a krylov subspace method for large scale second order cone linear complementarity problem in this paper we first show that the second order cone linear complementarity problem soclcp can be solved by finding a positive zero s in mathbb r of a particular rational function h s and we then propose a krylov subspace method to reduce h s to h ell s as in the model reduction the zero s of h s can be accurately approximated by that of h ell s 0 which itself can be cast as a small eigenvalue problem the new method is made possible by a complete characterization of the curve of h s and it has several advantages over the bisection newton sf bn iteration recently proposed by l h zhang and w h yang math comp 83 2013 pp 1701 1720 and shown to be very efficient for small to medium size problems the method is tested and compared against the sf bn iteration and two other state of the art packages sf sdpt3 and sf sedumi our numerical results show that the method is very efficient for both small to medium dense problems and large scale ones

a regularized sample average approximation method for stochastic mathematical programs with nonsmooth equality constraints we investigate a class of two stage stochastic programs where the second stage problem is subject to nonsmooth equality constraints parameterized by the first stage variant and a random vector we consider the case when the parametric equality constraints have more than one solution a regularization method is proposed to deal with the multiple solution problem and a sample average approximation method is proposed to solve the regularized problem we then investigate the convergence of stationary points of the regularized sample average approximation programs as the sample size increases the established results are applied to stochastic mathematical programs with p 0 variational inequality constraints preliminary numerical results are reported

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

smoothing technique of nonsmooth newton methods for control state constrained optimal control problems this paper is concerned with the numerical solution of optimal control problems subject to mixed control state constraints with differential algebraic equations the necessary conditions arising from a local minimum principle are transformed into an equivalent nonsmooth equation in appropriate banach spaces we then develop a parametric smoothing newton method for solving this nonsmooth equation the global and local convergence is proposed under suitable settings numerical examples are presented to demonstrate the efficiency of our global smoothing technique

a semi infinite programming algorithm for solving optimal power flow with transient stability constraints this paper proposes a new algorithm for solving a type of complicated optimal power flow opf problems in power systems i e opf problems with transient stability constraints ots the ots is converted into a semi infinite programming sip via some suitable function analysis then based on the kkt system of the reformulated sip a smoothing quasi newton algorithm is presented in which the numerical integration is used the convergence of the algorithm is established an ots problem in power system is tested which shows that the proposed algorithm is promising

a characterization of graphs with rank 5 abstract the rank of a graph g is defined to be the rank of its adjacency matrix in this paper we consider the following problem what is the structure of a connected graph g with rank 5 or equivalently what is the structure of a connected n vertex graph g whose adjacency matrix has nullity n 5 in this paper we completely characterize connected graphs g whose adjacency matrix has rank 5

a smooth penalty based sample average approximation method for stochastic complementarity problems sample average approximation method is one of the effective methods in the stochastic optimization a smooth penalty based sample average approximation method for stochastic nonlinear complementarity problems is presented in this paper based on a smooth penalty function a reformulation is proposed for the equivalent problem of ev formulation for stochastic complementary problems and it is proven that its solutions are existent under some mild assumptions an implementable sample average approximation method for the reformulation is further established and its convergence is analyzed the numerical results for some test examples are reported at last to show efficiency of the proposed method

a new unconstrained differentiable merit function for box constrained variational inequality problems and a damped gauss newton method in this paper we propose a new unconstrained differentiable merit function f for box constrained variational inequality problems vip l u f we study various desirable properties of this new merit function f and propose a gauss newton method in which each step requires only the solution of a system of linear equations global and superlinear convergence results for vip l u f are obtained key results are the boundedness of the level sets of the merit function for any uniform p function and the superlinear convergence of the algorithm without a nondegeneracy assumption numerical experiments confirm the good theoretical properties of the method

