lagrange multipliers in elastic plastic torsion problem for nonlinear monotone operators abstract the existence of lagrange multipliers as a radon measure is ensured for an elastic plastic torsion problem associated to a nonlinear strictly monotone operator a regularization of this result namely the existence of l p lagrange multipliers is obtained under strong monotonicity assumption on the operator moreover the relationships between elastic plastic torsion problem and the obstacle problem are investigated finally an example of the so called von mises functions is provided namely of solutions of the elastic plastic torsion problem associated to nonlinear monotone operators which are not obtained by means of the obstacle problem in the case f constant

bounded sets of lagrange multipliers for vector optimization problems in infinite dimension the aim of this paper is to point out some sufficient constraint qualification conditions ensuring the boundedness of a set of lagrange multipliers for vectorial optimization problems in infinite dimension in some smooth cases these conditions turn out to be necessary for the existence of multipliers as well

the fermat rule for multifunctions on banach spaces using variational analysis we study vector optimization problems with objectives being closed multifunctions on banach spaces or in asplund spaces in particular in terms of the coderivatives we present fermat s rules as necessary conditions for an optimal solution of the above problems as applications we also provide some necessary conditions in terms of clarke s normal cones or the limiting normal cones for pareto efficient points

a non conforming piecewise quadratic finite element on triangles it is shown in this paper that non conforming finite elements on the triangle using second degree polynomials can be easily built and used indeed they appear as an enriched version of the standard piecewise quadratic six node element this work is divided into two parts in the first we present the basic properties of the element namely how it can be built and basic error estimates we also show that this element exhibits a very peculiar regularity property in the second part we apply our element to the approximation of viscous incompressible flows and more generally to the approximation of incompressible materials

multiscale coupling in function space weak coupling between molecular dynamics and continuum mechanics we present a function space oriented coupling approach for the multiscale simulation of non linear processes in mechanics using finite elements and molecular dynamics concurrently the key idea is to construct a transfer operator between the different scales on the basis of weighted local averaging instead of using point wise taken values the local weight functions are constructed by assigning a partition of unity to the molecular degrees of freedom shepard s approach this allows for decomposing the micro scale displacements into a low frequency and a high frequency part by means of a weighted l2 projection numerical experiments illustrating the stabilizing effect of our coupling approach are given copyright 2009 john wiley sons ltd

treatments of discontinuity and bubble functions in the multigrid method when multilevel finite element spaces are not nested different intergrid transfer operators would lead to different multigrid algorithms it is proposed in this paper that discontinuous functions be averaged to continuous functions and that the bubble functions be discarded in the multigrid transferring applications of the techniques to various problems are presented with convergence analysis numeric comparisons with other existing methods are provided

on hardy s inequality abstract in this paper some new inequalities on the constant e are deduced as an application we build a strengthened hardy s inequality

quasivariational solutions for first order quasilinear equations with gradient constraint we prove the existence of solutions for a quasi variational inequality of evolution with a first order quasilinear operator and a variable convex set which is characterized by a constraint on the absolute value of the gradient that depends on the solution itself the only required assumption on the nonlinearity of this constraint is its continuity and positivity the method relies on an appropriate parabolic regularization and suitable a priori estimates we also obtain the existence of stationary solutions by studying the asymptotic behaviour in time in the variational case corresponding to a constraint independent of the solution we also give uniqueness results

a multigrid method for eigenvalue problem a multigrid method is proposed to solve the eigenvalue problem by the finite element method based on the combination of the multilevel correction scheme for the eigenvalue problem and the multigrid method for the boundary value problem in this scheme solving eigenvalue problem is transformed to a series of solutions of boundary value problems by the multigrid method on multilevel meshes and a series of solutions of eigenvalue problems on the coarsest finite element space besides the multigrid scheme all other efficient iteration methods can also serve as the linear algebraic solver for the associated boundary value problems the total computational work of this scheme can reach the optimal order as same as solving the corresponding boundary value problem therefore this type of iteration scheme improves the overall efficiency of the eigenvalue problem solving some numerical experiments are presented to validate the efficiency of the method

unstructured geometric multigrid in two and three dimensions on complex and graded meshes the use of multigrid and related preconditioners with the finite element method is often limited by the difficulty of applying the algorithm effectively to a problem especially when the domain has a complex shape or the mesh has adaptive refinement we introduce a simplification of a general topologically motivated mesh coarsening algorithm for use in creating hierarchies of meshes for geometric unstructured multigrid methods the connections between the guarantees of this technique and the quality criteria necessary for multigrid methods for nonquasi uniform problems are noted the implementation details in particular those related to coarsening remeshing and interpolation are discussed computational tests on pathological test cases from adaptive finite element methods show the performance of the technique

