generalized bundle methods we study a class of generalized bundle methods for which the stabilizing term can be any closed convex function satisfying certain properties this setting covers several algorithms from the literature that have been so far regarded as distinct under a different hypothesis on the stabilizing term and or the function to be minimized we prove finite termination asymptotic convergence and finite convergence to an optimal point with or without limits on the number of serious steps and or requiring the proximal parameter to go to infinity the convergence proofs leave a high degree of freedom in the crucial implementative features of the algorithm i e the management of the bundle of subgradients beta strategy and of the proximal parameter t strategy we extensively exploit a dual view of bundle methods which are shown to be a dual ascent approach to one nonlinear problem in an appropriate dual space where nonlinear subproblems are approximately solved at each step with an inner linearization approach this allows us to precisely characterize the changes in the subproblems during the serious steps since the dual problem is not tied to the local concept of varepsilon subdifferential for some of the proofs a generalization of inf compactness called compactness is required this concept is related to that of asymptotically well behaved functions

analysis of a unilateral contact problem taking into account adhesion and friction abstract in this paper we investigate a contact problem between a viscoelastic body and a rigid foundation when both the effects of the irreversible adhesion and of the friction are taken into account we describe the adhesion phenomenon in terms of a damage surface parameter according to fremond s theory and we model unilateral contact by signorini conditions and friction by a nonlocal coulomb law all the constraints on the internal variables as well as the contact and the friction conditions are rendered by means of subdifferential operators whence the highly nonlinear character of the resulting pde system our main result states the existence of a global in time solution to a suitable variational formulation of the related cauchy problem it is proved by an approximation procedure combined with time discretization

extreme values and the law of the iterated logarithm ifx takes values in a banach spaceb and is in the domain of attraction of a gaussian law onb thenx satisfies the compact law of the iterated logarithm lil with respect to a regular normalizing sequence n iffx satisfies a certain integrability condition the integrability condition is equivalent to the fact that the maximal term of the sample x 1 x 2 x n does not dominate the partial sums s n and here we examine the precise influence of these maximal terms and its relation to the compactlil in particular it is shown that if one deletes enough of the maximal terms there is always a compactlil with non trivial limit set

minimal representation of convex regions defined by analytic functions in this paper we are concerned with characterizing minimal representation of feasible regions defined by both linear and convex analytic constraints we say that a representation is minimal if every other representation has either more analytic nonlinear constraints or has the same number of analytic constraints and at least as many linear constraints we prove necessary and sufficient conditions for the representation to be minimal these are expressed in terms of the redundant constraints pseudo analytic constraints and implicit equality constraints in order to prove the minimal representation theorem we present results on the facets of the convex regions defined by analytic constraints finally we outline the steps of the procedure that could be used to determine a minimal representation

a penalty finite element approximation to a signorini two body contact problem in thermoelasticity in this work a numerical approximation by the finite element method of a quasi static frictionless unilateral contact problem between two thermoelastic bodies in two dimensions is considered the approximation scheme is based on a penalty formulation used to replace the signorini contact condition convergence results and error bounds are obtained and some numerical experiments presented

a quasistatic viscoelastic frictional contact problem with multivalued normal compliance unilateral constraint and material damage abstract the goal of this paper is to deal with a mathematical model which describes the quasistatic frictional contact between a viscoelastic body and a foundation we are interested in the foundation which is made of a hard material covered by a thin layer composed of a rigid crust and a viscoplastic material with finite thickness the mechanical damage caused by excessive stress or strain is emerged in the viscoelastic body during the contact process we provide the variational formulation of the model which is a system consisting of a history dependent variational hemivariational inequality a fully nonlinear integral equation and a parabolic variational inequality then in order to solve this complex system we deliver abstract results on a class of history dependent variational hemivariational inequalities finally our abstract results are applied to derive the unique weak solvability of the frictional contact problem under consideration

convex constrained programmes with unattained infima abstract we consider a problem of minimization of convex function f x over the convex region r where the objective function and the feasible region have a common direction of recession in cases when one of these directions is not in the constancy space of the objective function then the minimal solution is not achieved even if the function f x is bounded below over the region r many algorithms if applied to this class of programmes do not guarantee convergence to the global infimum our approach to this problem leads to derivation of the equation of the feasible parametrized curve c t such that the infimum of the logarithmic penalty function along this curve is equal to the global infimum of the objective function over the region r we show that if all functions defining the program are analytic then c t is also an analytic function the equation of the curve can be successfully used to determine the global infimum in particular unboundedness of the convex constrained programmes in cases when the application of classical methods such as the steepest descent method fails to converge to the global infimum

