an iterative scheme for a class of quasi variational inequalities an iterative scheme is given to obtain the approximate solution of a class of quasi variational inequalities it is shown that the approximate solution obtained by the iterative scheme converges strongly in the hilbert space to the exact solution as a special case we obtain the corresponding iterative scheme for variational inequalities

shape optimization in contact problems with coulomb friction the paper deals with a discretized problem of the shape optimization of elastic bodies in unilateral contact the aim is to extend existing results to the case of contact problems following the coulomb friction law mathematical modelling of the coulomb friction problem leads to a quasi variational inequality it is shown that for small coefficients of friction the discretized problem with coulomb friction has a unique solution and that this solution is lipschitzian as a function of a control variable describing the shape of the elastic body the shape optimization problem belongs to a class of so called mathematical programs with equilibrium constraints mpecs the uniqueness of the equilibria for fixed controls enables us to apply the so called implicit programming approach its main idea consists of minimizing a nonsmooth composite function generated by the objective and the single valued control state mapping in this paper the control state mapping is much more complicated than in most mpecs solved in the literature so far and the generalization of the relevant results is by no means straightforward numerical examples illustrate the efficiency and reliability of the suggested approach

a relaxed constant positive linear dependence constraint qualification and applications in this work we introduce a relaxed version of the constant positive linear dependence constraint qualification cpld that we call rcpld this development is inspired by a recent generalization of the constant rank constraint qualification by minchenko and stakhovski that was called rcrcq we show that rcpld is enough to ensure the convergence of an augmented lagrangian algorithm and that it asserts the validity of an error bound we also provide proofs and counter examples that show the relations of rcrcq and rcpld with other known constraint qualifications in particular rcpld is strictly weaker than cpld and rcrcq while still stronger than abadie s constraint qualification we also verify that the second order necessary optimality condition holds under rcrcq

on the constant positive linear dependence condition and its application to sqp methods in this paper we introduce a constant positive linear dependence condition cpld which is weaker than the mangasarian fromovitz constraint qualification mfcq and the constant rank constraint qualification crcq we show that a limit point of a sequence of approximating karush kuhn tucker kkt points is a kkt point if the cpld holds there we show that a kkt point satisfying the cpld and the strong second order sufficiency conditions ssosc is an isolated kkt point we then establish convergence of a general sequential quadratical programming sqp method under the cpld and the ssosc finally we apply these results to analyze the feasible sqp method proposed by panier and tits in 1993 for inequality constrained optimization problems we establish its global convergence under the ssosc and a condition slightly weaker than the mangasarian fromovitz constraint qualification and we prove superlinear convergence of a modified version of this algorithm under the ssosc and a condition slightly weaker than the linear independence constraint qualification

two new weak constraint qualifications and applications we present two new constraint qualifications cqs that are weaker than the recently introduced relaxed constant positive linear dependence rcpld cq rcpld is based on the assumption that many subsets of the gradients of the active constraints preserve positive linear dependence locally a major open question was to identify the exact set of gradients whose properties had to be preserved locally and that would still work as a cq this is done in the first new cq which we call the constant rank of the subspace component crsc cq this new cq also preserves many of the good properties of rcpld such as local stability and the validity of an error bound we also introduce an even weaker cq called the constant positive generator cpg which can replace rcpld in the analysis of the global convergence of algorithms we close this work by extending convergence results of algorithms belonging to all the main classes of nonlinear optimization methods sequential quadratic programming augmented lagrangians

local uniqueness and convergence of iterative methods for nonsmooth variational inequalities in this paper we study nonsmooth variational inequalities using nonsmooth analysis we characterize various monotonicity properties of locally lipschitzian functions and give some sufficient conditions to guarantee the local uniqueness of solutions to variational inequalities two iterative algorithms are also presented to solve nonsmooth variational inequalities

solving quasi variational inequalities via their kkt conditions we propose to solve a general quasi variational inequality by using its karush kuhn tucker conditions to this end we use a globally convergent algorithm based on a potential reduction approach we establish global convergence results for many interesting instances of quasi variational inequalities vastly broadening the class of problems that can be solved with theoretical guarantees our numerical testings are very promising and show the practical viability of the approach

