pac J Optim 2:667-678, 2006), Bot and Wanka (Nonlinear Anal to appear), Jeyakumar et al. (Applied Mathematics research report AMR04/8, 2004). Besides, as consequences of the main results, we obtain some new optimality conditions for DC and convex problems. [ABSTRACT FROM AUTHOR] more...
- Published
- 2010
- Full Text
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41. New extragradient-type methods for solving variational inequalities
- Author
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Bnouhachem, Abdellah, Fu, Xiao-Ling, Xu, M.H., and Zhaohan, Sheng
- Subjects
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MATHEMATICAL inequalities , *MONOTONE operators , *STOCHASTIC convergence , *ADAPTIVE control systems , *OPTIMAL designs (Statistics) , *NUMERICAL analysis - Abstract
Abstract: In this paper, we propose new methods for solving variational inequalities. The proposed methods can be viewed as a refinement and improvement of the method of He et al. [B.S. He, X.M. Yuan, J.J. Zhang, Comparison of two kinds of prediction–correction methods for monotone variational inequalities, Comp. Opt. Appl. 27 (2004) 247–267] by performing an additional projection step at each iteration and another optimal step length is employed to reach substantial progress in each iteration. Under certain conditions, the global convergence of the both methods is proved. Preliminary numerical experiments are included to illustrate the efficiency of the proposed methods. [Copyright &y& Elsevier] more...
- Published
- 2010
- Full Text
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42. A stochastic approach to a multivalued Dirichlet–Neumann problem
- Author
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Maticiuc, Lucian and Răşcanu, Aurel
- Subjects
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STOCHASTIC analysis , *DIRICHLET problem , *NEUMANN problem , *MANY-valued logic , *VISCOSITY solutions , *MATHEMATICAL inequalities , *BOUNDARY value problems , *NONLINEAR theories - Abstract
Abstract: We prove the existence and uniqueness of a viscosity solution of the parabolic variational inequality (PVI) with a mixed nonlinear multivalued Neumann–Dirichlet boundary condition: where and are subdifferential operators and is a second-differential operator given by The result is obtained by a stochastic approach. First we study the following backward stochastic generalized variational inequality: where is a continuous one-dimensional increasing measurable process, and then we obtain a Feynman–Kaç representation formula for the viscosity solution of the PVI problem. [Copyright &y& Elsevier] more...
- Published
- 2010
- Full Text
- View/download PDF
43. DISCONTINUOUS GALERKIN METHODS FOR SOLVING ELLIPTIC VARIATIONAL INEQUALITIES.
- Author
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Fei Wang, Wemin Han, and Xiao-Liang Cheng
- Subjects
- *
GALERKIN methods , *NUMERICAL analysis , *ELLIPTIC functions , *MATHEMATICAL inequalities , *BOUNDARY value problems , *ERROR analysis in mathematics - Abstract
We study discontinuous Galerkin methods for solving elliptic variational inequalities of both the first and second kinds. Analysis of numerous discontinuous Galerkin schemes for elliptic boundary value problems is extended to the variational inequalities. We establish a priori error estimates for the discontinuous Galerkin methods, which reach optimal order for linear elements. Results from some numerical examples are reported. [ABSTRACT FROM AUTHOR] more...
- Published
- 2010
- Full Text
- View/download PDF
44. VARIATIONAL INEQUALITIES AND OPTIMIZATION FOR MARKOV PROCESSES ASSOCIATED WITH SEMI-DIRICHLET FORMS.
- Author
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Tusheng Zhang
- Subjects
- *
MARKOV processes , *MARKOV operators , *DIRICHLET principle , *DIRICHLET forms , *OPTIMAL stopping (Mathematical statistics) , *MATHEMATICAL inequalities - Abstract
We consider an optimal stopping problem and an impulsive control for Markov processes associated with semi-Dirichlet forms. We show that the value functions are the maximum solutions of certain variational inequalities. Examples both in finite dimensions and infinite dimensions are given. [ABSTRACT FROM AUTHOR] more...
- Published
- 2009
- Full Text
- View/download PDF
45. The Elastic-Plastic Torsion Problem: A Posteriori Error Estimates for Approximate Solutions.
- Author
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Bildhauer, M., Fuchs, M., and Repin, S.
- Subjects
- *
MATHEMATICAL inequalities , *CALCULUS , *MEAN value theorems , *MATHEMATICAL functions , *DIRICHLET forms , *BOUNDARY value problems - Abstract
We consider the variational inequality that describes the torsion problem for a long elasto-plastic bar. Using duality methods of the variational calculus, we derive a posteriori estimates of functional type that provide computable and guaranteed upper bounds of the energy norm of the difference between the exact solution and any function from the corresponding energy space that satisfies the Dirichlet boundary condition. [ABSTRACT FROM AUTHOR] more...
- Published
- 2009
- Full Text
- View/download PDF
46. Proximal-Point Algorithm Using a Linear Proximal Term.
- Author
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He, B., Fu, X., and Jiang, Z.