on the rate of convergence of optimal solutions of monte carlo approximations of stochastic programs in this paper we discuss monte carlo simulation based approximations of a stochastic programming problem we show that if the corresponding random functions are convex piecewise linear and the distribution is discrete then an optimal solution of the approximating problem provides an exact optimal solution of the true problem with probability one for sufficiently large sample size moreover by using the theory of large deviations we show that the probability of such an event approaches one exponentially fast with increase of the sample size in particular this happens in the case of linear two or multi stage stochastic programming with recourse if the corresponding distributions are discrete the obtained results suggest that in such cases monte carlo simulation based methods could be very efficient we present some numerical examples to illustrate the ideas involved

smoothing projected gradient method and its application to stochastic linear complementarity problems a smoothing projected gradient spg method is proposed for the minimization problem on a closed convex set where the objective function is locally lipschitz continuous but nonconvex nondifferentiable we show that any accumulation point generated by the spg method is a stationary point associated with the smoothing function used in the method which is a clarke stationary point in many applications we apply the spg method to the stochastic linear complementarity problem slcp and image restoration problems we study the stationary point defined by the directional derivative and provide necessary and sufficient conditions for a local minimizer of the expected residual minimization erm formulation of slcp preliminary numerical experiments using the spg method for solving randomly generated slcp and image restoration problems of large sizes show that the spg method is promising

a superlinearly convergent method for a class of complementarity problems with non lipschitzian functions we consider a class of complementarity problems involving functions which are not lipschitz continuous in this paper we reformulate this class of non lipschitzian complementarity problems into a lipschitzian complementarity problem then we propose an inexact smoothing newton method to solve this lipschitzian complementarity problem we prove that our proposed method converges quadratically and globally under a mild condition numerical results show that this method is promising this method can solve these kinds of complementarity problems with one million variables in reasonable time on a pc with 1 gb of ram

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

biorthogonal basis functions in hp adaptive fem for elliptic obstacle problems abstract in this paper the discretization of a non symmetric elliptic obstacle problem with h p adaptive h 1 conforming finite elements is discussed for this purpose a higher order mixed finite element discretization is introduced where the dual space is discretized via biorthogonal basis functions the h p adaptivity is realized via automatic adaptive mesh refinement based on a residual a posteriori error estimation which is also derived in this paper the use of biorthogonal basis functions leads to unilateral box constraints and componentwise complementarity conditions enabling the highly efficient application of a quadratically converging semi smooth newton scheme which can be modified to ensure global convergence h p adaptivity usually implies meshes with hanging nodes and varying polynomial degrees which have to be handled appropriately within the h 1 conforming finite element discretization this is typically done by using so called connectivity matrices in this paper a procedure is proposed which efficiently computes these matrices for biorthogonal basis functions finally the applicability of the theoretical findings is demonstrated with several numerical experiments

a jacobian smoothing method for box constrained variational inequality problems in this paper we propose a jacobian smoothing algorithm for the solution of the box constrained variational inequality problems vip l u f the algorithm is based on a reformulation of a semismooth system of equations by using the fischer burmeister function h a b a 2 b 2 a b global and local superlinear convergence results for vip l u f are obtained numerical experiments confirm the good theoretical properties of the algorithm

a new smoothing newton method for solving constrained nonlinear equations in this paper a new smoothing newton method is proposed for solving constrained nonlinear equations we first transform the constrained nonlinear equations to a system of semismooth equations by using the so called absolute value function of the slack variables and then present a new smoothing newton method for solving the semismooth equations by constructing a new smoothing approximation function this new method is globally and quadratically convergent it needs to solve only one system of unconstrained equations and to perform one line search at each iteration numerical results show that the new algorithm works quite well

the quadratic convergence of a smoothing levenberg marquardt method for nonlinear complementarity problem abstract the nonlinear complementarity problem denoted by ncp f can be reformulated as the solution of a possibly inconsistent nonsmooth system of equations based on the ideas developed in smoothing newton methods we approximated the problem of the least l2 norm solution of the equivalent nonsmooth equations of ncp f with a family of parameterized optimization problem with twice continuously differentiable objective functions by making use of a new smoothing function then we presented a smoothing levenberg marquardt method to solve the parameterized smooth optimization problem by using the smooth and semismooth technique the local quadratic convergence of the proposed method is proved under some suitable assumptions