finite element prediction of wave motion in structural waveguides a method is presented by which the wavenumbers for a one dimensional waveguide can be predicted from a finite element fe model the method involves postprocessing a conventional but low order fe model the mass and stiffness matrices of which are typically found using a conventional fe package this is in contrast to the most popular previous waveguide fe approach sometimes termed the spectral finite element approach which requires new spectral element matrices to be developed in the approach described here a section of the waveguide is modeled using conventional fe software and the dynamic stiffness matrix formed a periodicity condition is applied the wavenumbers following from the eigensolution of the resulting transfer matrix the method is described estimation of wavenumbers energy and group velocity discussed and numerical examples presented these concern wave propagation in a beam and a simply supported plate strip for which analytical solutions exist and the more complex case of a viscoelastic laminate which involves postprocessing an ansys fe model the method is seen to yield accurate results for the wavenumbers and group velocities of both propagating and evanescent waves

finite element analysis of the vibrations of waveguides and periodic structures many structural components can be regarded as waveguides they are uniform in one direction so that the cross section of the waveguide has the same physical and geometric properties at all points along the axis of the waveguide in this paper a method is presented to calculate the forced response of such a structure using a combination of wave and finite element fe approaches the method involves post processing a conventional but low order fe model in which the mass and stiffness matrices are typically found using a conventional fe package a section of the waveguide is meshed and the eigenvalues and eigenvectors of the resulting transfer matrix found the eigenvectors form a set of basis functions for the analysis of the structure as a whole allowing the global dynamic stiffness matrix to be built easily and then the forced response to be calculated very efficiently the main advantage of the approach over the alternative waveguide fe approach often termed the spectral fe method is that conventional fe packages can be used to form the stiffness and mass matrices so that structures with complex geometries or material distributions can be analysed with relative ease to demonstrate the efficacy of the method examples of the forced response for a finite beam and plate strip are presented

strongly nonlinear quasivariational inequalities abstract in this paper we develop the algorithms for finding the approximate solution of a strongly nonlinear quasivariational inequality and a strongly nonlinear quasicomplementarity problem and we include as special cases known results in this area we also observe that these two problems are equivalent if the convex set involved in the formulation of these problem is a convex cone

generalized nash games and quasi variational inequalities abstract a generalized nash game is an n person noncooperative game with nondisjoint strategy sets other names for this game form include social equilibria and pseudo nash games this paper explores both the qualitative and quantitative properties of such games through the use of quasi variational inequality theory several interesting relationships between the variational and quasi variational inequality forms of this class of games are described and the practical implementation of generalized nash games are explored at length

augmented lagrangian methods under the constant positive linear dependence constraint qualification two augmented lagrangian algorithms for solving kkt systems are introduced the algorithms differ in the way in which penalty parameters are updated possibly infeasible accumulation points are characterized it is proved that feasible limit points that satisfy the constant positive linear dependence constraint qualification are kkt solutions boundedness of the penalty parameters is proved under suitable assumptions numerical experiments are presented

a representation of all solutions of the control algebraic riccati equation for infinite dimensional systems we obtain a representation of all self adjoint solutions of the control algebraic riccati equation associated to the infinite dimensional state linear system a b c under the following assumptions a generates a c0 group the system is output stabilizable strongly detectable and the dual riccati equation has an invertible self adjoint non negative solution

modified wilson s method for nonlinear programswith nonunique multipliers this paper first investigates newton type methods for generalized equations with compact solution sets the analysis of their local convergence behavior is based besides other conditions on the upper lipschitz continuity of the local solution set mapping of a simply perturbed generalized equation this approach is then applied to the kkt conditions of a nonlinear program with inequality constraints and leads to a modified version of the classical wilson method it is shown that the distances of the iterates to the set of kkt points converge q quadratically to zero under conditions that do not imply a unique multiplier vector additionally to the mangasarian fromovitz constraint qualification and to a second order sufficiency condition the local minimizer is required to fulfill a constant rank condition weaker than the constant rank constraint qualification one of janin and a so called weak complementarity condition