note on the carleman s inequality and hardy s inequality in this article using the properties of power mean and induction new strengthened carleman s inequality and hardy s inequality are obtained we also give an answer to the conjectures proposed by x yang in the literature yang 2001 5

a posteriori error estimates for a finite element approximation of transmission problems with sign changing coefficients we perform the a posteriori error analysis of residual type of transmission problem with sign changing coefficients according to bonnet bendhia et al 2010 9 if the contrast is large enough the continuous problem can be transformed into a coercive one we further show that a similar property holds for the discrete problem for any regular meshes extending the framework from bonnet bendhia et al 9 the reliability and efficiency of the proposed estimator are confirmed by some numerical tests

continued fraction inequalities related to 1 1x x abstract in this paper we provide some new continued fraction inequalities related to 1 1 x x

duality theory and optimality conditions for generalized complementarity problems abstract in this paper generalized complementarity problems are expressed in terms of suitable optimization problems and some necessary optimality conditions are given the infinite dimensional lagrangean and duality theories play an important role in order to achieve the main results

relative pareto minimizers for multiobjective problems existence and optimality conditions in this paper we introduce and study enhanced notions of relative pareto minimizers for constrained multiobjective problems that are defined via several kinds of relative interiors of ordering cones and occupy intermediate positions between the classical notions of pareto and weak pareto efficiency minimality using advanced tools of variational analysis and generalized differentiation we establish the existence of relative pareto minimizers for general multiobjective problems under a refined version of the subdifferential palais smale condition for set valued mappings with values in partially ordered spaces and then derive necessary optimality conditions for these minimizers as well as for conventional efficient and weak efficient counterparts that are new in both finite dimensional and infinite dimensional settings our proofs are based on variational and extremal principles of variational analysis in particular on new versions of the ekeland variational principle and the subdifferential variational principle for set valued and single valued mappings in infinite dimensional spaces

the infinite dimensional lagrange multiplier rule for convex optimization problems abstract in this paper an infinite dimensional generalized lagrange multipliers rule for convex optimization problems is presented and necessary and sufficient optimality conditions are given in order to guarantee the strong duality furthermore an application is presented in particular the existence of lagrange multipliers associated to the bi obstacle problem is obtained

on a conjecture by yang abstract the purpose of this paper is to prove a conjecture and settle an open problem raised by xiaojing yang 2001 j math anal appl 253 691 694

nonconforming multigrid method for nonsymmetric and indefinite problems abstract the convergence of w cycle multigrid algorithms applied to nonsymmetric and indefinite elliptic boundary value problems using p 1 nonconforming finite elements is proved

on zero duality gap and the farkas lemma for conic programming recently s a clark published an interesting duality result in linear conic programming dealing with a convex cone that is not closed in which the usual algebraic dual problem is replaced by a topological dual with the aim of having zero duality gap under certain usual hypotheses met in mathematical finance we present some examples to show that an extra condition is needed to reach a conclusion this supplementary condition is also provided we also give counterexamples for three results on hedging prices and simple proofs for two known solvability results see propositions 4 1 and 4 2

a multilevel algorithm for mixed problems we describe a multilevel algorithm for the numerical solution of symmetric indefinite problems which arise e g from mixed finite element approximations of the stokes equation the main difficulty besides the indefiniteness is the lack of regularity of the solution of the corresponding continuous problem this is overcome by introducing a scale of mesh dependent norms the convergence rate of the described algorithm is bounded independently of the meshsize for convenience we only discuss jacobi relaxation as smoothing operator in detail in the last section we comment on lanczos type smoothing procedures

conical regularization for multi objective optimization problems abstract this paper focuses on multi objective optimization problem with linear infinite dimensional constraints the main novelty of the contribution is that we do not assume that the ordering cone has a nonempty interior and consequently there might not exist kkt system to compute the solution numerically we devise a regularization framework which consists of replacing the ordering cone by a family of solid cones we provide general features of the original problem and the family of regularized problems using tools from the set valued analysis we prove that the weak outer limit of regularized solutions is the set of minimal elements of the original problem we scalarize the original problem as well as the regularized analogs and establish that the corresponding scalarized solutions parametrize the regularized minimal elements we demonstrate the convergence in the kuratowski mosco sense of the regularized solutions to the efficient set of the original problem we prove an a priori holder continuity estimate for the regularized efficient set valued map when the original problem is regular and a verifiable norm boundedness property of the multipliers holds we present detailed numerical examples to show the efficacy of the developed framework

optimal order nonnested multigrid methods for solving finite element equations i on quasi uniform meshes we prove that the multigrid method works with optimal computational order even when the multiple meshes are not nested when a coarse mesh is not a submesh of the finer one the coarse level correction usually does not have the a projection property and does amplify the iterative error in some components nevertheless the low frequency components of the error can still be caught by the coarse level correction since the amplified highfrequency errors will be damped out by the fine level smoothing efficiently the optimal work order of the standard multigrid method can still be maintained however unlike the case of nested meshes a nonnested multigrid method with one smoothing does not converge in general no matter whether it is a v cycle or a w cycle method it is shown numerically that the convergence rates of nonnested multigrid methods are not necessarily worse than those of nested ones since nonnested multigrid methods accept quite arbitrarily related meshes we may then combine the efficiencies of adaptive refinements and of multigrid algorithms