thermodynamics and analysis of rate independent adhesive contact at small strains we address a model for adhesive unilateral frictionless signorini type contact between bodies of heatconductive viscoelastic material in the linear kelvin voigt rheology undergoing thermal expansion the flow rule for debonding the adhesion is considered rate independent and unidirectional and a thermodynamically consistent model is derived and analysed as far as the existence of a weak solution is concerned

nonlinear error bounds for lower semicontinuous functions on metric spaces as a development of the theory of linear error bounds for lower semicontinuous functions defined on complete metric spaces introduced in aze et al nonlinear anal 49 643 670 2002 and refined in aze and corvellec esaim control optim calc var 10 409 425 2004 we propose a similar approach to nonlinear error bounds based on the notion of strong slope the variational principle and the change of metric principle the latter allowing to obtain sharp estimates for such error bounds through a reduction to the linear case

a reduced gradient method for quadratic programs with quadratic constraints and lp constrained lp approximation problems abstract a dual algorithm is developed for solving a general class of nonlinear programs that properly contains all convex quadratic programs with quadratic constraints and l p constrained l p approximation problems the general dual program to be solved has essentially linear constraints but the objective function is nondifferentiable when certain variables equal zero modifications to the reduced gradient method for linearly constrained problems are presented that overcome the numerical difficulties associated with the nondifferentiable objective function these modifications permit blocks of variables to move to and away from zero on certain iterations even though the objective function is nondifferentiable at points having a block of variables equal to zero

time dependent variational inequalities for viscoelastic contact problems abstract we consider a class of abstract evolutionary variational inequalities arising in the study of contact problems for viscoelastic materials we prove an existence and uniqueness result using standard arguments of time dependent elliptic variational inequalities and banach s fixed point theorem we then consider numerical approximations of the problem we use the finite element method to discretize the spatial domain and we introduce spatially semi discrete and fully discrete schemes for both schemes we show the existence of a unique solution and derive error estimates finally we apply the abstract results to the analysis and numerical approximations of a viscoelastic contact problem with normal compliance and friction

a consistent model coupling adhesion friction and unilateral contact a model considering both unilateral contact coulomb friction and adhesion is presented in the framework of continuum thermodynamics the contact zone is considered as a material boundary and the local constitutive laws are derived by choosing two specific surface potentials the free energy and the dissipation potential because of the non regular properties of these potentials convex analysis is used to derive the local behavior laws from the state and the complementary laws the adhesion is characterized by an internal variable introduced by fremond which represents the intensity of adhesion the continuous transition from a total adhesive condition to a possible pure frictional one is enforced by using elasticity coupled with damage for the interface non penetration conditions and coulomb law are strictly imposed without using any penalty the variational formulation for quasistatic problems is written as the coupling between an implicit variational inequality a variational inequality and a differential equation an increment formulation is proposed an existence result under a condition on the friction coefficient is given a numerical method is derived from the incremental formulation and various algorithms are implemented they solve a sequence of minimization problems under constraints the model is used to simulate a micro indentation experiment conducted to characterize the behavior of fiber matrix interface in a ceramic composite identification of the constitutive parameters is discussed

controlled perturbations for quadratically constrained quadratic programs consider a minimization problem of a convex quadratic function of several variables over a set of inequality constraints of the same type of function the duel program is a maximization problem with a concave objective function and a set of constrains that are essentially linear however the objective function is not differentiable over the constraint region in this paper we study a general theory of dual perturbations and derive a fundamental relationship between a perturbed dual program and the original problem based on this relationship we establish a perturbation theory to display that a well controlled perturbation on the dual program can overcome the nondifferentiability issue and generate an optimal dual solution for an arbitrarily small number a simple linear program is then constructed to make an easy conversion from the dual solution to a corresponding optimal primal solution moreover a numerical example is included to illustrate the potential of this controlled perturbation scheme

a quasistatic frictional problem with normal compliance a friction and contact problem will be considered for an elastic solid body which under the influence of volume and surface forces is in contact with a rigid foundation

mode jumping in the von karman equations we study numerical approximations of bifurcating solution curves of the von karman equations with simply supported and clamped boundary conditions respectively of special interest here is the splitting of a double bifurcation point into two simple bifurcation points and the tracing of the associated secondary solution branches which corresponds to the phenomenon of mode jumping in the buckling of a rectangular plate a continuation unsymmetric lanczos algorithm is utilized for curve tracking our numerical results verify the theoretical prediction of schaeffer and golubitsky namely mode jumping occurs when the boundary is partially clamped but not if it is merely simply supported

how to deal with the unbounded in optimization theory and algorithms the aim of this survey is to show how the unbounded arises in optimization problems and how it leads to fundamental notions which are not only useful for proving theoretical results such as convergence of algorithms and the existence of optimal solutions but also for constructing new methods