an efficient matrix splitting method for the second order cone complementarity problem given a symmetric and positive semi definite n by n matrix m and a vector in this paper we consider the matrix splitting method for solving the second order cone linear complementarity problem soclcp the matrix splitting method is among the most widely used approaches for large scale and sparse classical linear complementarity problems and its linear convergence is proved by luo and tseng siam j control optim 30 1992 pp 408 425 our first contribution is to prove that when the general matrix splitting algorithm is applied to soclcp with m symmetric and positive definite it also converges at least linearly numerically our second contribution is to propose a special and efficient matrix splitting algorithm the block successive overrelaxation method for solving the soclcp the algorithm makes good use of the underlying geometry of the soclcp and each iteration only involves solving triangular linear systems and requires o n 2 flops moreover the algorithm does not destroy

local convergence of quasi newton methods for b differentiable equations we study local convergence of quasi newton methods for solving systems of nonlinear equations defined by b differentiable functions we extend the classical linear and superlinear convergence results for general quasi newton methods as well as for broyden s method we also show how broyden s method may be applied to nonlinear complementarity problems and illustrate its computational performance on two small examples

stability analysis of two stage stochastic mathematical programs with complementarity constraints via nlp regularization this paper presents numerical approximation schemes for a two stage stochastic programming problem where the second stage problem has a general nonlinear complementarity constraint first the complementarity constraint is approximated by a parameterized system of inequalities with a well known regularization approach by scholtes siam j optim 11 2001 pp 918 936 in deterministic mathematical programs with equilibrium constraints the distribution of the random variables of the regularized two stage stochastic program is then approximated by a sequence of probability measures by treating the approximation problems as a perturbation of the original true problem we carry out a detailed stability analysis of the approximated problems including continuity and local lipschitz continuity of optimal value functions and outer semicontinuity and continuity of the set of optimal solutions and stationary points a particular focus is given to the case where the probability distribution is approximated by

a smoothing newton method for minimizing a sum of euclidean norms we consider the problem of minimizing a sum of euclidean norms f x sum i 1 m b i a i tx this problem is a nonsmooth problem because f is not differentiable at a point x when one of the norms is zero in this paper we present a smoothing newton method for this problem by applying the smoothing newton method proposed by qi sun and zhou math programming 87 2000 pp 1 35 directly to a system of strongly semismooth equations derived from primal and dual feasibility and a complementarity condition this method is globally and quadratically convergent as applications to this problem smoothing newton methods are presented for the euclidean facilities location problem and the steiner minimal tree problem under a given topology preliminary numerical results indicate that this method is extremely promising

smoothing methods for nonsmooth nonconvex minimization we consider a class of smoothing methods for minimization problems where the feasible set is convex but the objective function is not convex not differentiable and perhaps not even locally lipschitz at the solutions such optimization problems arise from wide applications including image restoration signal reconstruction variable selection optimal control stochastic equilibrium and spherical approximations in this paper we focus on smoothing methods for solving such optimization problems which use the structure of the minimization problems and composition of smoothing functions for the plus function x many existing optimization algorithms and codes can be used in the inner iteration of the smoothing methods we present properties of the smoothing functions and the gradient consistency of subdifferential associated with a smoothing function moreover we describe how to update the smoothing parameter in the outer iteration of the smoothing methods to guarantee convergence of the smoothing methods to a stationary point of the original minimization problem

bifurcation scenarios for bubbling transition dynamical systems with chaos on an invariant submanifold can exhibit a type of behavior called bubbling whereby a small random or fixed perturbation to the system induces intermittent bursting the bifurcation to bubbling occurs when a periodic orbit embedded in the chaotic attractor in the invariant manifold becomes unstable to perturbations transverse to the invariant manifold generically the periodic orbit can become transversely unstable through a pitchfork transcritical period doubling or hopf bifurcation in this paper a unified treatment of the four types of bubbling bifurcation is presented conditions are obtained determining whether the transition to bubbling is soft or hard that is whether the maximum burst amplitude varies continuously or discontinuously with variation of the parameter through its critical value for soft bubbling transitions the scaling of the maximum burst amplitude with the parameter is derived for both hard and soft transitions the scaling of the average interburst time with the bifurcation parameter is deduced both random noise and fixed mismatch perturbations are considered results of numerical experiments testing our theoretical predictions are presented

