confidence regions for stochastic variational inequalities the sample average approximation saa method is a basic approach for solving stochastic variational inequalities svi it is well known that under appropriate conditions the saa solutions provide asymptotically consistent point estimators for the true solution to an svi it is of fundamental interest to use such point estimators along with suitable central limit results to develop confidence regions of prescribed level of significance for the true solution however standard procedures are not applicable because the central limit theorem that governs the asymptotic behavior of saa solutions involves a discontinuous function evaluated at the true solution of the svi this paper overcomes such a difficulty by exploiting the precise geometric structure of the variational inequalities and by appealing to certain large deviations probability estimates and proposes a method to build asymptotically exact confidence regions for the true solution that are computable from the saa solutions we justify this method theoretically by establishing a precise limit theorem apply it to complementarity problems and test it with a linear complementarity problem

convergence and stability of iterative algorithms of generalized set valued variational like inclusions in banach spaces abstract in this paper we give the notion of p proximal mapping an extension of p proximal mapping given by ding and xia j comput appl math 147 2002 369 for a nonconvex lower semicontinuous subdifferentiable proper functional on banach space and prove its existence and lipschitz continuity further we consider a class of generalized set valued variational like inclusions in banach space and show its equivalence with a class of implicit wiener hopf equations using the concept of p proximal mapping using this equivalence we propose a new class of iterative algorithms for the class of generalized set valued variational like inclusions furthermore we prove the existence of solution of generalized set valued variational like inclusions and discuss the convergence criteria and the stability of the iterative algorithm

a degree theoretic approach to parametric nonsmooth equations with multivalued perturbed solution sets this paper studies the sensitivity of a parametric nonsmooth equation using degree theory and set valued analysis under the assumption of existence of an approximating function of a certain kind we investigate the existence continuity and differentiability of the set valued solutions of the parametric equation our analysis is a synthesis of the work of several authors king and rockafellar 1992 gowda and pang 1991 robinson 1991 applications of the derived results to the convergence of newton s method and to the sensitivity analysis of parametric complementarity problems and variational inequalities will be discussed

the binomial theorem and the extended pascal s triangle abstract with the binomial coefficients k n being defined for all integers n k several forms of the binomial theorem valid for all n are provided this is set up using algebraic means on infinitely long rows of numbers when diverging series result they are evaluated modulo an infinite number this modular arithmetic related to p adic arithmetic thus provides a new way to interpret diverging series

a semismooth equation approach to the solution of nonlinear complementarity problems in this paper we present a new algorithm for the solution of nonlinear complementarity problems the algorithm is based on a semismooth equation reformulation of the complementarity problem we exploit the recent extension of newton s method to semismooth systems of equations and the fact that the natural merit function associated to the equation reformulation is continuously differentiable to develop an algorithm whose global and quadratic convergence properties can be established under very mild assumptions other interesting features of the new algorithm are an extreme simplicity along with a low computational burden per iteration we include numerical tests which show the viability of the approach

lipschitz continuity of solutions of linear inequalities programs and complementarity problems it is shown that solutions of linear inequalities linear programs and certain linear complementarity problems e g those with p matrices or z matrices but not semidefinite matrices are lipschitz continuous with respect to changes in the right hand side data of the problem solutions of linear programs are not lipschitz continuous with respect to the coefficients of the objective function the lipschitz constant given here is a generalization of the role played by the norm of the inverse of a nonsingular matrix in bounding the perturbation of the solution of a system of equations in terms of a right hand side perturbation

lyapunov type inequality for a class of even order differential equations in this paper we generalize the well known lyapunov type inequality for second order linear differential equations to a class of even order linear differential equations the results of this paper are new and generalize some known inequalities in the literatures

on an interconnection between the lipschitz continuity of the solution map and the positive principal minor property in linear complementarity problems over euclidean jordan algebras in the setting of euclidean jordan algebras we study the lipschitz continuity of the solution map of linear complementarity problems we show that if the solution map is lipschitz continuous and if the linear transformation has the q property then the linear transformation has the positive principal minor property

lyapunov type inequalities for two classes of nonlinear systems with anti periodic boundary conditions in this paper we establish new lyapunov type inequalities for nonlinear cycled and strongly coupled systems with anti periodic boundary conditions