numerical issues concerning the wave and finite element method for free and forced vibrations of waveguides the wave and finite element wfe method is a numerical approach to the calculation of the wave properties of structures of arbitrary complexity the method starts from a finite element fe model of only a short segment of the structure typically by using existing element libraries and commercial fe packages the dynamic stiffness matrix of the segment is obtained a periodicity condition applied and an eigenvalue problem formed whose solution gives the dispersion equations and wave mode shapes these define a wave basis from which the forced response can be found straightforwardly although straightforward in application the wfe method is prone to numerical difficulties these are discussed in this paper and methods to avoid or remove them described attention is focused on 1 dimensional waveguide structures for which numerical problems are most severe three ways of phrasing the eigenvalue problem for free wave propagation are presented and a method based on singular value decomposition is proposed to determine eigenvectors at low frequencies discretisation errors are seen to occur if the segment is too large while round off errors occur if the segment is too small this can be overcome by forming a super segment from the concatenation of two or more segments the forced response is then considered the use of a reduced wave basis removes many problems direct calculation of the waves excited by a point force is very prone to poor numerical conditioning but can be circumvented by exploiting the orthogonality of the left and right eigenvectors numerical examples are presented

necessary conditions for exact controllability with a finite dimensional input space we derive necessary conditions for exact controllability of infinite dimensional systems described by x ax bu where the generator a has a riesz basis of eigenvectors and the input space is finite dimensional these conditions are in terms of the eigenvalues of a and the degree of unboundedness of b but no further information about b is needed our results easily imply lack of exact controllability for a wide range of distributed parameter systems it is known from partial differential equation pde examples that the addition of damping into an exactly controllable second order system can destroy exact controllability we discuss the effect of damping on such systems using our general conditions we also give simple new proofs for results about the lack of exact controllability of hyperbolic systems in more than one space dimension

a sequential minimization technique for elliptic quasi variational inequalities with gradient constraints a class of nonlinear elliptic quasi variational inequality qvi problems with gradient constraints in function space is considered problems of this type arise for instance in the mathematical description of the magnetization of superconductors in problems in elastoplasticity or in electrostatics as well as in game theory the paper addresses the iterative solution of the qvis by a sequential minimization technique relying on the repeated solution of variational inequality type problems a monotone operator theoretic approach is developed which does not resort to mosco convergence as is often done in connection with existence analysis for qvis for the numerical solution of the qvis a penalty approach combined with a semismooth newton iteration is proposed the paper ends with a report on numerical tests involving the p laplace operator and various types of nonlinear constraints

riesz bases for p subordinate perturbations of normal operators abstract for a class of unbounded perturbations of unbounded normal operators the change of the spectrum is studied and spectral criteria for the existence of a riesz basis with parentheses of root vectors are established a riesz basis without parentheses is obtained under an additional a priori assumption on the spectrum of the perturbed operator the results are applied to two classes of block operator matrices

global convergence of augmented lagrangian methods applied to optimization problems with degenerate constraints including problems with complementarity constraints we consider global convergence properties of the augmented lagrangian methods on problems with degenerate constraints with a special emphasis on mathematical programs with complementarity constrai

on the direct solution of wave propagation for repetitive structures abstract the direct method of solving the wave constants for a repetitive structure with given frequency is developed in this paper the analogy between structural mechanics and optimal control theory is applied the dynamic stiffness matrix approach is used to solve for the eigensolutions of wave propagation the weighted adjoint symplectic orthonormality relationship and the eigenvector expansion theorem are also established for this approach the symplectic eigenproblem is derived for skew symmetric matrices a cell triangular decomposition for skew symmetric matrices is introduced to reduce the generalized eigenproblem and the symplectic householder transformation is introduced to cell tri diagonalize the skew symmetric matrix to solve the eigenequation the problem of a wave impinging on the boundary is considered by the eigenvector expansion method which could also be applied to a wave scattering when passing over an abnormal part numerical examples show that resonance will occur for the localized vibration case