exchange price equilibria and variational inequalities the aim of this paper is to show the relevance of the concept and the theory of variational inequalities in the study of economic equilibria

conservative p 1 conforming and nonconforming galerkin fems effective flux evaluation via a nonmixed method approach given a p1 conforming or nonconforming galerkin finite element method gfem solution ph which approximates the exact solution p of the diffusion reaction equation nabla cdot mathcal k nabla p alpha p f with full tensor variable coefficient mathcal k we evaluate the approximate flux uh to the exact flux bf u mathcal k nabla p by a simple but physically intuitive formula over each finite element the flux is sought in the continuous in normal component or the discontinuous raviart thomas space a systematic way of deriving such a formula is introduced this direct method retains local conservation property at the element level typical of mixed methods finite element or finite volume type but avoids solving an indefinite linear system in short the present method retains the best of the gfem and the mixed method but without their shortcomings thus we view our method as a conservative gfem and demonstrate its equivalence to a certain mixed finite volume box method the equivalence theorems explain how the pressure can decouple basically cost free from the mixed formulation the accuracy in the flux is of first order in the h dive omega norm numerical results are provided to support the theory

on carleman s inequality abstract in this paper the result of p yan and g sun 1999 j math anal appl 240 290 293 is generalized and a conjecture is proposed

an optimal order multigrid method for p1 nonconforming finite elements etude d une methode multigrille d ordre optimal pour la resolution des problemes aux limites elliptiques du second ordre a l aide d elements finis non conformes

a measure type lagrange multiplier for the elastic plastic torsion abstract the existence of lagrange multiplier as a radon measure is ensured for an elastic plastic torsion problem associated to a linear operator the result is obtained by means of strong duality theory

some inequalities involving the constante and an application to carleman s inequality some inequalities involving the constanteare considered a strengthened carleman s inequality is proved

lagrange multipliers and non constant gradient constrained problem abstract the aim of the paper is to study a gradient constrained problem associated with a linear operator two types of problems are investigated the first one is the equivalence between a non constant gradient constrained problem and a suitable obstacle problem where the obstacle solves a hamilton jacobi equation in the viscosity sense the equivalence result is obtained under a condition on the gradient constraint the second problem is the existence of lagrange multipliers we prove that the non constant gradient constrained problem admits a lagrange multiplier which is a radon measure if the free term of the equation f l p p 1 if f is a positive constant we regularize the result namely we prove that the lagrange multipliers belong to l 2

degenerate nonlinear programming with a quadratic growth condition we show that the quadratic growth condition and the mangasarian fromovitz constraint qualification mfcq imply that local minima of nonlinear programs are isolated stationary points as a result when started sufficiently close to such points an l infty exact penalty sequential quadratic programming algorithm will induce at least r linear convergence of the iterates to such a local minimum we construct an example of a degenerate nonlinear program with a unique local minimum satisfying the quadratic growth and the mfcq but for which no positive semidefinite augmented lagrangian exists we present numerical results obtained using several nonlinear programming packages on this example and discuss its implications for some algorithms

approximation formulas for the constant e and an improvement to a carleman type inequality abstract we give an explicit formula for the determination of the coefficients c j appearing in the expansion x 1 j 1 q c j x j x 1 2 1 x e o 1 x q 1 for x and q n 1 2 we also derive a pair of recurrence relations for the determination of the constants l and l in the expansion 1 1 x x e 1 l 1 l x l 2 l 1 as x based on this expansion we establish an inequality for 1 1 x x as an application we give an improvement to a carleman type inequality

variational and quasivariational inequalities with first order constraints abstract we study the existence of solutions of stationary variational and quasivariational inequalities with curl constraint neumann type boundary condition and a p curl type operator these problems are studied in bounded not necessarily simply connected domains with a special geometry and the functional framework is the space of divergence free functions with curl in l p and null tangential or normal traces the analogous variational or quasivariational inequalities with gradient constraint are also studied considering neumann or dirichlet non homogeneous boundary conditions the existence of a generalized solution for a lagrange multiplier problem with homogeneous dirichlet boundary condition and the equivalence with the variational inequality is proved in the linear case for an arbitrary gradient constraint