convergence analysis of sample average approximation methods for a class of stochastic mathematical programs with equality constraints in this paper we discuss the sample average approximation saa method for a class of stochastic programs with nonsmooth equality constraints we derive a uniform strong law of large numbers for random compact set valued mappings and use it to investigate the convergence of karush kuhn tucker points of saa programs as the sample size increases we also study the exponential convergence of global minimizers of the saa problems to their counterparts of the true problem the convergence analysis is extended to a smoothed saa program finally we apply the established results to a class of stochastic mathematical programs with complementarity constraints and report some preliminary numerical test results

on almost smooth functions and piecewise smooth functions abstract piecewise smooth ps functions are perhaps the best known examples of semismooth functions which play key roles in the solution of nonsmooth equations and nonsmooth optimization recently there have emerged other examples of semismooth functions including the p norm function 1 p defined on r n with n 2 ncp functions smoothing penalty functions and integral functions these semismooth functions share the special property that their smooth point sets are locally connected around their nonsmooth points by extending a result of rockafellar we show that the smooth point set of a ps function cannot have such a property this shows that the above functions though semismooth are not ps we call such functions almost smooth as we show that the b subdifferential of an as function at a point has either one or infinitely many elements which contrasts with ps functions whose b subdifferential at a point has only a finite number of elements we derive other useful properties of as functions and sufficient conditions for a function to be as these results are then applied to various smoothing penalty functions and integral functions

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 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

a combined smoothing and regularization method for monotone second order cone complementarity problems the second order cone complementarity problem soccp is a wide class of problems containing the nonlinear complementarity problem ncp and the second order cone programming problem socp recently fukushima luo and tseng siam j optim 12 2001 pp 436 460 extended some merit functions and their smoothing functions for ncp to soccp moreover they derived computable formulas for the jacobians of the smoothing functions and gave conditions for the jacobians to be invertible in this paper we propose a globally and quadratically convergent algorithm which is based on smoothing and regularization methods for solving monotone soccp in particular we study strong semismoothness and jacobian consistency which play an important role in establishing quadratic convergence of the algorithm furthermore we examine the effectiveness of the algorithm by means of numerical experiments

on the characterizing properties of the permanental polynomials of graphs abstract let g be a graph and a g the adjacency matrix of g the permanental polynomial of g is defined as g x per x i a g if two graphs g and h have the same permanental polynomial then g is called a per cospectral mate of h a graph g is said to be characterized by its permanental polynomial if all the per cospectral mates of g are isomorphic to g it is shown that complete graphs stars regular complete bipartite graphs and odd cycles are characterized by their permanental polynomials we prove that in general the permanental polynomial cannot characterize the paths and even cycles in particular for each l 1 and m 2 we can find non isomorphic per cospectral mates of p 4 l 3 and c 4 m respectively when we restrict our consideration to connected graphs both the paths and even cycles c 4 l 2 are characterized by their permanental polynomials

improved smoothing type methods for the solution of linear programs we consider a smoothing type method for the solution of linear programs its main idea is to reformulate the primal dual optimality conditions as a nonlinear and nonsmooth system of equations and to apply a newton type method to a smooth approximation of this nonsmooth system the method presented here is a predictor corrector method and is closely related to some methods recently proposed by burke and xu on the one hand and by the authors on the other hand however here we state stronger global and or local convergence properties moreover we present quite promising numerical results for the whole netlib test problem collection

