global asymptotic and exponential stability of a dynamic neural system with asymmetric connection weights recently a dynamic neural system was presented and analyzed due to its good performance in optimization computation and low complexity for implementation the global asymptotic stability of such a dynamic neural system with symmetric connection weights was well studied in this note based on a new lyapunov function we investigate the global asymptotic stability of the dynamic neural system with asymmetric connection weights since the dynamic neural system with asymmetric weights is more general than that with symmetric ones the new results are significant in both theory and applications specially the new result can cover the asymptotic stability results of linear systems as special cases

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

a projection algorithm for general variational inequalities with perturbed constraint sets we consider the problem of general variational inequalities gvi with nonmonotone operator in a finite dimensional space we propose a method to solve gvi that at each iteration considers only one projection on an easy approximation of the constraint set which is important from a practical point of view we analyse the convergence of the algorithm under a weak cocoercivity condition using variational metric analysis computational experience is reported and comparative analysis with other two algorithms is also given for the monotone case

global exponential periodicity of a class of recurrent neural networks with oscillating parameters and time varying delays in this paper we present the analytical results on the global exponential periodicity of a class of recurrent neural networks with oscillating parameters and time varying delays sufficient conditions are derived for ascertaining the existence uniqueness and global exponential periodicity of the oscillatory solution of such recurrent neural networks by using the comparison principle and mixed monotone operator method the periodicity results extend or improve existing stability results for the class of recurrent neural networks with and without time delays

a novel neural network for nonlinear convex programming in this paper we present a neural network for solving the nonlinear convex programming problem in real time by means of the projection method the main idea is to convert the convex programming problem into a variational inequality problem then a dynamical system and a convex energy function are constructed for resulting variational inequality problem it is shown that the proposed neural network is stable in the sense of lyapunov and can converge to an exact optimal solution of the original problem compared with the existing neural networks for solving the nonlinear convex programming problem the proposed neural network has no lipschitz condition no adjustable parameter and its structure is simple the validity and transient behavior of the proposed neural network are demonstrated by some simulation results

approximation by unitaries with finite spectrum in purely infinite c algebras abstract we prove that a unital purely infinite simple c algebra has c exponential rank at most 1 and therefore has the property weak fu every element of the identity component of the unitary group is norm limit of unitaries with finite spectrum we also prove that the appropriate modification of this statement for nonunital algebras

global and superlinear convergence of the smoothing newton method and its application to general box constrained variational inequalities the smoothing newton method for solving a system of nonsmooth equations f x 0 which may arise from the nonlinear complementarity problem the variational inequality problem or other problems can be regarded as a variant of the smoothing method at the kth step the nonsmooth function f is approximated by a smooth function f and the derivative of f at x k is used as the newton iterative matrix the merits of smoothing methods and smoothing newton methods are global convergence and convenience in handling in this paper we show that the smoothing newton method is also superlinearly convergent if f is semismooth at the solution and f satisfies a jacobian consistency property we show that most common smooth functions such as the gabriel more function have this property as an application we show that for box constrained variational inequalities if the involved function is p uniform the iteration sequence generated by the smoothing newton method will converge to the unique solution of the problem globally and superlinearly quadratically

effect of transmission delay on the rate of convergence of a class of nonlinear contractive dynamical systems we investigate the qualitative properties of a general class of contractive dynamical systems with time delay by using a unified analysis approach for any p contraction with p spl isin 1 spl infin it is proved that the delayed contractive dynamical system is always globally exponentially stable no matter how large the time delay is while the rate of convergence of the delayed system is reduced as the time delay increases a lower bound on the rate of convergence of the delayed contractive dynamical system is obtained which is the unique positive solution of a nonlinear equation with three parameters namely the time delay the time constant and the p contraction constant in the system we show that the previously established results in the literature about the global asymptotic or exponential stability independent of delay for hopfield type neural networks can actually be deduced by recasting the network model into the general framework of contractive dynamical systems with some p contraction p spl isin 1 spl infin under the given delay independent stability conditions numerical simulation examples are also presented to illustrate the obtained theoretical results

a novel neural network for variational inequalities with linear and nonlinear constraints variational inequality is a uniform approach for many important optimization and equilibrium problems based on the sufficient and necessary conditions of the solution this paper presents a novel neural network model for solving variational inequalities with linear and nonlinear constraints three sufficient conditions are provided to ensure that the proposed network with an asymmetric mapping is stable in the sense of lyapunov and converges to an exact solution of the original problem meanwhile the proposed network with a gradient mapping is also proved to be stable in the sense of lyapunov and to have a finite time convergence under some mild conditions by using a new energy function compared with the existing neural networks the new model can be applied to solve some nonmonotone problems has no adjustable parameter and has lower complexity thus the structure of the proposed network is very simple since the proposed network can be used to solve a broad class of optimization problems it has great application potential the validity and transient behavior of the proposed neural network are demonstrated by several numerical examples

on a generalization of a normal map and equation the class of normal maps was recently investigated by robinson and ralph in connection with the study of a variational inequality defined on a polyhedral set in this paper a generalization of such a map is considered and the associated generalized normal equation is studied the latter provides a unified formulation of several generalized variational inequality and complementarity problems using degree theory some sufficient conditions for the existence of a zero of a generalized normal map are established and the stability of a generalized normal equation at a solution is analyzed specializations of the results to various applications are discussed