lyapunov type inequalities for nonlinear systems in this paper by using elementary analysis we establish some new lyapunov type inequalities for nonlinear systems of differential equations special cases of which contain the well known equations such as emden fowler type and half linear equations the inequalities obtained here can be used as handy tools in the study of qualitative behaviour of solutions of the associated equations

lower bounds for generalized eigenvalues of the quasilinear systems abstract in this paper we establish several new lyapunov type inequalities for two classes of one dimensional quasilinear elliptic systems of resonant type which generalize or improve all related existing ones then we use the lyapunov type inequalities obtained in this paper to derive a better lower bound for the generalized eigenvalues of the one dimensional quasilinear elliptic system with the dirichlet boundary conditions

lyapunov inequalities for discrete linear hamiltonian systems abstract in this paper we present some lyapunov type inequalities for discrete linear scalar hamiltonian systems when the coefficient c t is not necessarily nonnegative valued and when the end points are not necessarily usual zeros but rather generalized zeros applying these inequalities we obtain some disconjugacy and stability criteria for discrete hamiltonian systems

stability criteria for linear periodic hamiltonian systems in this paper we obtain new stability criteria for linear periodic hamiltonian systems a lyapunov type inequality is established our results improve the existing works in the literature

on lyapunov type inequality for quasilinear systems in this paper we sketch some recent developments in the theory of lyapunov type inequalities and present some new results relating to a quasilinear system special cases of which contain some well known differential equations such as half linear and linear equations our result generalize the lyapunov type inequality given in 18

mathematical foundations of nonsmooth embedding methods this paper establishes a mathematical foundation for application of the well known classical embedding approach to a class of nonsmooth functions using a recently developed analogue of the derivative in cases where the functions involved fail to be differentiable in the usual sense as part of this development we show how to obtain an extension to this case of the classical hadamard theorem giving conditions for a map of n to itself to be a homeomorphism

variational conditions under the constant rank constraint qualification this paper studies solution properties of a parametric variational condition under the constant rank constraint qualification crcq and properties of its underlying set we start by showing that if the crcq holds at a point in a fixed set then there exists a one to one correspondence between the collection of nonempty faces of the normal cone to the set at that point and the collection of active index sets at points nearby we then study the behavior of the euclidean projector and prove under the crcq that the set of multipliers associated with the euclidean projection is locally a continuous multifunction following that we apply the degree theory to a localized normal map to show that the combination of the crcq and the so called strong coherent orientation condition suffices for the parametric variational condition to have a locally unique continuous solution which is selected from finitely many c1 functions applications of this result under additional assumptions extend or recover some earlier results on parametric variational inequalities

nonsingularity and symmetry for linear normal maps normal maps are single valued generally nonsmooth functions expressing conditions for the solution of variational problems such as those of optimization or equilibrium normal maps arising from linear transformations are particularly important both in their own right and as predictors of the behavior of related nonlinear normal maps they are called locally or globally nonsingular if the functions appearing in them are local or global homeomorphisms satisfying a lipschitz condition we show here that when the linear transformation giving rise to such a normal map has a certain symmetry property the necessary and sufficient condition for nonsingularity takes a particularly simple and convenient form being simply a positive definiteness condition on a certain subspace

solution dependence on initial conditions in differential variational inequalities in the first part of this paper we establish several sensitivity results of the solution x t to the ordinary differential equation ode initial value problem ivp dx dt f x x 0 as a function of the initial value for a nondifferentiable f x specifically we show that for xi t equiv x t xi 0 0 leq t leq t a if f is b differentiable on xi t then so is the solution operator x t at 0 b if f is semismooth on xi t then so is x t at 0 c if f has a linear newton approximation on xi t then so does x t at 0 moreover the linear newton approximation of the solution operator can be obtained from the solution of a linear differential inclusion in the second part of the paper we apply these ode sensitivity results to a differential variational inequality dvi and discuss a the existence uniqueness and lipschitz dependence of solutions to subclasses of the dvi subject to boundary conditions via an implicit function theorem for semismooth equations and b the convergence of a nonsmooth shooting method for numerically computing such boundary value solutions

a homotopy continuation method for solving normal equations in this paper we present a continuation method for solving normal equations generated byc 2 functions and polyhedral convex sets we embed the normal map into a homotopyh and study the existence and characteristics of curves inh 1 0 starting at a specificd point we prove the convergence of such curves to a solution of the normal equation under some conditions on the polyhedral convex setc and the functionf we prove that the curve will have finite are length if the normal map associated with the derivative df and the critical conek is coherently oriented at each zero of the normal mapf c inside a certain ball of n 1998 the mathematical programming society inc published by elsevier science b v