a continuation method for strongly monotone variational inequalities we consider the variational inequality problem denoted by vip x f wheref is a strongly monotone function and the convex setx is described by some inequality and possibly equality constraints this problem is solved by a continuation or interior point method which solves a sequence of certain perturbed variational inequality problems these perturbed problems depend on a parameter 0 it is shown that the perturbed problems have a unique solution for all values of 0 and that any sequence generated by the continuation method converges to the unique solution of vip x f under a well known linear independence constraint qualification licq we also discuss the extension of the continuation method to monotone variational inequalities and present some numerical results obtained with a suitable implementation of this method 1998 the mathematical programming society inc published by elsevier science b v

permanental polynomials of graphs abstract the first section surveys recent results on the permanental polynomial of a square matrix a i e per xi a the second section concerns the permanental polynomial of the adjacency matrix of a graph the final section is an introduction to the permanental polynomial of the laplacian matrix of a graph an appendix lists some of these latter polynomials

law of large numbers for random sets and allocation processes in this paper we establish a strong law of large numbers for unbounded random sets and then apply it to an optimization problem arising in allocation processes under uncertainty

analysis of sample path optimization sample path optimization is a method for optimizing limit functions occurring in stochastic modeling problems such as steady state functions in discrete event dynamic systems it is closely related to retrospective optimization techniques and to m estimation the method has been computationally tested elsewhere on problems arising in production and in project planning with apparent success in this paper we provide a mathematical justification for sample path optimization by showing that under certain assumptions which hold for the problems just mentioned the method will almost surely find a point that is in a specified sense sufficiently close to the set of optimizers of the limit function

regular pseudo smooth ncp and bvip functions and globally and quadratically convergent generalized newton methods for complementarity and variational inequality problems the nonlinear complementarity problem ncp can be reformulated as a system of semismooth equations by some ncp functions a well known ncp function is the fischer burmeister function which is a strongly semismooth function it is smooth everywhere except at the origin the generalized newton direction of the system of semismooth equations formulated with the fischer burmeister function is always a descent direction at a nonsolution point the generalized jacobian of the system is nonsingular under mild conditions efficient algorithms have been developed based upon these nice properties in this paper we define a class of ncp functions called regular pseudo smooth ncp functions and show that they have these nice properties regular pseudo smooth ncp functions can be easily identified they include the fischer burmeister function the tseng luo ncp function family and the kanzow kleinmichel ncp function family we give two new regular pseudo smooth ncp function families the ratio generated ncp function family and the c curve generated ncp function family we then discuss the box constrained variational inequality problem bvip we define a class of bvip functions called regular pseudo smooth bvip functions and show that they have these nice properties too we present three different approaches to generate regular pseudo smooth bvip functions from regular pseudo smooth ncp functions globally and quadratically convergent generalized newton methods are established for solving the ncp and the bvip based upon regular pseudo smooth ncp and bvip functions

superlinear convergence of smoothing quasi newton methods for nonsmooth equations abstract we study local convergence of smoothing quasi newton methods for solving a system of nonsmooth nondifferentiable equations in r n the feature of smoothing quasi newton methods is to use a smooth function to approximate the nonsmooth mapping and update the quasi newton matrix at each step convergence results are given under directional derivative consistence property without differentiability we establish a dennis more type superlinear convergence theorem for smoothing quasi newton methods and we prove linear convergence of the smoothing broyden method furthermore we propose a superlinear convergent smoothing newton broyden method without using the generalized jacobian and the semismooth assumption we illustrate the smoothing approach on box constrained variational inequalities

convergence analysis of some algorithms for solving nonsmooth equations this paper presents convergence analysis of some algorithms for solving systems of nonlinear equations defined by locally lipschitzian functions for the directional derivative based and the generalized jacobian based newton methods both the iterates and the corresponding function values are locally superlinearly convergent globally a limiting point of the iterate sequence generated by the damped directional derivative based newton method is a zero of the system if and only if the iterate sequence converges to this point and the stepsize eventually becomes one provided that the system is strongly bd regular and semismooth at this point in this case the convergence is superlinear a general attraction theorem is presented which can be applied to two algorithms proposed by han pang and rangaraj a hybrid method which is both globally convergent in the sense of finding a stationary point of the norm function of the system and locally quadratically convergent is also presented