a general methodology for designing globally convergent optimization neural networks we present a general methodology for designing optimization neural networks we prove that the neural networks constructed by using the proposed method are guaranteed to be globally convergent to solutions of problems with bounded or unbounded solution sets in contrast with the gradient methods whose convergence is not guaranteed we show that the proposed method contains both the gradient methods and nongradient methods employed in existing optimization neural networks as special cases based on the theoretical results of the proposed method we study the convergence and stability of general gradient models in the case of unisolated solutions using the proposed method we derive some new neural network models for a very large class of optimization problems in which the equilibrium points correspond to exact solutions and there is no variable parameter finally some numerical examples show the effectiveness of the method

a neural network model for monotone linear asymmetric variational inequalities a linear variational inequality is a uniform approach for some important problems in optimization and equilibrium problems we give a neural network model for solving asymmetric linear variational inequalities the model is based on a simple projection and contraction method computer simulation is performed for linear programming lp and linear complementarity problems lcp the test results for the lp problem demonstrate that our model converges significantly faster than the three existing neural network models examined in a comparative study paper

common nature of learning between back propagation and hopfield type neural networks for generalized matrix inversion with simplified models in this paper two simple structure neural networks based on the error back propagation bp algorithm i e bp type neural networks bpnns are proposed developed and investigated for online generalized matrix inversion specifically the bpnn l and bpnn r models are proposed and investigated for the left and right generalized matrix inversion respectively in addition for the same problem solving task two discrete time hopfield type neural networks hnns are developed and investigated in this paper similar to the classification of the presented bpnn l and bpnn r models the presented hnn l and hnn r models correspond to the left and right generalized matrix inversion respectively comparing the bpnn weight updating formula with the hnn state transition equation for the specific i e left or right generalized matrix inversion we show that such two derived learning expressions turn out to be the same in mathematics although the bp and hopfield type neural networks are evidently different from each other a great deal in terms of network architecture physical meaning and training patterns numerical results with different illustrative examples further demonstrate the efficacy of the presented bpnns and hnns for online generalized matrix inversion and more importantly their common natures of learning

high order differential approximants we introduce a new form of differential approximant for the summation of power series the method is a special type of pade hermite approximant it consists of a high order linear differential equation with polynomial coefficients that is satisfied approximately by the partial sum of the power series this method is able to reproduce the polylogarithmic functions exactly numerical evidence suggests that this is currently one of the best methods of singularity analysis for many problems

global convergence of a class of trust region algorithms for optimization with simple bounds this paper extends the known excellent global convergence properties of trust region algorithms for unconstrained optimization to the case where bounds on the variables are present weak conditions on the accuracy of the hessian approximations are considered it is also shown that when the strict complementarily condition holds the proposed algorithms reduce to an unconstrained calculation after finitely many iterations allowing a fast asymptotic rate of convergence

dynamic system methods for solving mixed linear matrix inequalities and linear vector inequalities and equalities a novel idea is proposed for solving a system of mixed linear matrix inequalities and linear vector inequalities and equalities first the problem is converted into an unconstrained minimization problem with a continuously differentiable convex objective function then a continuous time dynamic system and a discrete time dynamic system are proposed for solving it under some mild conditions the proposed dynamic systems are shown to be globally convergent to a solution of the problem the merits of the methods refer to their simple numerical implementations and capability for handling nonstrict lmis easily in addition the methods are promising in neural circuits realization and therefore have potential applications in many online control problems several numerical examples are presented to illustrate the performance of the methods and substantiate the theoretical results

an efficient dynamic model for solving a portfolio selection with uncertain chance constraint models this paper presents a neural network model for solving maximization programming model with chance constraint in which the security returns are uncertain variables the main idea is to replace the portfolio selection models when the uncertain returns are chosen as some special cases such as linear uncertain variables trapezoidal uncertain variables and normal uncertain variables with a linear programming lp problem according to the saddle point theorem optimization theory convex analysis theory lyapunov stability theory and lasalle invariance principle the equilibrium point of the proposed neural network is proved to be equivalent to the optimal solution of the original problem it is also shown that the proposed neural network model is stable in the sense of lyapunov and it is globally convergent to an exact optimal solution of the portfolio selection problem with uncertain returns two illustrative examples are provided to show the feasibility and the efficiency of the proposed method in this paper

estimate of exponential convergence rate and exponential stability for neural networks estimates of exponential convergence rate and exponential stability are studied for a class of neural networks which includes hopfield neural networks and cellular neural networks both local and global exponential convergence are discussed theorems for estimation of the exponential convergence rate are established and the bounds on the rate of convergence are given the domains of attraction in the case of local exponential convergence are obtained simple conditions are presented for checking exponential stability of the neural networks

the barzilai and borwein gradient method for the large scale unconstrained minimization problem the barzilai and borwein gradient method for the solution of large scale unconstrained minimization problems is considered this method requires few storage locations and very inexpensive computations furthermore it does not guarantee descent in the objective function and no line search is required recently the global convergence for the convex quadratic case has been established however for the nonquadratic case the method needs to be incorporated in a globalization scheme in this work a nonmonotone line search strategy that guarantees global convergence is combined with the barzilai and borwein method this strategy is based on the nonmonotone line search technique proposed by grippo lampariello and lucidi siam j numer anal 23 1986 pp 707 716 numerical results to compare the behavior of this method with recent implementations of the conjugate gradient method are presented these results indicate that the global barzilai and borwein method may allow some significant reduction in the number of line searches and also in the number of gradient evaluations