analysis of an adhesive contact problem for viscoelastic materials with long memory abstract the goal of this paper is to study a mathematical model which describes the adhesive contact between a deformable body and a foundation the body consists of a viscoelastic material with long memory and the process is assumed to be quasistatic the adhesive contact condition on the normal plane is modeled by a version of normal compliance condition with unilateral constraint in which adhesion is taken into account the adhesive contact condition on the tangential plane is described by an adhesive clarke subdifferential condition and the evolution of the bonding field is described by an ordinary differential equation we derive the variational formulation of this problem which is a system of a variational hemivariational inequality for the displacement field and an ordinary differential equation for the bonding field then we obtain existence and uniqueness results on abstract inclusions and abstract variational hemivariational inequalities finally we apply the abstract results to prove the existence of a unique weak solution to the contact problem

cone of recession and unboundedness of convex functions abstract we consider the problem of determining whether or not a convex function f x is bounded below over r n our focus is on investigating the properties of the vectors in the cone of recession 0 f of f x which are related to the unboundedness of the function

perturbation analysis of a condition number for convex inequality systems and global error bounds for analytic systems in this paper several types of perturbations on a convex inequality system are considered and conditions are obtained for the system to be well conditioned under these types of perturbations where the well conditionedness of a convex inequality system is defined in terms of the uniform boundedness of condition numbers under a set of perturbations it is shown that certain types of perturbations can be used to characterize the well conditionedness of a convex inequality system in which either the system has a bounded solution set and satisfies the slater condition or an associated convex inequality system which defines the recession cone of the solution set for the system satisfies the slater condition finally sufficient conditions are given for the existence of a global error bound for an analytic system it is shown that such a global error bound always holds for any inequality system defined by finitely many convex analytic functions when the zero vector is in the relative interior of the domain of an associated convex conjugate function

variational and numerical analysis of a quasistatic viscoelastic problem with normal compliance friction and damage we consider a model for quasistatic frictional contact between a viscoelastic body and a foundation the material constitutive relation is assumed to be nonlinear the mechanical damage of the material caused by excessive stress or strain is described by the damage function the evolution of which is determined by a parabolic inclusion the contact is modeled with the normal compliance condition and the associated version of coulomb s law of dry friction we derive a variational formulation for the problem and prove the existence of its unique weak solution we then study a fully discrete scheme for the numerical solutions of the problem and obtain error estimates on the approximate solutions

elastic beam in adhesive contact abstract dynamic and quasistatic processes of contact with adhesion between an elastic or viscoelastic beam and a foundation are considered the contact is modeled with the signorini condition when the foundation is rigid and with normal compliance when it is deformable the adhesion is modeled by introducing the bonding function the evolution of which is described by an ordinary differential equation the existence and uniqueness of the weak solution for each of the problems is established using the theory of variational inequalities fixed point arguments and the existence and uniqueness result in commun contemp math 1 1 1999 87 123 the numerical approximations of the quasistatic problem with normal compliance are considered based on semi discrete and fully discrete schemes the convergence of the solutions of the discretized schemes is proved and error estimates for these approximate solutions are derived

global error bounds for convex inequality systems in banach spaces we study conditions under which a global error bound in terms of a natural residual exists for a convex inequality system specifically we obtain an error bound result which unifies many existing results assuming a slater condition we also derive two characterizations for a convex inequality system to possess a global error bound one is in terms of metric regularity and the other is in terms of an associated convex inequality system as a consequence we show that in font bbbabstract msbm10 at 8pt mbox bbbabstract r n a global error bound holds for such a system under the assumption of the zero vector in the relative interior of the domain of an associated conjugate function along with metric regularity at every point of the feasible set defined by the system finally we discuss some applications of these results to convex programs

existence results for dynamic adhesive contact of a rod the existence and uniqueness of the weak solution to the model for the dynamics of a viscoelastic rod which is in adhesive contact with an obstacle is established the model consists of a hyperbolic equation for the vibrations of the rod coupled with a nonlinear ordinary differential equation ode for the evolution of the bonding function the model allows for failure i e complete debonding in finite time the existence of the weak solution is established by using an existence result for odes and the schauder fixed point theorem the limit of an elastic rod when the viscosity vanishes is studied too

the influence of extremes on the law of the iterated logarithm we investigate the law of the iterated logarithm for the partial sum of independent identically distributed random variables when the r n largest summands are excluded

asymptotic derivation of quasistatic frictional contact models with wear for elastic rods abstract the aim of this paper is to derive mathematical models for the bending stretching of an elastic rod in contact with a moving foundation when the resulting wear is taken into account the process is assumed to be quasistatic the contact is modeled with normal compliance and the evolution of the wear function is described with archard s law to derive the models we start with the corresponding 3d problem introduce a change of variable together with the scaling of the unknowns and then we use arguments of asymptotic analysis to obtain error estimates and a convergence result to the limit model