estimation of derivatives of nonsmooth performance measures in regenerative systems we investigate the problem of estimating derivatives of expected steady state performance measures in parametric systems unlike most of the existing work in the area we allow those functions to benonsmooth and study the estimation ofdirectional derivatives for the class of regenerative markovian systems we provide conditions under which we can obtainconsistent estimators of those directional derivatives an example illustrates that the conditions imposed must be different from those in the differentiable case the result also allows us to derive necessary and sufficient conditions for differentiability of the expected steady state function we then analyze the process formed by thesubdifferentials of the original process and show that the subdifferential set of the expected steady state function can be expressed as an average of integrals of multifunctions which is the approach commonly found in the literature for integrals of sets the latter result can also be viewed as a limit theorem for more general compact convex multivalued processes

a globally convergent successive approximation method for severely nonsmooth equations this paper presents a globally convergent successive approximation method for solving f x 0 where f is a continuous function at each step of the method f is approximated by a smooth function f k with pa f k f pa rightarrow 0 as k rightarrow infty the direction f k x k 1 f x k is then used in a line search on a sum of squares objective the approximate function f k can be constructed for nonsmooth equations arising from variational inequalities maximal monotone operator problems nonlinear complementarity problems and nonsmooth partial differential equations numerical examples are given to illustrate the method

convergence of stationary points of sample average two stage stochastic programs a generalized equation approach this paper presents an asymptotic analysis of a monte carlo method variously known as sample average approximation saa or sample path optimization spo for a general two stage stochastic minimization problem we study the case when the second stage problem may have multiple local optima or stationary points that are not global solutions and saa is implemented using a general nonlinear programming solver that is only guaranteed to find stationary points new optimality conditions are developed for both the true problem and its saa problem to accommodate karush kuhn tucker points because the optimality conditions are essentially stochastic generalized equations the asymptotic analysis is carried out for the generalized equations first and then applied to optimality conditions for this purpose we analyze piecewise continuous pc0 stochastic mappings to understand when their expectations are piecewise continuous and thereby derive exponential convergence of saa it is shown under moderate conditions that with probability one an accumulation point of the saa stationary points satisfies a relaxed stationary condition for the true problem and further that with probability approaching one exponentially fast with increasing sample size a stationary point of saa converges to the set of relaxed stationary points these results strengthen or complement existing results where the second stage problem is often assumed to have a unique solution and the exponential convergence is focused on how fast a solution of the true problem becomes an approximate solution of an saa problem rather than the other way around

per spectral characterizations of graphs with extremal per nullity abstract a graph g is said to be determined by its permanental spectrum if any graph having the same permanental spectrum as g is isomorphic to g in this paper we introduce the permanental nullity of a graph the multiplicity of the number zero in the permanental spectrum of a graph to study graphs determined by their permanental spectra first we determine all graphs of order n whose permanental nullities are n 2 n 3 n 4 and n 5 respectively then we show that all graphs with the permanental nullity n 2 n 3 or n 5 and all non bipartite graphs with the permanental nullity n 4 are determined by their permanental spectra in particular we prove that the complete bipartite graphs are determined by their permanental spectra

a regularized smoothing method for fully parameterized convex problems with applications to convex and nonconvex two stage stochastic programming we present an approach to regularize and approximate solution mappings of parametric convex optimization problems that combines interior penalty log barrier solutions with tikhonov regularization because the regularized mappings are single valued and smooth under reasonable conditions they can be used to build a computationally practical smoothing for the associated optimal value function the value function in question while resulting from parameterized convex problems need not be convex one motivating application of interest is two stage possibly nonconvex stochastic programming we show that our approach being computationally implementable provides locally bounded upper bounds for the subdifferential of the value function of qualified convex problems as a by product of our development we also recover that in the given setting the value function is locally lipschitz continuous numerical experiments are presented for two stage convex stochastic programming problems comparing the approach with the bundle method for nonsmooth optimization