stability analysis of dynamical neural networks in this paper we use the matrix measure technique to study the stability of dynamical neural networks testable conditions for global exponential stability of nonlinear dynamical systems and dynamical neural networks are given it shows how a few well known results can be unified and generalized in a straightforward way local exponential stability of a class of dynamical neural networks is also studied we point out that the local exponential stability of any equilibrium point of dynamical neural networks is equivalent to the stability of the linearized system around that equilibrium point from this some well known and new sufficient conditions for local exponential stability of neural networks are obtained

solving complementarity and variational inequalities problems using neural networks in this paper we propose a recurrent neural network model for solving a class of monotone variational inequalities problem with linear constraints the neural network is stable in the sense of lyapunov and globally convergent to an optimal solution compared with the existing convergence results the present proof do not require lipschitz continuity condition on the objective function this neural network model has no adjustable parameter thus its structure is very simple variational inequalities problem with general set of constraints plus a general form of the complementarity problems are solved using the proposed neural networks some examples demonstrated to show the applicability of the proposed neural networks to solve various nonlinear optimization problems numerically

a case study of methods of series summation kelvin helmholtz instability of finite amplitude we compute the singularities of the solution of the birkhoff rott equation that governs the evolution of a planar periodic vortex sheet our approach uses the taylor series obtained by meiron et al j fluid mech 114 1982 283 for a flat sheet subject initially to a sinusoidal disturbance of amplitude a the series is then summed by using various generalisations of the pade method we find approximate values for the location and type of the principal singularity as a ranges from zero to infinity finally the results are used as a basis to guide the choice of methods of summing series arising from problems in fluid mechanics

a dual neural network for redundancy resolution of kinematically redundant manipulators subject to joint limits and joint velocity limits in this paper a recurrent neural network called the dual neural network is proposed for online redundancy resolution of kinematically redundant manipulators physical constraints such as joint limits and joint velocity limits together with the drift free criterion as a secondary task are incorporated into the problem formulation of redundancy resolution compared to other recurrent neural networks the dual neural network is piecewise linear and has much simpler architecture with only one layer of neurons the dual neural network is shown to be globally exponentially convergent to optimal solutions the dual neural network is simulated to control the pa10 robot manipulator with effectiveness demonstrated

parameter identification of tsk neuro fuzzy models abstract in the framework of the tsk neuro fuzzy model a combination of the two well known identification methods are employed for parameter estimation of the neuro fuzzy inference system namely the series parallel and the parallel configurations the presented paper proposes two new possible configurations for identifying the parameters of the tsk neuro fuzzy model using the combinations of these two existing configurations one of the proposed configurations constitutes the series parallel configuration to the premise part and the parallel configuration to the consequent part of the neuro fuzzy model termed as ps p configuration the second one is composed of the series parallel configuration to the consequent part and the parallel configuration to the premise part of the neuro fuzzy model termed as cs p configuration the presented work mainly deals with a comparative study of the proposed configurations and the existing configurations in the context of parameter identification of the tsk neuro fuzzy model on three different benchmark examples moreover it investigates upper bound of the learning rates using the lyapunov stability theorem to assure the stability and the convergence of the model learning process implementation of the modified mountain clustering mmc and the cluster validity function yields initial models to restrict the upper bound during the learning process it also presents a two phase adaptive learning rate

a novel neural network for generally constrained variational inequalities this paper presents a novel neural network for solving generally constrained variational inequality problems by constructing a system of double projection equations by defining proper convex energy functions the proposed neural network is proved to be stable in the sense of lyapunov and converges to an exact solution of the original problem for any starting point under the weaker cocoercivity condition or the monotonicity condition of the gradient mapping on the linear equation set furthermore two sufficient conditions are provided to ensure the stability of the proposed neural network for a special case the proposed model overcomes some shortcomings of existing continuous time neural networks for constrained variational inequality and its stability only requires some monotonicity conditions of the underlying mapping and the concavity of nonlinear inequality constraints on the equation set the validity and transient behavior of the proposed neural network are demonstrated by some simulation results

exponential stability of globally projected dynamic systems in this paper we further analyze and prove the stability and convergence of the dynamic system proposed by friesz et al 1994 whose equilibria solve the associated variational inequality problems two sufficient conditions are provided to ensure the asymptotic stability of this system with a monotone and asymmetric mapping by means of an energy function meanwhile this system with a monotone and gradient mapping is also proved to be asymptotically stable using another energy function furthermore the exponential stability of this system is also shown under strongly monotone condition some obtained results improve the existing ones and the given conditions can be easily checked in practice since this dynamic system has wide applications the obtained results are significant in both theory and applications