a dynamic model with friction and adhesion with applications to rocks abstract dynamic frictional contact with adhesion of a viscoelastic body and a foundation is formulated as a hemivariational inequality this may model the dynamics of rock layers the normal stress displacement relation on the contact boundary is nonmonotone and nonconvex because of the adhesion process a sequence of regularized problems is considered the necessary a priori estimates are obtained and the existence of a weak solution for the hemivariational inequality is established by passing to the limit as the regularization parameter vanishes

a fast and robust iterative solver for nonlinear contact problems using a primal dual active set strategy and algebraic multigrid for extending the usability of implicit fe codes for large scale forming simulations the computation time has to be decreased dramatically in principle this can be achieved by using iterative solvers in order to facilitate the use of this kind of solvers one needs a contact algorithm which does not deteriorate the condition number of the system matrix and therefore does not slow down the convergence of iterative solvers like penalty formulations do additionally an algorithm is desirable which does not blow up the size of the system matrix like methods using standard lagrange multipliers the work detailed in this paper shows that a contact algorithm based on a primal dual active set strategy provides these advantages and therefore is highly efficient with respect to computation time in combination with fast iterative solvers especially algebraic multigrid methods copyright 2006 john wiley sons ltd

a newton gmres approach for the analysis of the postbuckling behavior of the solutions of the von karman equations we propose a newton gmres type algorithm to solve the discrete von karman problem we show that this algorithm is efficient both in memory and computation time and robust in the neighborhood of the singular points of the bifurcation diagrams placing ourselves in the context of the schaeffer and golubitsky theory we use this algorithm to study the postbuckling behavior of a rectangular plate clamped and compressed along its four sides

convex functions with unbounded level sets and applications to duality theory a class of convex functions with unbounded level sets but good behavior at infinity analyse non lineaire gauthier villars paris 1989 pp 101 122 is investigated characterizations and properties are given the results are then applied to studying sequential approximation schemes for optimization problems and to duality theory when the involved functions have unbounded level sets in particular the convergence properties of stationary sequences for the dual of a convex program are studied and methods for associating with it a primal sequence converging to a solution of the primal problem are demonstrated

a frictionless contact problem for elastic viscoplastic materials with normal compliance numerical analysis and computational experiments in this paper we consider a frictionless contact problem between an elastic viscoplastic body and an obstacle the process is assumed to be quasistatic and the contact is modeled with normal compliance we present a variational formulation of the problem and prove the existence and uniqueness of the weak solution using strongly monotone operators arguments and banach s fixed point theorem we also study the numerical approach to the problem using spatially semi discrete and fully discrete finite elements schemes with implicit and explicit discretization in time we show the existence of the unique solution for each of the schemes and derive error estimates on the approximate solutions finally we present some numerical results involving examples in one two and three dimensions

buckling and postbuckling analysis of rectangular plates resting on elastic foundations with the use of the spectral method abstract buckling and contact effects for a von karman s plate unilaterally connected with a nonlinear elastic tensionless foundation are investigated the foundation is modeled as nonlinear elastic winkler type and shear deformable pasternak type medium compressive and tensile loading forces along the edges of the plate are considered the spectral method is developed for the discretization of the boundary value problems a parametric analysis of compressive and tensile loadings is given bifurcation phenomena of the solution of the equations are explored the iterative scheme with the path following algorithm is employed to solve the resulting system of nonlinear algebraic equations numerical results are presented

convex analysis of spectrally defined matrix functions the purpose of this work is to carry out a systematic study of a special class of convex functions defined over the space s n of symmetric matrices of order n ti n the functions under consideration phi s n ra r cup infty are spectrally defined in the sense that the value phi a depends only on the spectrum lam 1 a ldots lam n a of the matrix a in s n fenchel legendre conjugation first and second order subdifferentiability asymptotic behavior and other concepts of convex analysis are the main ingredients of our exposition

duality methods for solving variational inequalities abstract methods of maximal monotone operators are used in order to study from a general point of view duality numerical algorithms for solving variational inequalities with classical algorithms such as uzawa s method for the standard and augmented lagrangian this paper presents some new algorithms which appear to have very good numerical performances

a frictionless contact problem for elastic viscoplastic materials with normal compliance and damage abstract we study a quasistatic frictionless contact problem with normal compliance and damage for elastic viscoplastic bodies the mechanical damage of the material caused by excessive stress or strain is described by a damage function whose evolution is modelled by an inclusion of parabolic type we provide a variational formulation for the mechanical problem and sketch a proof of the existence of a unique weak solution to the model we then introduce and study a fully discrete scheme for the numerical solutions of the problem an optimal order error estimate is derived for the approximate solutions under suitable solution regularity numerical examples are presented to show the performance of the method