a primal dual algorithm for minimizing a sum of euclidean norms we study the problem of minimizing a sum of euclidean norms this nonsmooth optimization problem arises in many different kinds of modern scientific applications in this paper we first transform this problem and its dual problem into a system of strongly semismooth equations and give some uniqueness theorems for this problem we then present a primal dual algorithm for this problem by solving this system of strongly semismooth equations preliminary numerical results are reported which show that this primal dual algorithm is very promising

a newton like method for nonsmooth variational inequalities abstract we present new results for the local convergence of the newton like method to a unique solution of nondifferentiable variational inclusions in a banach space setting using the lipschitz like property of set valued mappings and the concept of slant differentiability hypothesis on the operator involved as was introduced by x chen z nashed and l qi the linear convergence of the newton like method is also established our results extend the applicability of the newton like method argyros and hilout 2009 5 and chen nashed and qi 2000 7 to variational inclusions

mechanisms for the hard bubbling transition in symmetrically coupled chaotic systems we investigate mechanisms for the hard bubbling transition in symmetrically coupled one dimensional maps a transition from strong to weak synchronization occurs via a first period doubling or pitchfork transverse bifurcation of a periodic saddle embedded in the synchronous chaotic attractor the consequence of such transverse bifurcations depends on the existence of an absorbing area controlling the global dynamics in the presence of an absorbing area a subcritical transverse bifurcation is found to give rise to abrupt appearance of transient intermittent bursts with large amplitude for this case any small parameter mismatch that is unavoidable in real systems induces a hard bubbling transition via an interior crisis through a detailed numerical analysis we present explicit examples for the subcritical period doubling and pitchfork transverse bifurcations leading to hard bubbling

analysis of nonsmooth symmetric matrix valued functions with applications to semidefinite complementarity problems for any function f from mathbb r to mathbb r one can define a corresponding function on the space of n n block diagonal real symmetric matrices by applying f to the eigenvalues of the spectral decomposition we show that this matrix valued function inherits from f the properties of continuity local lipschitz continuity directional differentiability frechet differentiability continuous differentiability as well as rho order semismoothness our analysis uses results from nonsmooth analysis as well as perturbation theory for the spectral decomposition of symmetric matrices we also apply our results to the semidefinite complementarity problem addressing some basic issues in the analysis of smoothing semismooth newton methods for solving this problem

a new smoothing quasi newton method for nonlinear complementarity problems the nonlinear complementarity problem can be reformulated as a nonsmooth equation in this paper we propose a new smoothing quasi newton method for the solution of the nonlinear complementarity problems by constructing a new smoothing approximation function global and local superlinear convergence results of the proposed algorithm are established under suitable conditions some numerical results are reported in the paper

modified newton methods for solving a semismooth reformulation of monotone complementarity problems in this paper we propose a newton type method for solving a semismooth reformulation of monotone complementarity problems in this method a direction finding subproblem which is a system of linear equations is uniquely solvable at each iteration moreover the obtained search direction always affords a direction of sufficient decrease for the merit function defined as the squared residual for the semismooth equation equivalent to the complementarity problem we show that the algorithm is globally convergent under some mild assumptions next by slightly modifying the direction finding problem we propose another newton type method which may be considered a restricted version of the first algorithm we show that this algorithm has a superlinear or possibly quadratic rate of convergence under suitable assumptions finally some numerical results are presented

entropic approximation for mathematical programs with robust equilibrium constraints in this paper we consider a class of mathematical programs with robust equilibrium constraints represented by a system of semi infinite complementarity constraints sicc we propose a numerical scheme for tackling sicc specifically by relaxing the complementarity constraints and then randomizing the index set of sicc we employ the well known entropic risk measure to approximate the semi infinite constraints with a finite number of stochastic inequality constraints under some moderate conditions we quantify the approximation in terms of the feasible set and the optimal value the approximation scheme is then applied to a class of two stage stochastic mathematical programs with complementarity constraints in combination with the polynomial decision rules finally we extend the discussion to a mathematical program with distributionally robust equilibrium constraints which is essentially a one stage stochastic program with semi infinite stochastic constraints indexed by some probability measures from