a new recurrent neural network for solving convex quadratic programming problems with an application to the k winners take all problem in this paper a new recurrent neural network is proposed for solving convex quadratic programming qp problems compared with existing neural networks the proposed one features global convergence property under weak conditions low structural complexity and no calculation of matrix inverse it serves as a competitive alternative in the neural network family for solving linear or quadratic programming problems in addition it is found that by some variable substitution the proposed network turns out to be an existing model for solving minimax problems in this sense it can be also viewed as a special case of the minimax neural network based on this scheme a k winners take all k wta network with o n complexity is designed which is characterized by simple structure global convergence and capability to deal with some ill cases numerical simulations are provided to validate the theoretical results obtained more importantly the network design method proposed in this paper has great potential to inspire other competitive inventions along the same line

on the resolution of linearly constrained convex minimization problems the problem of minimizing a twice differentiable convex function f is considered subject to ax b x geq 0 where a in mathbb r m times n m n are large and the feasible region is bounded it is proven that this problem is equivalent to a primal dual box constrained problem with 2n m variables the equivalent problem involves neither penalization parameters nor ad hoc multiplier estimators this problem is solved using an algorithm for bound constrained minimization that can deal with many variables numerical experiments are presented

an improved dual neural network for solving a class of quadratic programming problems and its k winners take all application this paper presents a novel recurrent neural network for solving a class of convex quadratic programming qp problems in which the quadratic term in the objective function is the square of the euclidean norm of the variable this special structure leads to a set of simple optimality conditions for the problem based on which the neural network model is formulated compared with existing neural networks for general convex qp the new model is simpler in structure and easier to implement the new model can be regarded as an improved version of the dual neural network in the literature based on the new model a simple neural network capable of solving the k winners take all k wta problem is formulated the stability and global convergence of the proposed neural network is proved rigorously and substantiated by simulation results

alternative theorems for nonlinear projection equations and applications to generalized complementarity problems let k be a closed convex subset in rn and let f g and h be three functions from rn into itself we consider the following nonlinear projection equation npe h x k g x f x x r 1 where k is the orthogonal projection operator onto the set k this equation provides a uni ed formulation of several interesting and important special cases in the case when h x g x for any x rn then 1 reduces to the equation studied by pang and yao 17 under the name of generalized normal equation which is equivalent to the following generalized variational inequality gvi g x k y g x f x 0 for all y k 2

a recurrent neural network for solving bilevel linear programming problem in this brief based on the method of penalty functions a recurrent neural network nn modeled by means of a differential inclusion is proposed for solving the bilevel linear programming problem blpp compared with the existing nns for blpp the model has the least number of state variables and simple structure using nonsmooth analysis the theory of differential inclusions and lyapunov like method the equilibrium point sequence of the proposed nns can approximately converge to an optimal solution of blpp under certain conditions finally the numerical simulations of a supply chain distribution model have shown excellent performance of the proposed recurrent nns

exponentials in simple z stable c algebras abstract let a be a unital separable simple z stable c algebra which has rational tracial rank at most one and let u u 0 a where u 0 a is the connected component of the unitary group of a containing the identity we show that for any 0 there exists a self adjoint element h a such that 0 1 u exp i h but there is no control of h in general let cu a be the closure of the commutator subgroup of the unitary group of a and let u cu a we prove that there exists a self adjoint element h a such that 0 2 u exp i h and h 2 examples are given that the bound 2 for h is optimal in general for the jiang su algebra z we show that if u u 0 z and 0 there exists a real number t and a self adjoint element h z with h 2 such that e i t u exp i h

a new projection based neural network for constrained variational inequalities this paper presents a new neural network model for solving constrained variational inequality problems by converting the necessary and sufficient conditions for the solution into a system of nonlinear projection equations five sufficient conditions are provided to ensure that the proposed neural network is stable in the sense of lyapunov and converges to an exact solution of the original problem by defining a proper convex energy function the proposed neural network includes an existing model and can be applied to solve some nonmonotone and nonsmooth problems the validity and transient behavior of the proposed neural network are demonstrated by some numerical examples

an alternative theorem for generalized variational inequalities and solvability of nonlinear quasi p m complementarity problems in this paper an alternative theorem and hence a sufficient solution condition is established for generalized variational inequality problems the concept of exceptional family for generalized variational inequality is introduced this concept is general enough to include as special cases the notions of exceptional family of elements and the d orientation sequence for continuous functions particularly we apply the alternative theorem for investigating the solvability of the nonlinear complementarity problems with so called quasi p m maps which are broad enough to encompass the quasi monotone maps and p maps as the special cases an existence theorem for this class of complementarity problems is established which significantly generalizes several previous existence results in the literature

a neural network for a class of convex quadratic minimax problems with constraints in this paper we propose a neural network for solving a class of convex quadratic minimax problems with constraints four sufficient conditions are provided to ensure the asymptotic stability of the proposed network furthermore the exponential stability of the proposing network is also proved under certain conditions the results obtained here can be further extended to the globally projected dynamical system in addition some new stability conditions for the system are also obtained since our stability conditions can be easily checked in practice these results becomes more attractive in real applications

reformulation of variational inequalities on a simplex and compactification of complementarity problems many variational inequality problems vips can be reduced by a compactification procedure to a vip on the canonical simplex reformulations of this problem are studied including smooth reformulations with simple constraints and unconstrained reformulations based on the penalized fischer burmeister function it is proved that bounded level set results hold for these reformulations under quite general assumptions on the operator therefore it can be guaranteed that minimization algorithms generate bounded sequences and under monotonicity conditions these algorithms necessarily find solutions of the original problem some numerical experiments are presented