- Subjects
- *
MATHEMATICAL optimization , *ALGORITHMS , *MATHEMATICAL inequalities , *LAGRANGE equations , *MATHEMATICAL analysis , *MATHEMATICS - Abstract
Proximal-point algorithms (PPAs) are classical solvers for convex optimization problems and monotone variational inequalities (VIs). The proximal term in existing PPAs usually is the gradient of a certain function. This paper presents a class of PPA-based methods for monotone VIs. For a given current point, a proximal point is obtained via solving a PPA-like subproblem whose proximal term is linear but may not be the gradient of any functions. The new iterate is updated via an additional slight calculation. Global convergence of the method is proved under the same mild assumptions as the original PPA. Finally, profiting from the less restrictions on the linear proximal terms, we propose some parallel splitting augmented Lagrangian methods for structured variational inequalities with separable operators. [ABSTRACT FROM AUTHOR] more...
- Published
- 2009
- Full Text
- View/download PDF
47. Strong Convergence of Iterative Algorithms for Variational Inequalities in Banach Spaces.
- Author
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Ceng, L., Schaible, S., and Yao, J.
- Subjects
- *
BANACH spaces , *STOCHASTIC convergence , *ALGORITHMS , *MATHEMATICAL inequalities , *MATHEMATICAL mappings , *MATHEMATICS - Abstract
Let C be a nonempty closed convex subset of a Banach space E with the dual E*, let T: C→ E* be a Lipschitz continuous mapping and let S: C→ C be a relatively nonexpansive mapping. In this paper, by employing the notion of generalized projection operator, we study the following variational inequality (for short, VI( T− f, C)): find x∈ C such that where f∈ E* is a given element. Utilizing the modified Ishikawa iteration and the modified Halpern iteration for relatively nonexpansive mappings, we propose two modified versions of J.L. Li’s (J. Math. Anal. Appl. 295:115–126, ) iterative algorithm for finding approximate solutions of VI( T− f, C). Moreover, it is proven that these iterative algorithms converge strongly to the same solution of VI( T− f, C), which is also a fixed point of S. [ABSTRACT FROM AUTHOR] more...
- Published
- 2009
- Full Text
- View/download PDF
48. Robinson’s implicit function theorem and its extensions.
- Author
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Dontchev, A. and Rockafellar, R.
- Subjects
- *
GENERALIZED spaces , *EQUATIONS , *MATHEMATICAL functions , *MATHEMATICAL inequalities , *APPROXIMATION theory , *FUNCTIONAL analysis - Abstract
S. M. Robinson published in 1980 a powerful theorem about solutions to certain “generalized equations” corresponding to parameterized variational inequalities which could represent the first-order optimality conditions in nonlinear programming, in particular. In fact, his result covered much of the classical implicit function theorem, if not quite all, but went far beyond that in ideas and format. Here, Robinson’s theorem is viewed from the perspective of more recent developments in variational analysis as well as some lesser-known results in the implicit function literature on equations, prior to the advent of generalized equations. Extensions are presented which fully cover such results, translating them at the same time to generalized equations broader than variational inequalities. Robinson’s notion of first-order approximations in the absence of differentiability is utilized in part, but even looser forms of approximation are shown to furnish significant information about solutions. [ABSTRACT FROM AUTHOR] more...
- Published
- 2009
- Full Text
- View/download PDF
49. Approximations of Nash equilibria.
- Author
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Gürkan, Gül and Pang, Jong-Shi
- Subjects
- *
MATHEMATICAL optimization , *APPROXIMATION theory , *NASH equilibrium , *STOCHASTIC processes , *MATHEMATICAL inequalities , *STOCHASTIC convergence - Abstract
Inspired by previous works on approximations of optimization problems and recent papers on the approximation of Walrasian and Nash equilibria and on stochastic variational inequalities, the present paper investigates the approximation of Nash equilibria and clarifies the conditions required for the convergence of the approximate equilibria via a direct approach, a variational approach, and an optimization approach. Besides directly addressing the issue of convergence of Nash equilibria via approximation, our investigation leads to a deeper understanding of various notions of functional convergence and their interconnections; more importantly, the investigation yields improved conditions for convergence of the approximate Nash equilibria via the variational approach. An illustrative application of our results to the approximation of a Nash equilibrium in a competitive capacity expansion model under uncertainty is presented. [ABSTRACT FROM AUTHOR] more...
- Published
- 2009
- Full Text
- View/download PDF
50. On singular stochastic control and optimal stopping of spectrally negative jump diffusions.
- Author
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Alvarez, Luis H. R. and Rakkolainen, Teppo A.
- Subjects
- *
STOCHASTIC control theory , *DIFFUSION , *VARIATIONAL inequalities (Mathematics) , *MATHEMATICAL inequalities , *MATHEMATICS - Abstract
We consider a broad class of singular stochastic control problems of spectrally negative jump diffusions in the presence of potentially nonlinear state-dependent exercise payoffs. We analyse these problems by relying on associated variational inequalities and state a set of sufficient conditions under which the value of the considered problems can be explicitly derived in terms of the increasing minimal r-harmonic map. We also present a set of inequalities bounding the value of the optimal policy and prove that increased policy flexibility increases both the value of the optimal strategy as well as the rate at which this value grows. [ABSTRACT FROM AUTHOR] more...
- Published
- 2009
- Full Text
- View/download PDF
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