on the sensitivity analysis of hoffman constants for systems of linear inequalities relying on a general variational method developed by the authors and lucchetti nonlinear anal to appear the origin of which goes back to ioffe trans amer math soc 251 1979 pp 61 69 we give a formula for the best hoffman constant sigma inf x notin p a b frac ax b infty d x p a b where p a b x ax le b is a nonempty polyhedron in mathbb r n we also sharpen some results of luo and tseng siam j matrix anal appl 15 1994 pp 636 659 by characterizing the continuity set of some hoffman constants and by pointing out their locally lipschitzian character we apply these results to the study of the behavior of the solution set of a linear program inf ax le b u scriptscriptstyle t x with respect to a b u

tracing the buckling of a rectangular plate with the block gmres method we investigate bifurcation scenario of the von karman equations with partially clamped boundary conditions defined in rectangular domains first we study how the preconditioned block gmres method can be used in the context of continuation methods for tracing solution curves of large systems of nonlinear equations next we discuss linear stabilities of the von karman equations with partially clamped boundary conditions in particular we calculate the first seven eigenvalues and its associated eigenfunctions of the linearized von karman equations via computer algebra the block gmres method is used to solve linear systems and to detect singularity along solution paths of the discrete problem sample numerical results are reported

elasto plastic contact problems with heat exchange and fatigue abstract we deal with a one dimensional temperature dependent model for fatigue accumulation in a moving visco elasto plastic material in contact with an elasto plastic obstacle the problem for the unknown displacement and temperature is formulated using hysteresis operators as solution operators of the underlying variational inequalities the existence result for this problem consisting of the momentum and energy balance equations and an evolution equation for the fatigue is obtained using a priori estimates established for the space discretized problem the uniqueness result follows from the lipschitz continuity of the nonlinearities

regularity of solutions to a dynamic frictionless contact problem with normal compliance the regularity in time of solutions to a problem of frictionless dynamic contact between a viscoelastic body and a reactive foundation is established contact is modeled with the normal compliance condition it is shown that the differentiability of the normal compliance function its behavior at the onset of contact determines the regularity of the solution and additional differentiability of the data and appropriate compatibility on the initial conditions yield higher solutions differentiability unlike the case when the signorini condition is used in this problem there is no intrinsic regularity ceiling a simple example shows that these results are close to being optimal

on infeasibility of systems of convex analytic inequalities abstract in this paper we address the problem of the infeasibility of systems defined by convex analytic inequality constraints in particular we investigate properties of irreducible infeasible sets and provide an algorithm that identifies a set of all constraints k that may affect the feasibility status of the system after some perturbation of the right hand sides we analyze properties of the irreducible sets as well as infeasibility sets in connection with the set k showing in particular that every infeasible system contains an inconsistent subsystem of cardinality not greater than the number of variables plus one the results presented in this paper are generalizations of a theory developed for the systems of quadratic and linear inequality constraints

a membrane in adhesive contact a model for the process of quasi static evolution of an elastic membrane in adhesive contact with a rigid obstacle is developed analyzed and numerically simulated the model consists of an elliptic variational inequality for the membrane displacements and a nonlinear ordinary differential equation for the evolution of the adhesion field by using regularity results from the theory of elliptic variational inequalities and a fixed point argument the system is shown to have a unique weak solution a fully discrete algorithm is described and shown to converge and its error estimates are derived in this process we make critical use of the regularity properties of the solution finally the results of numerical simulations based on the fully discrete algorithm are presented

obstacle problems with cohesion a hemivariational inequality approach and its efficient numerical solution motivated by an obstacle problem for a membrane subject to cohesion forces constrained minimization problems involving a nonconvex and nondifferentiable objective functional representing the total potential energy are considered the associated first order optimality system leads to a hemivariational inequality which can also be interpreted as a special complementarity problem in function space besides an analytical investigation of first order optimality a primal dual active set solver is introduced it is associated to a limit case of a semismooth newton method for a regularized version of the underlying problem class for the numerical algorithms studied in this paper global as well as local convergence properties are derived and verified numerically

variational and numerical analysis of a dynamic frictionless contact problem with adhesion we study a dynamic frictionless contact problem between a viscoelastic body and an obstacle the so called foundation the contact is subjected to an adhesion effect whose evolution is described by an ordinary differential equation for the variational formulation of the contact problem we present and prove an existence and uniqueness result a fully discrete scheme is introduced to solve the problem under certain solution regularity assumptions we derive an optimal order error estimate some numerical examples are included to show the performance of the method