neural network for quadratic optimization with bound constraints a recurrent neural network is presented which performs quadratic optimization subject to bound constraints on each of the optimization variables the network is shown to be globally convergent and conditions on the quadratic problem and the network parameters are established under which exponential asymptotic stability is achieved through suitable choice of the network parameters the system of differential equations governing the network activations is preconditioned in order to reduce its sensitivity to noise and to roundoff errors the optimization method employed by the neural network is shown to fall into the general class of gradient methods for constrained nonlinear optimization and in contrast with penalty function methods is guaranteed to yield only feasible solutions

on the analysis of a recurrent neural network for solving nonlinear monotone variational inequality problems we investigate the qualitative properties of a recurrent neural network rnn for solving the general monotone variational inequality problems vips defined over a nonempty closed convex subset which are assumed to have a nonempty solution set but need not be symmetric the equilibrium equation of the rnn system simply coincides with the nonlinear projection equation of the vip to be solved we prove that the rnn system has a global and bounded solution trajectory starting at any given initial point in the above closed convex subset which is positive invariant for the rnn system for general monotone vips we show by an example that the trajectory of the rnn system can converge to a limit cycle rather than an equilibrium in the case that the monotone vips are not symmetric contrary to this for the strictly monotone vips it is shown that every solution trajectory of the rnn system starting from the above closed convex subset converges to the unique equilibrium which is also locally asymptotically stable in the sense of lyapunov no matter whether the vips are symmetric or nonsymmetric for the uniformly monotone vips the aforementioned solution trajectory of the rnn system converges to the unique equilibrium exponentially

piecewise smoothness local invertibility and parametric analysis of normal maps this paper is concerned with properties of the euclidean projection map onto a convex set defined by finitely many smooth convex inequalities and affine equalities under a constant rank constraint qualification we show that the projection map is piecewise smooth pc1 hence bouligand differentiable or directionally differentiable and a relatively simple formula is given for the b derivative these properties of the projection map are used to obtain inverse and implicit function theorems for associated normal maps using a new characterization of invertibility of a pc1 function in terms of its b derivative an extension of the implicit function theorem which does not require local uniqueness is also presented degree theory plays a major role in the analysis of both the locally unique case and its extension

a nonfeasible gradient projection recurrent neural network for equality constrained optimization problems in this paper a recurrent neural network for both convex and nonconvex equality constrained optimization problems is proposed which makes use of a cost gradient projection onto the tangent space of the constraints the proposed neural network constructs a generically nonfeasible trajectory satisfying the constraints only as t rarr infin local convergence results are given that do not assume convexity of the optimization problem to be solved global convergence results are established for convex optimization problems an exponential convergence rate is shown to hold both for the convex case and the nonconvex case numerical results indicate that the proposed method is efficient and accurate

on the global stability of delayed neural networks lyapunov functional methods combining with some inequality techniques are employed to study the global asymptotic stability of delayed neural networks without assuming lipschitz conditions on the activation functions a new sufficient condition is established such criteria allows us to include non lipschitzian activation functions in the design of delayed neural networks the result presented here is also discussed from the point of view of its relationship to some previous results

solving generally constrained generalized linear variational inequalities using the general projection neural networks generalized linear variational inequality glvi is an extension of the canonical linear variational inequality in recent years a recurrent neural network nn called general projection neural network gpnn was developed for solving glvis with simple bound often box type or sphere type constraints the aim of this paper is twofold first some further stability results of the gpnn are presented second the gpnn is extended for solving glvis with general linear equality and inequality constraints a new design methodology for the gpnn is then proposed furthermore in view of different types of constraints approaches for reducing the number of neurons of the gpnn are discussed which results in two specific gpnns moreover some distinct properties of the resulting gpnns are also explored based on their particular structures numerical simulation results are provided to validate the results

a delayed projection neural network for solving linear variational inequalities in this paper a delayed projection neural network is proposed for solving a class of linear variational inequality problems the theoretical analysis shows that the proposed neural network is globally exponentially stable under different conditions by the proposed linear matrix inequality lmi method the monotonicity assumption on the linear variational inequality is no longer necessary by employing lagrange multipliers the proposed method can resolve the constrained quadratic programming problems finally simulation examples are given to demonstrate the satisfactory performance of the proposed neural network

feasible descent algorithms for mixed complementarity problems in this paper we consider a general algorithmic framework for solving nonlinear mixed complementarity problems the main features of this framework are a it is well defined for an arbitrary mixed complementarity problem b it generates only feasible iterates c it has a strong global convergence theory and d it is locally fast convergent under standard regularity assumptions this framework is applied to the path solver in order to show viability of the approach numerical results for an appropriate modification of the path solver indicate that this framework leads to substantial computational improvements

the bounded smooth reformulation and a trust region algorithm for semidefinite complementarity problems in this paper we present a new merit function for the semidefinite complementarity problem sdcp by extending the bounded smooth reformulation for variational inequality problems we prove that the merit function has a bounded level set and any stationary point of it is a global minimizer without the assumption of monotonicity moreover we present a trust region algorithm for solving the minimization problem with semidefinite constraints the trust region subproblem is solved by the truncated conjugate gradient method and the global convergence is established even without requiring the existence of an accumulation point of the generated sequence