splitting methods for pseudomonotone mixed variational inequalities we consider and analyze some new splitting methods for solving monotone mixed variational inequalities by using the technique of updating the solution the modified methods converge for monotone and pseudomonotone continuous operators the new splitting methods differ from the existing splitting methods the new results are versatile and proof of convergence is very simple

linearly perturbed polyhedral normal cone mappings and applications abstract under a mild regularity assumption we derive an exact formula for the frechet coderivative and some estimates for the mordukhovich coderivative of the normal cone mappings of perturbed polyhedra in reflexive banach spaces our focus point is a positive linear independence condition which is a relaxed form of the linear independence condition employed recently by henrion et al 2010 1 and nam 2010 3 the formulae obtained allow us to get new results on solution stability of affine variational inequalities under linear perturbations thus our paper develops some aspects of the work of henrion et al 2010 1 nam 2010 3 qui in press 12 and yao and yen 2009 6 7

homeomorphism conditions for coherently oriented piecewise affine mappings this article is mainly concerned with the homeomorphism problem for piecewise affine mappings pa maps i e mappings which coincide with an affine mapping on each polyhedron of some finite polyhedral subdivision of rn in the first part we prove that a pa map can be defined without referring to a subdivision of rn as a continuous mapping which coincides at every point x rn with at least one function from a finite collection of affine functions the second part studies the recession function of a pa map it is shown that the recession function is piecewise linear and that a coherently oriented pa map is a homeomorphism if and only if its recession function is a homeomorphism in the last part we prove that a coherently oriented pa map is a homeomorphism if it admits a corresponding polyhedral subdivision of rn such that for some number k 2 n every face of codimension k is contained in at most 2k polyhedra provided the subdivision contains at least one face of codimension k

linear stability of generalized equations part i basic theory dedicated to anthony v fiacco on the occasion of his 65th birthday we consider a class of generalized equations involving set valued maps which formulate many problems from mathematical programming complementarity theory and mathematical economics many results concerning the stability behavior of the solution sets of this class of generalized equations have been established mainly focusing on qualitative characterizations we develop a theory concentrating on quantitative characterizations of the stability behavior of solutions of generalized equations and establish conditions which ensure the solution set of a generalized equation is quantitatively stable in part ii of the paper we use the concepts and methods developed here to treat norlinear programming problems

pascal s triangle on a torus and on a binary tree abstract the extension of pascal s triangle by defining binomial coefficients rn for all integers n r has an interesting interpretation in terms of mod p arithmetic for p an infinite prime integer the different uses and interpretations of the classical binomial coefficients can be extended in varying degrees to the new ones two useful tools in this regard are a torus and a binary tree

a new proof of robinson s homeomorphism theorem for pl normal maps abstract we study a certain class of piecewise linear functions from r n to r n namely robinson s normal maps induced by linear mappings and polyhedral convex sets called pl normal maps robinson s homeomorphism theorem characterizes the pl normal maps which are homeomorphisms as those which have nonzero determinants of the same sign on all pieces of linearity this paper presents a new shorter proof of the result pl normal systems include many optimization and equilibrium problems they arise from variational inequalities or equivalently generalized equations specified by linear maps and polyhedral convex sets unique continuous solvability of these systems which is important in theory and computation is captured by the homeomorphism result

lyapunov type inequalities for second order half linear differential equations in this paper we establish some new lyapunov type inequalities for second order half linear differential equations which almost generalize and extend all related existing results in the literature

some global uniqueness and solvability results for linear complementarity problems over symmetric cones the purpose of this paper is to give a numerical treatment for a class of quasi linear elliptic equations under nonlinear boundary conditions including the three basic types of linear boundary conditions the quasi linear equation is discretized by the finite difference method and the method of upper lower solutions and its associated monotone iteration are used to compute the solutions of the finite difference system this method leads to monotone iterative schemes for the computation of numerical solutions as well as some comparison results among the monotone iterative schemes it also leads to the existence of a maximal and a minimal finite difference solution including the uniqueness of the solution and the convergence of the finite difference solution to the corresponding continuous solution applications are given to two physical problems in heat conduction and combustion theory and numerical results for the heat conduction problem are given and are compared with the known true continuous solution