minimal representation of quadratically constrained convex feasible regions in this paper we are concerned with characterizing minimal representations of feasible regions defined by both linear and convex quadratic constraints we say that representation is minimal if every other representation has either more quadratic constraints or has the same number of quadratic constraints and at least as many linear constraints we will prove that a representation is minimal if and only if it contains no redundant constraints no pseudo quadratic constraints and no implicit equality constraints we define a pseudo quadratic constraint as a quadratic constraint that can be replaced by a finite number of linear constraints in order to prove the minimal representation theorem we also prove that if the surfaces of two quadratic constraints match on a ball then they match everywhere in this paper we also provide algorithms that can be used to detect implicit equalities and pseudoquadratic constraints the redundant constraints can be identified using the hypersphere directions hd method

numerical approximation of the elastic viscoplastic contact problem with non matching meshes this work deals with the approximation of a time dependent variational inequality modelling the unilateral contact problem of elastic viscoplastic bodies in a bidimensional context the problem is approximated in the space variable with nonconforming finite element methods which allow the handling of nonmatching meshes on the contact zone several error estimates are established and the corresponding numerical experiments are achieved

a numerical algorithm for a signorini problem associated with maxwell norton materials by using generalized newton s methods abstract the goal of this work is to present new numerical approaches to obtain computationally efficient adaptive procedures to solve contact problems associated with maxwell norton materials we will describe in detail the numerical methods used to solve such problems and derive iterative algorithms to validate the proposed algorithms we will present some test examples with known analytical solutions finally we will compare the efficiency of the algorithms presented in this paper with others previously proposed by the authors

lower cs closed sets and functions abstract in this work we introduce a class of convex sets lcsc f x of a locally convex separated and not necessarily separable topological vector space x they are called the lower cs closed sets this class contains the cs closed sets satisfies the property core c int c c lcsc f x when x is metrizable barrelled and is stable under many operations among them the projection and the denumerable intersection we characterize the lower cs closed functions i e the functions who have a lower cs closed epigraph as marginal functions of cs closed ones and show that they are very stable too we establish an open mapping and a closed graph theorem for the lower cs closed relations finally we show that every real extended valued lower cs closed function defined on a metrizable barrelled space is continuous on the interior of its domain this result allows us to extend classical theorems of convex duality by replacing lower semicontinuous functions by lower cs closed ones more than that it systematizes and extends some methods of convex analysis

variational and numerical analysis of a quasistatic viscoelastic contact problem with adhesion a model for the adhesive quasistatic and frictionless contact between a viscoelastic body and a deformable foundation is described the adhesion process is modelled by a bonding field on the contact surface and contact is described by a modified normal compliance condition the problem is formulated as a coupled system of a variational equality for the displacements and a differential equation for the bonding field the existence of a unique weak solution for the problem and its continuous dependence on the adhesion parameters are established then the numerical analysis of the problem is conducted for the fully discrete approximation the convergence of the scheme is established and error estimates derived finally representative numerical simulations are presented depicting the evolution of the state of the system and in particular the evolution of the bonding field

an algorithm to determine boundedness of quadratically constrained convex quadratic programmes abstract in this paper we present an algorithm which can be used to determine whether or not a convex quadratic objective function is bounded from below over a feasible region defined by convex quadratic constraints if there are m constraints and n variables the algorithm terminates after at most min m 1 n 1 iterations each iteration requires the identification of implicit equality constraints in a system of homogeneous linear inequality and equality constraints and the identification can be completed by the solution of linear programmes in addition the algorithm has the advantage of providing a mechanism to reduce both the number of constraints and the dimension of the problem

numerical solving of frictionless contact problems in perfectly plastic bodies abstract we consider the frictionless unilateral contact problems of a perfectly plastic solid with a rigid foundation and two perfectly plastic solids under an analogous variational formulation for each one we introduce a discrete approximation via the finite element method and in due course prove convergence also we introduce a converging iterative algorithm to the solution of the discrete problems and present some numerical experiences

a constrained least squares regularization method for nonlinear ill posed problems this paper deals with a method for solving ill posed nonlinear hilbert space operator equations f x y regularization is obtained by solving a constrained least squares regularization problem min f x y 2 quad text subject to j x leqq beta 2 beta serves as a regularization parameter and j x is a quadratic penalty functional to robustly and efficiently solve this regularization problem we apply a trust region method at each iteration the quadratic penalty constraint is retained a gauss newton approximation to the objective functional is taken and we add a quadratic trust region constraint the resulting quadratic subproblem is then reformulated as a nonlinear complementarily problem and solved using newton s method this paper applies methods to find approximate solutions to a severely ill posed nonlinear first kind integral equation arising in geophysics the method of generalized cross validation gcv is used to pick the regularization parameter when random er