design of recurrent neural networks for solving constrained least absolute deviation problems recurrent neural networks for solving constrained least absolute deviation lad problems or l1 norm optimization problems have attracted much interest in recent years but so far most neural networks can only deal with some special linear constraints efficiently in this paper two neural networks are proposed for solving lad problems with various linear constraints including equality two sided inequality and bound constraints when tailored to solve some special cases of lad problems in which not all types of constraints are present the two networks can yield simpler architectures than most existing ones in the literature in particular for solving problems with both equality and one sided inequality constraints another network is invented all of the networks proposed in this paper are rigorously shown to be capable of solving the corresponding problems the different networks designed for solving the same types of problems possess the same structural complexity which is due to the fact these architectures share the same computing blocks and only differ in connections between some blocks by this means some flexibility for circuits realization is provided numerical simulations are carried out to illustrate the theoretical results and compare the convergence rates of the networks

an improved goldstein s type method for a class of variant variational inequalities this paper aims at presenting an improved goldstein s type method for a class of variant variational inequalities in particular the iterate computed by an existing goldstein s type method he a goldstein s type projection method for a class of variant variational inequalities j comput math 17 4 1999 425 434 is used to construct a descent direction and thus the new method generates the new iterate by searching the optimal step size along the descent direction some restrictions on the involving functions of the existing goldstein s type methods are relaxed while the global convergence of the new method is proved without additional assumptions the computational superiority of the new method is verified by the comparison to some existing methods

sensitivity analysis of composite piecewise smooth equations this paper is a contribution to the sensitivity analysis of piecewise smooth equations a piecewise smooth function is a lipschitzian homeomorphism near a given point if and only if it is coherently oriented and has an invertible b derivative at this point we emphasise the role of functions of the typef g h whereg is piecewise smooth andh is smooth and present verifiable conditions which ensure that the functionf g tilde h is a lipschitzian homeomorphism near a given point for every tilde h sufficiently close toh with respect to thec1 topology

smooth approximations to nonlinear complementarity problems it is well known that a nonlinear complementarity problem ncp can be formulated as a system of nonsmooth equations chen and mangasarian comput optim appl 5 1996 pp 97 138 proposed a class of parametric smooth functions by twice integrating a probability density function as a result the nonsmooth equations can be approximated by smooth equations this paper refines the smooth functions proposed by chen and mangasarian and investigates their structural properties the refinement allows us to establish the existence uniqueness and limiting properties of the trajectory defined by the solutions of these smooth equation approximations in addition global error bounds for the ncp with a uniform p function are obtained

summation of asymptotic expansions of multiple valued functions using algebraic approximants application to anharmonic oscillators the divergent rayleigh schrodinger perturbation expansions for energy eigenvalues of cubic quartic sextic and octic oscillators are summed using algebraic approximants these approximants are generalized pade approximants that are obtained from an algebraic equation of arbitrary degree numerical results indicate that given enough terms in the asymptotic expansion the rate of convergence of the diagonal staircase approximant sequence increases with the degree different branches of the approximants converge to different branches of the function the success of the high degree approximants is attributed to their ability to model the function on multiple sheets of the riemann surface and to reproduce the correct singularity structure in the limit of large perturbation parameter an efficient recursive algorithm for computing the diagonal approximant sequence is presented

numerical study of bifurcations by analytic continuation of a function defined by a power series a novel computational approach to the investigation of bifurcations relying on the use of power series in the bifurcation parameter for a particular solution branch is presented the first part of the paper is devoted to the description of a series summation technique based on the assumption that the given series is the local representation of a function algebraic in the independent variable the procedure leads to a special type of hermite pade approximant although no mathematical analysis is presented the numerical evidence suggests that the error decays faster than exponentially with the number of terms of the series used the procedure s chief merit is its ability to reveal solution branches of the underlying problem in addition to the one represented by the original series in the final part of the paper an algorithm is described for numerically generating the required power series where standard perturbation methods are inadequate thus it is shown how path following techniques may be combined

global exponential stability of neural networks with globally lipschitz continuous activations and its application to linear variational inequality problem this paper investigates the existence uniqueness and global exponential stability ges of the equilibrium point for a large class of neural networks with globally lipschitz continuous activations including the widely used sigmoidal activations and the piecewise linear activations the provided sufficient condition for ges is mild and some conditions easily examined in practice are also presented the ges of neural networks in the case of locally lipschitz continuous activations is also obtained under an appropriate condition the analysis results given in the paper extend substantially the existing relevant stability results in the literature and therefore expand significantly the application range of neural networks in solving optimization problems as a demonstration we apply the obtained analysis results to the design of a recurrent neural network rnn for solving the linear variational inequality problem vip defined on any nonempty and closed box set which includes the box constrained quadratic programming and the linear complementarity problem as the special cases it can be inferred that the linear vip has a unique solution for the class of lyapunov diagonally stable matrices and that the synthesized rnn is globally exponentially convergent to the unique solution some illustrative simulation examples are also given

exponential rank of c algebras with real rank zero and the brown pedersen conjectures abstract we show that every c algebra with real rank zero has exponential rank 1 consequently c algebras with real rank zero have the property weak fu we also show that if a is a unital c algebra with real rank zero stable rank one and trivial k1 group then its multiplier algebra has real rank zero if a is a unital stable c algebra with stable rank one we show that its multiplier algebra has real rank zero if and only if a has real rank zero and k1 a 0