iterative algorithms for a new class of extended general nonconvex set valued variational inequalities abstract some new classes of extended general nonconvex set valued variational inequalities and the extended general wiener hopf inclusions are introduced by the projection technique equivalence between the extended general nonconvex set valued variational inequalities and the fixed point problems as well as the extended general nonconvex wiener hopf inclusions is proved then by using this equivalent formulation we discuss the existence of solutions of the extended general nonconvex set valued variational inequalities and construct some new perturbed finite step projection iterative algorithms with mixed errors for approximating the solutions of the extended general nonconvex set valued variational inequalities we also verify that the approximate solutions obtained by our algorithms converge to the solutions of the extended general nonconvex set valued variational inequalities the results presented in this paper extend and improve some known results from the literature

a class of linear generalized equations solution stability of a class of linear generalized equations in finite dimensional euclidean spaces is investigated by means of generalized differentiation exact formulas for the frechet and the mordukhovich coderivatives of the normal cone mappings of perturbed euclidean balls are obtained necessary and sufficient conditions for the local lipschitz like property of the solution maps of such linear generalized equations are derived from these coderivative formulas since the trust region subproblems in nonlinear programming can be regarded as linear generalized equations these conditions lead to new results on stability of the parametric trust region subproblems

non zenoness of a class of differential quasi variational inequalities the zeno phenomenon of a switched dynamical system refers to the infinite number of mode switches in finite time the absence of this phenomenon is crucial to the numerical simulation of such a system by time stepping methods and to the understanding of the behavior of the system trajectory extending a previous result for a strongly regular differential variational inequality this paper establishes that a certain class of non strongly regular differential variational inequalities is devoid of the zeno phenomenon the proof involves many supplemental results that are of independent interest specialized to a frictional contact problem with local compliance and polygonal friction laws this non zenoness result is of fundamental significance and the first of its kind

existence and iterative approximation of solutions of generalized mixed equilibrium problems in this paper we consider a generalized mixed equilibrium problem involving non monotone set valued mappings in real hilbert space we extend the notions of the yosida approximation and its corresponding regularized operator given by moudafi and thera a moudafi m thera proximal and dynamical approaches to equilibrium problems in lecture notes in econom and math system vol 477 springer verlag berlin 2002 pp 187 201 and discuss some of their properties further we consider a generalized wiener hopf equation problem and show its equivalence with the generalized mixed equilibrium problem using a fixed point formulation of the generalized wiener hopf equation problem we construct an iterative algorithm furthermore we extend the notion of stability given by harder and hick a m harder t l hicks stability results for fixed point iteration procedures math japonica 33 5 1998 693 706 we prove the existence of a solution and discuss the convergence and stability of the iterative algorithm for the generalized wiener hopf equation problem since the generalized mixed equilibrium problems include variational inequalities as special cases the results presented in this paper continue to hold for these problems

lyapunov type inequalities for higher order half linear differential equations the lyapunov inequality for second order linear differential equations has been extended to higher order linear differential equations in various forms it has also been extended to second and third order half linear equations however it has not yet been developed for higher order half linear differential equations due to the nonlinear feature of the equation and the complexity caused by the multiple p laplacian operators in the equation in this paper by subtle applications of forward and backward inductions we establish several lyapunov type inequalities for even and odd order half linear differential equations these inequalities are applied to determine the nonexistence of nontrivial solutions of higher order boundary value problems and to estimate eigenvalues for higher order eigenvalue problems our results cover many results in the literature when the equations become linear

solving variational inequality problems via smoothing nonsmooth reformulations it has long been known that variational inequality problems can be reformulated as nonsmooth equations recently locally high order convergent newton methods for nonsmooth equations have been well established via the concept of semismoothness when the constraint set of the variational inequality problem is a rectangle several locally convergent newton methods for the reformulated nonsmooth equations can also be globalized in this paper our main aim is to provide globally and locally high order convergent newton methods for solving variational inequality problems with general constraints to achieve this we first prove via convolution that these nonsmooth equations can be well approximated by smooth equations which have desirable properties for the design of newton methods we then reformulate the variational inequality problems as equivalent smoothing nonsmooth equations and apply newton type methods to solve the latter systems and so the variational inequality problems stronger convergence results have been obtained