eigendeformation based reduced order homogenization for failure analysis of heterogeneous materials the manuscript presents a novel model reduction approach for periodic heterogeneous media which combines the multiple scale asymptotic msa expansion method with the transformation field analysis tfa to reduce the computational cost of a direct homogenization approach without significantly compromising on solution accuracy the evolution of failure in micro phases and interfaces is modeled using eigendeformation adaptive model improvement strategy incorporating a hierarchical sequence of computational homogenization models is employed to control the accuracy of the model we present the model formulation and the computational details along with verification with respect to direct homogenization and validation with respect to physical experiments studies

pseudomonotone variational inequality problems existence of solutions necessary and sufficient conditions for the set of solutions of a pseudomonotone variational inequality problem to be nonempty and compact are given

minimizing and stationary sequences of constrained optimization problems we study some fundamental asymptotic properties associated with the well posedness of constrained optimization problems emphasis is placed on the relation between minimizing and stationary sequences and their characterizations in terms of a set of asymptotic karush kuhn tucker kkt optimality conditions unlike the subdifferential approach used by auslender cominetti and crouzeix we use a residual function approach that is closely tied to the theory of error bounds this approach handles constraints explicitly and allows the effective treatment of infeasible stationary sequences the asymptotic kkt conditions provide an asymptotic optimality certificate for inequality constrained programs specializations of the results to convex quadratically constrained convex quadratic spline minimization problems and convex programs with holderian minima are discussed an application of the results to the convergence of a family of newton type iterative descent methods involving singular quasi newton matrices for solving a constrained minimization problem is also presented

optimality condition analysis of macro hybrid mixed variational inclusions abstract optimality conditions of stationary macro hybrid mixed variational inclusions governing constrained optimal control problems are analyzed in a setting of reflexive banach functional spaces primal and dual macro hybrid mixed variational resolvent fixed point methods are applied to establish solvability results for the state systems optimal control macro hybrid mixed optimality conditions are determined via a perturbation conjugate duality analysis on the basis of mixed variational evolutionary results recently published by the author mechanical applications to dual nonlinear distributed control steady diffusion processes as well as to primal boundary constrained static elastoviscoplastic deformation systems illustrate the theory

infeasibility analysis for systems of quadratic convex inequalities in this paper we address the problem of the infeasibility of systems defined by quadratic convex inequality constraints in particular we investigate properties of irreducible infeasible sets and provide an algorithm that identifies a set of all constraints k that may affect the feasibility status of the system after some perturbation of the right hand sides we show that all irreducible sets as well as infeasibility sets are subsets of the set k and that every infeasible system contains an inconsistent subsystem of cardinality not greater than the number of variables plus one the results presented in this paper are also applicable to linear systems

condition numbers and error bounds in convex programming after a brief survey on condition numbers for linear systems of equalities we analyse error bounds for convex functions and convex sets the canonical representation of a convex set is defined other representations of a convex set by a convex function are compared with the canonical representation then condition numbers are introduced for convex sets and their convex representations

a method of analytic centers for quadratically constrained convex quadratic programs an interior point method is developed for maximizing a concave quadratic function under convex quadratic constraints the algorithm constructs a sequence of nested convex sets and finds their approximate centers using a partial newton step given the first convex set and its approximate center the total arithmetic operations required to converge to an approximate solution are of order o sqrt m m n n 2 ln varepsilon where m is the number of constraints n is the number of variables and varepsilon is determined by the desired tolerance of the optimal value and the size of the first convex set a method to initialize the algorithm is also proposed so that the algorithm can start from an arbitrary perhaps infeasible point

a computational contact formulation based on surface potentials this work presents the theory and numerical implementation of a contact formulation based on surface potentials the new theory formulates contact based on distance dependent surface interaction potentials and distinguishes between three interaction classes point interaction short range surface interaction and long range surface interaction here the focus is placed on frictionless contact although the first class readily admits frictional sticking contact as is also shown the proposed contact theory provides a unified framework for various contact formulations ranging from numerical constraint formulations like penalty barrier cross constrained and augmented lagrange multiplier methods to physical interaction formulations like cohesive zone models as well as electrostatic gravitational and van der waals interaction apart from recovering classical penalty and barrier formulations the new theory also naturally leads to a modified penalty and barrier method the formulation also recovers classical one pass contact algorithms however the real advantages lie in a novel two pass contact algorithm denoted the two half pass contact algorithm since each pass only accounts for the contact forces acting on the slave body this implies that traction continuity is only satisfied in theory but not imposed a priori in the algorithm instead it is obtained naturally to high accuracy as is demonstrated by several 2d and 3d numerical examples these include sliding contact peeling contact and electrostatic attraction between deformable solids among the examples is a detailed comparison between the new formulation and classical one pass approaches it is further shown that the new contact formulation passes the contact patch test