a new necessary and sufficient condition for the strong duality and the infinite dimensional lagrange multiplier rule abstract throughout this paper the authors introduce a new condition defined by assumption s which establishes a necessary and sufficient condition for the validity of the strong duality between a convex optimization problem and its lagrange dual this work will be focused on the context of emptiness of the interior of the ordering cone and convexity of the equality constraints moreover this new condition will be necessary and sufficient for the infinite dimensional lagrange multiplier rule this new principle will find application to the elastic plastic torsion problem to the continuum model of transportation and to a problem with quadratic equality constraint with connected to evolutionary illumination and visibility problems

higher dimensional nonnested multigrid methods nonnested multigrid methods are shown to be optimal order solvers for systems of finite element equations arising from elliptic boundary problems in any space dimension results are derived for lagrange type elements of arbitrary degree

a generalized bpx multigrid framework covering nonnested v cycle methods more than a decade ago bramble pasciak and xu developed a framework in analyzing the multigrid methods with nonnested spaces or non inherited quadratic forms it was subsequently known as the bpx multigrid framework which was widely used in the analysis of multigrid and domain decomposition methods however the framework has an apparent limit in the analysis of nonnested v cycle methods and it produces a variable v cycle or nonuniform convergence rate v cycle methods or other nonoptimal results in analysis thus far this paper completes a long time effort in extending the bpx multigrid framework so that it truly covers the nonnested v cycle we will apply the extended bpx framework to the analysis of many v cycle nonnested multigrid methods some of them were proven previously only for two level and w cycle iterations some numerical results are presented to support the theoretical analysis of this paper

a new convergence proof for the multigrid method including the v cycle for a positive definite finite element equation we describe a multigrid iteration and prove convergence under natural assumptions on the discretization and the elliptic problem hitherto existing convergence proofs require a sufficiently large number of smoothing iterations and exclude the v cycle the presented proof applies to procedures with any number of smoothing iterations and to the v cycle

competition and efficiency in congested markets we study the efficiency of oligopoly equilibria oe in congested markets the motivating examples are the allocation of network flows in a communication network or of traffic in a transportation network we show that increasing competition among oligopolists can reduce efficiency measured as the difference between users willingness to pay and delay costs we characterize a tight bound of 5 6 on efficiency in pure strategy equilibria when there is zero latency at zero flow and a tight bound of 2 2 2 with positive latency at zero flow these bounds are tight even when the numbers of routes and oligopolists are arbitrarily large

an optimal order process for solving finite element equations a k level iterative procedure for solving the algebraic equations which arise from the finite element approximation of elliptic boundary value problems is presented and analyzed the work estimate for this procedure is proportional to the number of unknowns an optimal order result general geometry is permitted for the underlying domain but the shape plays a role in the rate of convergence through elliptic regularity finally a short discussion of the use of this method for parabolic problems is presented

lagrangean variables in infinite dimensional spaces for a dynamic economic equilibrium problem abstract this paper is focused on the study of a dynamic competitive equilibrium by using lagrangean multipliers this mathematical formulation allows us the improve the walrasian model by considering the common possibility of an uncharged delayed payment in a given time for example by using a credit card firstly the economic equilibrium problem is reformulated as an evolutionary variational problem then the lagrangean theory in infinite dimensional spaces is applied thanks to the application of this theory we obtain the existence of lagrangean multipliers which allows us to give a computational procedure for the equilibrium solutions

multigrid computation of vector potentials in this paper we analyze the use of edge finite elements and the multigrid method to approximate the problem of computing a static magnetic field in a cavity we show how the mixed finite element discrete problem may be reformulated as a symmetric positive definite system this involves the proof of a discrete analogue of the second kind friedrichs inequality then we show how the multigrid method may be applied to compute the magnetic vector potential the difficulty here is that the bilinear forms on different meshes are not related in the standard way two different intergrid transfer operators are defined in the paper for the multigrid method the convergence theory and the numerical tests include both algorithms

an extended extremal principle with applications to multiobjective optimization we develop an extended version of the extremal principle in variational analysis that can be treated as a variational counterpart to the classical separation results in the case of nonconvex sets and which plays an important role in the generalized differentiation theory and its applications to optimization related problems the main difference between the conventional extremal principle and the extended version developed below is that instead of the translation of sets involved in the extremal systems we allow deformations the new version seems to be more flexible in various applications and covers in particular multiobjective optimization problems with general preference relations in this way we obtain new necessary optimality conditions for constrained problems of multiobjective optimization with nonsmooth data and also for multiplayer multiobjective games

infinite dimensional duality and applications the usual duality theory cannot be applied to infinite dimensional problems because the underlying constraint set mostly has an empty interior and the constraints are possibly nonlinear in this paper we present an infinite dimensional nonlinear duality theory obtained by using new separation theorems based on the notion of quasi relative interior which in all the concrete problems considered is nonempty we apply this theory to solve the until now unsolved problem of finding in the infinite dimensional case the lagrange multipliers associated to optimization problems or to variational inequalities as an example we find the lagrange multiplier associated to a general elastic plastic torsion problem