z transformations on proper and symmetric cones z transformations motivated by the similarities between the properties of z matrices on r n and lyapunov and stein transformations on the semidefinite cone mathcal s n we introduce and study z transformations on proper cones we show that many properties of z matrices extend to z transformations we describe the diagonal stability of such a transformation on a symmetric cone by means of quadratic representations finally we study the equivalence of q and p properties of z transformations on symmetric cones in particular we prove such an equivalence on the lorentz cone

on branching numbers of normal manifolds kuhn and lowen l and robinson 2 have provided deep and elegant results that characterize the homeomorphic members of certain classes of piecewise linear pl mappings the purpose of this paper is to show that the class of mappings examined by robinson is contained in the class of mappings examined by kuhn and lowen let 312 be a finite piecewise linear pl manifold of r and p r r be pl with respect to 3n i e p is affine on each set in 9x the branching number of 9x l is the maximum number of sets in 317 that can contain any common face of codimension 2 in l theorem 5 31 it is shown that if 312 has a branching number less than or equal to 4 then p is a homeomorphism if and only if it is coherently oriented i e the determinants of p on each set in 312 have the same nonzero sign let c be a nonempty polyhedral convex set in real n dimensional space ir and a belong to the space of real n x n matrices f robinson 2 defines a finite pl manifold 32 of k 2 proposition 2 41 called the normal manifold of c and the normal map a i t r induced by a c which is pl with respect to e 2 proposition 2 51 the main result 2 theorem 4 31 shows that ac is homeomorphic if and only if it is coherently oriented see ralph 3 for an alternative proof and robinson 4 for a much simpler proof in the special case that a is symmetric it is pointed out in 2 section l that if xc could be shown to have a branching number less than or equal to 4 then 2 theorem 4 31 would follow from l theorem 5 31 indeed showing that normal manifolds have this branching number property is precisely our goal throughout the paper c and a will be as above let q be the euclidean projection onto c that is rcc maps each point of ir to its nearest point in c with respect to the euclidean norm the identity in ir x is i

lipschitz continuity of solutions of variational inequalities with a parametric polyhedral constraint it is proved that the metric projection from a point onto a moving polyhedron is lipschitz continuous with respect to the perturbations on the right hand sides of the linear inequalities defining the polyhedron the property leads to a simple sufficient condition for lipschitz continuity of a locally unique solution of parametric variational inequalities with a moving polyhedral constraint set applications of these results to traffic network equilibrium problems are given in detail

lyapunov type inequality for a class of dirichlet quasilinear systems involving the p1 p2 pn laplacian abstract we state and prove a generalized lyapunov type inequality for one dimensional dirichlet quasilinear systems involving the p 1 p 2 p n laplacian our result generalize the lyapunov type inequality given in napoli and pinasco 2006 12

piecewise affine bijections of rn and the equation sx tx y we prove two criteria for bijectivity of piecewise affine maps f of rn it is assumed that det f has the same nonzero sign everywhere the first criterion is based solely on the branching properties of the partition of rn into domains of linearity of f the second one concerns the map itself and depends on the images under f of certain two dimensional angles this generalizes and improves results of r schramm the methods are topological our results imply k g murty s solution of the linear complementarity problem and some other partly known facts which we show are equivalent to it for example the map x sx tx is bijective iff all principal minors of s 1t are strictly positive

normal maps inducted by linear transformations we study a certain piecewise linear manifold which we call the normal manifold associated with a polyhedral convex set and a family of continuous functions called normal maps that are induced on this manifold by continuous functions from rn to rn these normal maps occur frequently in optimization and equilibrium problems and the subclass of normal maps induced by linear transformations plays a key role our main result is that the normal map induced by a linear transformation is a lipschitzian homeomorphism if and only if the determinant of the map in each n cell of the normal manifold has the same nonzero sign

variational inequalities for fuzzy mappings ii in this paper we establish the equivalence between the variational inequalities for fuzzy mappings and the wiener hopf equations for fuzzy mappings we use this equivalence to suggest a number of new iterative algorithms for solving the variational inequalities for fuzzy mappings we also study the convergence criteria of these iterative algorithms the auxiliary principle technique is used to study the existence of a solution of the variational inequality for fuzzy mappings and to suggest a novel and innovative iterative algorithm for computing the approximate solution the results proved in this paper represent an improvement of previously known results