degree theory for generalized variational inequalities and applications in this paper a degree theory for finite dimensional generalized variational inequalities is built and employed to prove some results on solution existence and solution stability

some developments in general variational inequalities general variational inequalities provide us with a unified natural novel and simple framework to study a wide class of equilibrium problems arising in pure and applied sciences in this paper we present a number of new and known numerical techniques for solving general variational inequalities using various techniques including projection wiener hopf equations updating the solution auxiliary principle inertial proximal penalty function dynamical system and well posedness we also consider the local and global uniqueness of the solution and sensitivity analysis of the general variational inequalities as well as the finite convergence of the projection type algorithms our proofs of convergence are very simple as compared with other methods our results present a significant improvement of previously known methods for solving variational inequalities and related optimization problems since the general variational inequalities include quasi variational inequalities and quasi implicit complementarity problems as special cases results presented here continue to hold for these problems several open problems have been suggested for further research in these areas

a new regression estimator with neural network realization a new regression estimator viewed as the solution of a strictly convex quadratic programming problem is introduced in this paper two recurrent neural networks in continuous time and discrete time respectively are proposed to solve the quadratic programming problem in real time the continuous time neural network is shown to have a global stability including the global asymptotic and exponential stability the discrete time neural network is shown to have a global convergence with a fixed step length this fixed step length can be independent of the regression problem size by scaling a design parameter since the sizes of the proposed neural networks depend only on the constraints of the optimization problems the proposed new regression estimator can obtained by two novel neural networks with lower implementation costs than the conventional methods our simulation results confirm that the proposed neural networks are effective in solving various kind of regression problems

exceptional families and finite dimensional variational inequalities over polyhedral convex sets a new concept of exceptional family for nonlinear variational inequalities over polyhedral sets is introduced in this paper it generalizes the concepts for complementarity problem introduced by smith and isac we applied the new analytical tool to the study of existence problem for variational inequalities it is shown that our existence condition is weaker than most of the sufficient conditions which have been known

day to day dynamic network disequilibria and idealized traveler information systems in this paper we present tatonnement models for calculating static wardropian user equilibria on congested networks with fully general demand and cost structures we present both a qualitative analysis of stability and numerical studies which show that such an approach provides a reliable means for determining static user equilibria we also describe circumstances for which these models depict day to day adjustments from one realizable disequilibrium state to another and how these adjustment processes differ depending on the quality of the information being provided by abstract traveler information systems specifically we demonstrate that such dynamic adjustment processes settle down to equilibria both when information is complete and when it is incomplete

on the classification of nuclear c algebras two of the most influential works on c algebras from the mid seventies brown douglas and fillmore s 6 and elliott s 21 both contain uniqueness and existence results in the now standard sense which we shall outline below these papers served as keystones for two separate theories kk theory and the classification program which for many years parted ways with only moderate interaction but with this common origin in mind it is not surprising that recent years have seen a fruitful interaction which has been one of the main engines behind rapid progress in the classification program in the present paper we take this interaction even further combining a concept of quasidiagonality for representations and a new characterization of equivalence in the cuntz picture of kk theory we achieve a general uniqueness result and from the quasidiagonality notion we derive a corresponding general existence result by comparing well known realizations of the kk groups these results are then employed to obtain new classification results for certain classes of quasidiagonal c algebras introduced by h lin an important novel feature of these classes is that they are defined by a certain local approximation property rather than by an inductive limit construction we emphasize that our fundamental uniqueness result does not depend on the universal coefficient theorem uct nor on nuclearity of the c algebras involved on the other hand when we refine the uniqueness result for use in classification we need to gradually add such conditions on the c algebras it is to be expected that requirements of this type are necessary but we have found that holding them back as long as possible leads to more conceptual proofs while clarifying the role of the uct and nuclearity in classification further since uniqueness results answer fundamental questions about kk and k theory they are of interest in themselves we have pursued this theme in 14 and defer to this paper the proof of the new characterization of kk theory which lies behind our uniqueness results our existence and uniqueness results are in the spirit of the classical ext theory from 6 the main complication overcome in the paper is to control the stabilization which is necessary when one works with finite c algebras in the infinite case where programs of this type have already been successfully carried out stabilization is unnecessary yet our methods are sufficiently versatile to allow us to reprove from a handful of basic results the classification of purely infinite nuclear c algebras of kirchberg and phillips

on the solution of the extended linear complementarity problem abstract the extended linear complementarity problem xlcp has been introduced in a recent paper by mangasarian and pang in the present research minimization problems with simple bounds associated to this problem are defined when the xlcp is solvable their solutions are global minimizers of the associated problems sufficient conditions that guarantee that stationary points of the associated problems are solutions of the xlcp will be proved these theoretical results support the conjecture that local methods for box constrained optimization applied to the associated problems could be efficient tools for solving the xlcp