the symmetric boundary element method for unilateral contact problems abstract on the basis of the boundary integral equation method in its symmetric formulation the frictionless unilateral contact between two elastic bodies has been studied a boundary discretization by boundary elements leads to an algebraic formulation in the form of a linear complementarity problem in this paper the process of contact or detachment is obtained through a step by step analysis by using generalized weighted quantities as the check elements the detachment or the contact phenomenon may happen when the weighted traction or the weighted displacement is greater than the weighted cohesion or weighted minimum reference gap respectively the applications are performed by using the karnak sgbem program made by some of the authors of this paper

quadratically constrained convex quadratic programmes faulty feasible regions abstract we are concerned with the problem of minimizing a convex quadratic function subject to convex quadratic constraints in particular we are concerned with the application of the method of analytic centres to problems having a faulty feasible region i e a region that is either unbounded or not full dimensional each iteration of a method of centres must determine an analytic centre of a truncated feasible region if the feasible region is faulty then the truncated region may have no interior it may have no analytic centre it may have an infinity of centres or it may have a unique centre thus the difficulty caused by faulty feasible regions is that even when the problem has a solution the method of centres may fail typically faulty regions have been dealt with by the direct method of adding more constraints and variables often involving a big m constant our method is unique in that it results in a reduction in the number of constraints and effectively in the number of variables

unboundedness of a convex quadratic function subject to concave and convex quadratic constraints abstract we present conditions for the existence of upper and lower bounds on convex quadratic objective functions subject to concave and convex quadratic constraints we also present techniques for determining whether or not the conditions are satisfied

non classical law of the iterated logarithm behaviour for trimmed sums we study the law of the iterated logarithm for the partial sum of i i d random variables when the r n largest summands are excluded where r n o log logn this complements earlier work in which the case log logn o rn was considered a law of the iterated logarithm is again seen to prevail for a wide class of distributions but for reasons quite different from previously

existence and uniqueness results for a class of dynamic elasto plastic contact problems abstract this paper focuses on a dynamical model for the motion of a visco elasto plastic body in contact with an elasto plastic obstacle the elastoplastic constitutive laws as well as the contact boundary condition are stated in terms of hysteresis operators under appropriate regularity assumptions on the initial data we show that the resulting partial differential equation with hysteresis possesses a unique solution which is constructed by galerkin approximations and the minty trick in the 1d case the existence and uniqueness proof can be carried out without the viscosity assumption and the necessary a priori estimates are derived from a hysteresis second order energy inequality

noncoercive optimization problems this paper modifies a recent sufficient condition for the existence of optimal solutions proposed by buttazzo and tomarelli to obtain a condition that is necessary and sufficient applications to mathematical programming are given and stability results established

numerical analysis of a frictionless viscoelastic contact problem with normal damped response abstract we consider a mathematical model which describes the frictionless contact between a viscoelastic body and a deformable foundation we model the material s behavior with a nonlinear kelvin voigt constitutive law the process is assumed to be quasistatic and the contact is modeled with a general normal damped response condition we present the variational formulation of the problem including the existence of a unique weak solution to the model we then study the numerical approach to the problem using a fully discrete finite elements scheme with an explicit discretization in time we state the existence of the unique solution for the scheme and derive an error estimate on the approximate solutions finally we present some numerical results involving examples in one and two dimensions

evolutionary variational inequalities arising in viscoelastic contact problems we consider a class of evolutionary variational inequalities arising in various frictional contact problems for viscoelastic materials under the smallness assumption of a certain coefficient we prove an existence and uniqueness result using banach s fixed point theorem we then study two numerical approximation schemes of the problem a semidiscrete scheme and a fully discrete scheme for both schemes we show the existence of a unique solution and derive error estimates finally all these results are applied to the analysis and numerical approximations of a viscoelastic frictional contact problem with the finite element method used to discretize the spatial domain

numerical analysis of a frictionless contact problem for elastic viscoplastic materials abstract we consider a mathematical model which describes the unilateral quasistatic contact of two elastic viscoplastic bodies the contact is without friction and it is modeled by the classical signorini boundary conditions the model consists of an evolution equation coupled with a time dependent variational inequality it has been shown that the variational problem of the model has a unique solution here we consider numerical approximations of the problem we use the finite element method to discretize the spatial domain spatially semi discrete and fully discrete schemes are studied for both schemes we show the existence of a unique solution and derive error estimates under appropriate regularity assumptions of the solution we have the optimal order convergence

on minimizing and critical sequences in nonsmooth optimization let f be a bounded below lower semicontinuous function from a banach space into r cup infty we study the relationships between minimizing and critical sequences of f where the criticality condition is given in terms of some subdifferential partial here the objective function f is not supposed to be convex or smooth our work extends that of auslender and crouzeix and that of chou ng and pang

mathematical justification of viscoelastic beam models by asymptotic methods abstract the authors derive and justify two models for the bending stretching of a viscoelastic rod by using the asymptotic expansion method