generalized multivalued quasi variational inequalities in this paper we introduce and study a new class of variational inequalities known as generalized multivalued quasi variational inequalities this class is the most general and unifying one using essentially the projection technique we establish the equivalence between this class and the implicit wiener hopf equations this equivalence is used to suggest a number of iterative algorithms for solving the generalized multivalued quasi variational inequalities the convergence analysis of these algorithms is also studied the results proved in this paper represent a significant improvement and refinement of the previously known results in this area

variational inequalities over perturbed polyhedral convex sets this paper provides conditions for existence of a locally unique lipschitzian solution of a linear variational inequality posed over a polyhedral convex set in rn under perturbation of either or both of the constant term in the variational inequality and the right hand side of the system of linear constraints defining its feasible set conditions for perturbation of just the constant term are well known here we show that a suitable extension of those conditions suffices for the more general case in which the right hand side of the constraints varies also as a consequence we obtain existence uniqueness and lipschitz continuity properties of solutions of nonlinear variational inequalities posed over perturbed polyhedral convex sets

extending the binomial coefficients to preserve symmetry and pattern abstract we show how to extend the domain of thee binomial coefficients rn so that n and r may take any integer value we argue from two directions on the one hand we wish to preserve symmetry and pattern within pascal s triangle thus creating pascal s hexagon and on the other hand we wish the binomial coefficients to preserve their algebraic role in terms of the taylor series and laurent series expansions of 1 x n valid when x 1 respectively a geometric configuration within the pascal triangle called the pascal flower has some extraordinary properties these properties persist into the hexagon moreover the binomial coefficients may by the use of the function even be extended to all real or complex values of n and r with the conservation of their principal properties

strong lipschitz stability of stationary solutions for nonlinear programs and variational inequalities the stationary solution map x of a canonically perturbed nonlinear program or variational condition is studied we primarily aim at characterizing x to be locally single valued and lipschitz near some stationary point x0 of an initial problem where our focus is on characterizations which are explicitly given in terms of the original functions and assigned quadratic problems since such criteria are closely related to a nonsingularity property of the strict graphical derivative of x explicit formulas for this derivative are presented too it turns out that even for convex polynomial problems our stability and the aubin property does not depend only on the derivatives up to some fixed order of the problem functions at x0 this is in contrast to various other stability concepts further we clarify completely the relations to kojima s strong stability and present simplifications for linearly and certain linearly quadratically constrained problems convex programs and for the map of global minimizers as well

three step iterative algorithms for multivalued quasi variational inclusions in this paper we suggest and analyze some new classes of three step iterative algorithms for solving multivalued quasi variational inclusions by using the resolvent equations technique new iterative algorithms include the ishikawa mann and noor iterations for solving variational inclusions inequalities and optimization problems as special cases the results obtained in this paper represent an improvement and a significant refinement of previously known results

on existence and uniqueness of stationary points in semi infinite optimization the paper deals with semi infinite optimization problems which are defined by finitely many equality constraints and infinitely many inequality constraints we generalize the concept of strongly stable stationary points which was introduced by kojima for finite problems it refers to the local existence and uniqueness of a stationary point for each sufficiently small perturbed problem where perturbations up to second order are allowed under the extended mangasarian fromovitz constraint qualification we present equivalent conditions for the strong stability of a considered stationary point in terms of first and second derivatives of the involved functions in particular we discuss the case where the reduction approach is not satisfied

extending pascal s triangle an alternative is given to hilton and pedersen s method of defining binomial coefficients rn for any integers n r it involves using the usual factorial formula for rn via the defining of the factorial ratio n r for any integers n r in the process defining 00 1 turns out to have several useful applications an attempt to extend the binomial theorem to negative exponents suggests viewing the extended pascal s triangle as binomial coefficients modulo an infinite number

existence of three solutions for a class of dirichlet quasilinear elliptic systems involving the p1 pn laplacian abstract in this paper we prove the existence of at least three weak solutions for the quasilinear elliptic systems p 1 u 1 f u 1 x u 1 u 2 u n 0 in p 2 u 2 f u 2 x u 1 u 2 u n 0 in p n u n f u n x u 1 u 2 u n 0 in u i 0 for 1 i n on our main tool is a recent three critical points theorem of ricceri b ricceri on a three critical points theorem arch math basel 75 2000 220 226