on controllability of diagonal systems with one dimensional input space abstract this paper deals with diagonal systems on a hilbert state space with a one dimensional input space and a possibly unbounded control operator a priori it is not assumed that the input operator is admissible necessary and sufficient conditions for different notions of controllability such as null controllability exact controllability and approximate controllability are presented these conditions which are given in terms of the eigenvalues of the diagonal operator and in terms of the control operator are linked with the theory of interpolation in hardy spaces

backward sdes with constrained jumps and quasi variational inequalities we consider a class of backward stochastic differential equations bsdes driven by brownian motion and poisson random measure and subject to constraints on the jump component we prove the existence and uniqueness of the minimal solution for the bsdes by using a penalization approach moreover we show that under mild conditions the minimal solutions to these constrained bsdes can be characterized as the unique viscosity solution of quasi variational inequalities qvis which leads to a probabilistic representation for solutions to qvis such a representation in particular gives a new stochastic formula for value functions of a class of impulse control problems as a direct consequence this suggests a numerical scheme for the solution of such qvis via the simulation of the penalized bsdes

penalty methods for the solution of generalized nash equilibrium problems the generalized nash equilibrium problem gnep is an extension of the classical nash equilibrium problem where both the objective functions and the constraints of each player may depend on the rivals strategies this class of problems has a multitude of important engineering applications and yet solution algorithms are extremely scarce in this paper we analyze in detail a globally convergent penalty method that has favorable theoretical properties we also consider strengthened results for a particular subclass of problems very often considered in the literature basically our method reduces the gnep to a single penalized and nonsmooth nash equilibrium problem we suggest a suitable method for the solution of the latter penalized problem and present extensive numerical results

hamiltonians and riccati equations for linear systems with unbounded control and observation operators in this paper we construct infinitely many selfadjoint solutions of the control algebraic riccati equation using invariant subspaces of the associated hamiltonian we do this under the assumptions that the system operator is normal and has compact inverse and that the hamiltonian possesses a riesz basis of invariant subspaces

on the multiplier penalty approach for quasi variational inequalities the multiplier penalty approach is one of the classical methods for the solution of constrained optimization problems this method was generalized to the solution of quasi variational inequalities by pang and fukushima comput manag sci 2 21 56 2005 based on the recent improvements achieved for the multiplier penalty approach for optimization we generalize the method by pang and fukushima for quasi variational inequalities in several respects a we allow to compute inexact kkt points of the resulting subproblems b we improve the existing convergence theory c we investigate some special classes of quasi variational inequalities where the resulting subproblems turn out to be easy to solve some numerical results indicate that the corresponding method works quite reliable in practice

sensitivity analysis of generalised eigenproblems and application to wave and finite element models abstract the first and second order sensitivity analysis of the eigenvalue problem of generalised non symmetric matrices using perturbation theory is developed these results are then applied to sensitivity analysis of wave propagation in structures modelled using the wave and finite element wfe method three formulations of the wfe eigenvalue problem are considered the transfer matrix method the projection method and zhong s method the sensitivities with respect to system parameters of wavenumbers and wave mode shapes are derived expressions for the group velocity are presented numerical results for a thin beam a foam core panel and a cross laminated timber panel are used to demonstrate the proposed approach it is shown that sensitivities can be calculated at negligible computational cost

the reciprocal variational approach to the signorini problem with friction approximation results in this paper a new variational formulation of the signorini problem with friction is given in terms of the contact stresses the method corresponds to the direct integral equation approach in classical elastostatic problems first the displacement and mixed problems are briefly described together with some numerical results next the displacements are eliminated by the use of green s function and a constrained minimum problem with respect to the normal and tangential tractions on the contact boundary is derived then the resulting approximation procedure is studied and certain convergence results are proved finally some remarks on the signorini problem with coulomb friction are presented numerical results illustrate the theory

interaction of eigenvalues in multi parameter problems the paper presents a new general theory of interaction of eigenvalues of matrix operators depending on parameters both complex and real eigenvalues are considered strong and weak interactions are distinguished and their geometric interpretation on the complex plane is given mechanical examples are presented and discussed in detail