exponential stability and oscillation of hopfield graded response neural network both exponential and stochastic stabilities of the hopfield neural network are analyzed the results are especially useful for analyzing the stabilities of asymmetric neural networks a constraint on the connection matrix has been found under which the neural network has a unique and exponentially stable equilibrium given any connection matrix this constraint can be satisfied through the adjustment of the gains of the amplifiers and the resistances in the neural net circuit a one to one and smooth map between input currents and the equilibria of the neural network can be set up the above results can be applied to the master slave net to prove that the master net can find the best connection matrix for the slave net for the neural network disturbed by some noise the stochastic stability of the network is also analyzed a special asymmetric neural network formed by a closed chain of formal neurons is also studied for its stability and oscillation both stable and oscillatory dynamics are obtained in the closed chain network through the adjustment of the gains and resistances of the amplifiers

a nonsmooth version of newton s method newton s method for solving a nonlinear equation of several variables is extended to a nonsmooth case by using the generalized jacobian instead of the derivative this extension includes the b derivative version of newton s method as a special case convergence theorems are proved under the condition of semismoothness it is shown that the gradient function of the augmented lagrangian forc2 nonlinear programming is semismooth thus the extended newton s method can be used in the augmented lagrangian method for solving nonlinear programs

a truly globally convergent newton type method for the monotone nonlinear complementarity problem the josephy newton method for solving a nonlinear complementarity problem consists of solving possibly inexactly a sequence of linear complementarity problems under appropriate regularity assumptions this method is known to be locally superlinearly convergent to enlarge the domain of convergence of the newton method some globalization strategy based on a chosen merit function is typically used however to ensure global convergence to a solution some additional restrictive assumptions are needed these assumptions imply boundedness of level sets of the merit function and often even global uniqueness of the solution we present a new globalization strategy for monotone problems which is not based on any merit function our linesearch procedure utilizes the regularized newton direction and the monotonicity structure of the problem to force global convergence by means of a computationally explicit projection step which reduces the distance to the solution set of the problem the resulting algorithm is truly globally convergent in the sense that the subproblems are always solvable and the whole sequence of iterates converges to a solution of the problem without any regularity assumptions in fact the solution set can even be unbounded each iteration of the new method has the same order of computational cost as an iteration of the damped newton method under natural assumptions the local superlinear rate of convergence is also achieved

a discrete time neural network for optimization problems with hybrid constraints recurrent neural networks have become a prominent tool for optimizations including linear or nonlinear variational inequalities and programming due to its regular mathematical properties and well defined parallel structure this brief presents a general discrete time recurrent network for linear variational inequalities and related optimization problems with hybrid constraints in contrary to the existing discrete time networks this general model can operate not only on bound constraints but also on hybrid constraints comprised of inequality equality and bound constraints the model has dynamical properties of global convergence asymptotical and exponential convergences under some weaker conditions numerical examples demonstrate its efficacy and performance

inverse variational inequalities with projection based solution methods abstract an inverse variational inequality is defined as to find a vector u r n such that f u v f u t u 0 v if an inverse function u f 1 x exists the above inverse variational inequality could be transformed as a regular variational inequality however in reality it is not uncommon that the inverse function of f 1 x does not have explicit form although its functional values can be observed existing line search algorithms cannot be applied directly to solve such inverse variational inequalities in this paper we propose two projection based methods using the co coercivity of mapping f a self adaptive strategy is developed to determine the step sizes efficiently when the co coercivity modulus is unknown the convergence of the proposed methods is proved rigorously

square lattice ising model susceptibility series expansion method and differential equation for χ 3 in a previous paper 2004 j phys a math gen 37 9651 68 we have given the fuchsian linear differential equation satisfied by 3 the three particle contribution to the susceptibility of the isotropic square lattice ising model this paper gives the details of the calculations with some useful tricks and tools allowing one to obtain long series in polynomial time the method is based on series expansion in the variables that appear in the n 1 dimensional integrals representing the n particle contribution to the isotropic square lattice ising model susceptibility the integration rules are straightforward due to remarkable formulae we derived for these variables we obtain without any numerical approximation 3 as a fully integrated series in the variable w s 2 1 s2 where s sh 2k with k j kt the conventional ising model coupling constant we also give some perspectives and comments on these results

exponential rank and exponential length of operators on hilbert c modules in this article we consider unitary operators on the hilbert c module ha associated with a large class of c algebras let l x denote the algebra consisting of all bounded linear operators on a separable hilbert space 7 and let k denote the algebra of all compact operators on r halmos and kakutani proved that every unitary operator on r is a product of four symmetries cf ha 143 haka for the case when x is a compact hausdorff space ringrose r recently considered the c algebra c x l q consisting of all norm continuous maps from x to l x and proved that every unitary element of c x l x can be written as a product of at most three exponentials in other words a norm continuous map u from x to the unitary group of l x forms a product of three norm continuous maps written as exp ihi exp ih2 exp ih3 where hl h2 and h3 are maps from x to the space of self adjoint operators in l x phillips and ringrose phr conjectured that every unitary of c x l x can even be approximated in the norm by products of two exponentials using techniques in the theory of c algebras but by a different approach we generalized ringrose s result to hilbert c modules associated with all a unital in particular all unital c algebras cf z8 we also proved in z8 similar results for the c algebra consisting of norm continuous maps from x to a purely infinite simple c algebra the purpose of this article is to settle the phillips ringrose conjecture in the affirmative and to prove a generalized version of the halmos kakutani theorem in haka we consider the group of unitary operators on hilbert c modules associated with the following two specific classes which together comprise the majority of interesting c algebras