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

lyapunov stability of complementarity and extended systems a linear complementarity system lcs is a piecewise linear dynamical system consisting of a linear time invariant ordinary differential equation ode parameterized by an algebraic variable that is required to be a solution to a finite dimensional linear complementarity problem lcp whose constant vector is a linear function of the differential variable continuing the authors recent investigation of the lcs from the combined point of view of system theory and mathematical programming this paper addresses the important system theoretic properties of exponential and asymptotic stability for an lcs with a c 1 state trajectory the novelty of our approach lies in our employment of a quadratic lyapunov function that involves the auxiliary algebraic variable of the lcs when expressed in the state variable alone the lyapunov function is piecewise quadratic and thus nonsmooth the nonsmoothness feature invalidates standard stability analysis that is based on smooth lyapunov functions in addition to providing sufficient conditions for exponential stability we establish a generalization of the well known lasalle invariance theorem for the asymptotic stability of a smooth dynamical system to the lcs which is intrinsically a nonsmooth system sufficient matrix theoretic copositivity conditions are introduced to facilitate the verification of the stability properties properly specialized the latter conditions are satisfied by a passive like lcs and certain hybrid linear systems having common quadratic lyapunov functions we provide numerical examples to illustrate the stability results we also develop an extended local exponential stability theory for nonlinear complementarity systems and differential variational inequalities based on a new converse theorem for odes with b differentiable right hand sides the latter theorem asserts that the existence of a b differentiable lyapunov function is a necessary and sufficient condition for the exponential stability of an equilibrium of such a differential system

a discrete analogue of lyapunov type inequalities for nonlinear systems in this paper by using elementary analysis we establish some new lyapunov type inequalities for nonlinear systems of difference equations when the coefficent b 2 t is not necessarily nonnegative valued and when the end points are not necessarily usual zeros but rather generalized zeros applying these inequalities we obtain a disconjugacy criterion and boundedness for the solution of our system some special cases of our results contain recently developed lyapunov inequalities for discrete linear hamiltonian systems the inequalities obtained here can be used as handy tools in the study of the qualitative behaviour of solutions of the associated equations

iterative solving of generalized equations with calm solution mappings abstract we present an iterative procedure for solving in finite dimensions generalized equations of the form 0 f x f x where f is a continuous function while f stands for a closed set valued mapping assuming that f belongs to a class of functions admitting a certain type of approximation and that the solution set of satisfies a calmness type property we show that the method we consider is superlinearly convergent to a solution of

lyapunov type integral inequalities for certain higher order differential equations in this paper first we will give a short survey of the most basic results on lyapunov inequality and next we obtain this type integral inequalities for certain higher order differential equations our results are sharper than some results of yang 2003 20

solution continuity in monotone affine variational inequalities in this paper we study the behavior of solutions of finite dimensional monotone affine variational inequalities posed over graph convex polyhedral multifunctions we identify precisely the class of positive semidefinite linear transformations appearing in these variational inequalities for which the solution sets will be lipschitzian in the argument of the underlying multifunction this class is that of cocoercive linear transformations which include but are not limited to those appearing in problems of linear or of convex quadratic programming

new extragradient type methods for general variational inequalities in this paper we consider and analyze a new class of extragradient type methods for solving general variational inequalities the modified methods converge for pseudomonotone operators which is weaker condition than monotonicity our proof of convergence is very simple as compared with other methods the proposed methods include several new and known methods as special cases our results present a significant improvement of previously known methods for solving variational inequalities and related optimization problems

a class of projection methods for general variational inequalities abstract in this paper we consider and analyze a new class of projection methods for solving pseudomonotone general variational inequalities using the wiener hopf equations technique the modified methods converge for pseudomonotone operators our proof of convergence is very simple as compared with other methods the proposed methods include several known methods as special cases

homotopy method for solving ball constrained variational inequalities abstract in this paper we present a smoothing homotopy method for solving ball constrained variational inequalities by utilizing a similar chen harker kanzow smale function to smooth robinson s normal equation without any monotonicity condition on the defining map f for the starting point chosen almost everywhere in r n the existence and convergence of the homotopy pathway are proven numerical experiments illustrate that the method is feasible and effective

on the lipschitzian properties of polyhedral multifunctions in this paper we show that for a polyhedral multifunctionf rn rm with convex range the inverse functionf 1 is locally lower lipschitzian at every point of the range off equivalently lipschitzian on the range off if and only if the functionf is open as a consequence we show that for a piecewise affine functionf rn rn f is surjective andf 1 is lipschitzian if and only iff is coherently oriented an application via robinson s normal map formulation leads to the following result in the context of affine variational inequalities the solution mapping as a function of the data vector is nonempty valued and lipschitzian on the entire space if and only if the solution mapping is single valued this extends a recent result of murthy parthasarathy and sabatini proved in the setting of linear complementarity problems

on variational inequalities over polyhedral sets the results on regularity behavior of solutions to variational inequalities over polyhedral sets proved in a series of papers by robinson ralph and dontchev rockafellar in the 90s has long become classics of variational analysis but the available proofs are very complicated and practically do not use techniques of variational analysis the only exception is the proof by dontchev and rockafellar of their critical face regularity criterion in the paper we offer a different approach completely based on polyhedral geometry and a few basic principles of metric regularity theory it leads to new proofs that look simpler and shorter and in addition gives some clarifying geometrical information

a direct complementarity approach for the elastoplastic analysis of plane stress and plane strain structures this paper presents a direct complementarity approach for carrying out the elastoplastic analysis of plane stress and plane strain structures founded on a traditional finite step formulation our approach however avoids the typically cumbersome implementation of iterative predictorcorrector procedures associated with the ubiquitous return mapping algorithm instead at each predefined step the governing formulationcast in its most natural mathematical programming format known as a mixed complementarity problemis directly solved by using a complementarity solver run from within a mathematical modeling system we have chosen the industry standard general algebraic modeling system path mixed complementarity problem solver that is called from within the general algebraic modeling system environment we consider both von mises and tresca materials with perfect or hardening kinematic and isotropic behaviors our numerical tests five benchmark examples of which are presented in this paper have been carried out using models constructed from the mixed finite element of capsoni and corradi comput methods appl mech eng 1997 141 6793 which beneficially offers a locking free behavior and coarse mesh accuracy the results indicate in addition to an isochoric locking free behavior good accuracy and the ability to circumvent the difficult singularity problem associated with the corners of tresca yield surfaces

a note on tang and he s paper abstract in this paper we establish new lyapunov type inequalities for two classes of one dimensional quasilinear elliptic systems of resonant type which improve the recent results of tang and he x h tang x he lower bounds for generalized eigenvalues of the quasilinear systems j math anal appl 385 2012 72 85 when 1 p i i 1 2 n

a proof of the branching number bound for normal manifolds abstract we prove that every face of dimension n 2 of the normal manifold induced by a polyhedron in r n is contained in precisely four distinct n dimensional polyhedra of the normal manifold this result implies that the branching number of the normal manifold is 4 and refines ralph s result which only gave 4 as an upper bound the proof is based on the well known fact that every ridge of a polyhedron is contained in precisely two facets

lyapunov type inequality for a class of odd order differential equations in this paper we give a generalization of the well known lyapunov type inequality for a class of odd order differential equations the result of this paper is new and generalizes some early results on this topic

semismooth homeomorphisms and strong stability of semidefinite and lorentz complementarity problems based on an inverse function theorem for a system of semismooth equations this paper establishes several necessary and sufficient conditions for an isolated solution of a complementarity problem defined on the cone of symmetric positive semidefinite matrices to be strongly regular stable we show further that for a parametric complementarity problem of this kind if a solution corresponding to a base parameter is strongly stable then a semismooth implicit solution function exists whose directional derivatives can be computed by solving certain affine problems on the critical cone at the base solution similar results are also derived for a complementarity problem defined on the lorentz cone the analysis relies on some new properties of the directional derivatives of the projector onto the semidefinite cone and the lorentz cone

on bilevel programming part i general nonlinear cases this paper is concerned with general nonlinear nonconvex bilevel programming problems blpp we derive necessary and sufficient conditions at a local solution and investigate the stability and sensitivity analysis at a local solution in the blpp we then explore an approach in which a bundle method is used in the upper level problem with subgradient information from the lower level problem two algorithms are proposed to solve the general nonlinear blpp and are shown to converge to regular points of the blpp under appropriate conditions the theoretical analysis conducted in this paper seems to indicate that a sensitivity based approach is rather promising for solving general nonlinear blpp