smoothing methods and semismooth methods for nondifferentiable operator equations we consider superlinearly convergent analogues of newton methods for nondifferentiable operator equations in function spaces the superlinear convergence analysis of semismooth methods for nondifferentiable equations described by a locally lipschitzian operator in rn is based on rademacher s theorem which does not hold in function spaces we introduce a concept of slant differentiability and use it to study superlinear convergence of smoothing methods and semismooth methods in a unified framework we show that a function is slantly differentiable at a point if and only if it is lipschitz continuous at that point an application to the dirichlet problems for a simple class of nonsmooth elliptic partial differential equations is discussed

spatial patterns of desynchronization bursts in networks we adapt a previous model and analysis method the master stability function extensively used for studying the stability of the synchronous state of networks of identical chaotic oscillators to the case of oscillators that are similar but not exactly identical we find that bubbling induced desynchronization bursts occur for some parameter values these bursts have spatial patterns which can be predicted from the network connectivity matrix and the unstable periodic orbits embedded in the attractor we test the analysis of bursts by comparison with numerical experiments in the case that no bursting occurs we discuss the deviations from the exactly synchronous state caused by the mismatch between oscillators

a new filter levenberg marquardt method with disturbance for solving nonlinear complementarity problems recently filter methods are extensively studied to handle nonlinear programming problems because of good numerical results filter techniques are attached importance to the nonlinear complementarity problem can be reformulated as the least l 2 norm solution of an optimization problem in this paper basing on the filter technique and the new smoothing function we present a new filter levenberg marquardt method for solving the equation system j z k m z k l ki dz k b kz with the disturbance b kz under the assumption that the lever set of the problem is compact we prove its global convergence

a globally convergent levenberg marquardt method for the least l2 norm solution of nonlinear inequalities abstract the least l 2 norm solution for a possibly inconsistent system of nonlinear inequalities is studied in this paper by introducing a new smoothing function the problem is approximated by a family of parameterized optimization problems with twice continuously differentiable objective functions then a smoothing levenberg marquardt method is applied to solve the parameterized optimization problems the global convergence of the proposed method is established under suitable assumptions

a globally convergent levenberg marquardt method for solving nonlinear complementarity problem the nonlinear complementarity problem denoted by ncp f can be reformulated as the least l2 norm solution of a optimization problem by introducing a new smoothing function the problem is approximated by a family of parameterized optimization problems with twice continuously differentiable objective functions then a smoothing levenberg marquardt method is applied to solve the parameterized optimization problems the global convergence of the proposed method is proved under an assumption that the level set of the problem is compact 2007 elsevier inc all rights reserved

a new smoothing quasi newton method for nonlinear complementarity problems a new smoothing quasi newton method for nonlinear complementarity problems is presented the method is a generalization of thomas method for smooth nonlinear systems and has similar properties as broyden s method local convergence is analyzed for a strictly complementary solution as well as for a degenerate solution presented numerical results demonstrate quite similar behavior of thomas and broyden s methods

semi smooth newton methods for nonlinear complementarity formulation of compositional two phase flow in porous media abstract simulating compositional multiphase flow in porous media is a challenging task especially when phase transition is taken into account the main problem with phase transition stems from the inconsistency of the primary variables such as phase pressure and phase saturation i e they become ill defined when a phase appears or disappears recently a new approach for handling phase transition has been developed whereby the system is formulated as a nonlinear complementarity problem ncp unlike the widely used primary variable switching pvs method which requires a drastic reduction of the time step size when a phase appears or disappears this approach is more robust and allows for larger time steps one way to solve an ncp system is to reformulate the inequality constraints as a non smooth equation using a complementary function c function because of the non smoothness of the constraint equations a semi smooth newton method needs to be developed in this work we consider two methods for solving ncp systems used to model multiphase flow 1 a semi smooth newton method for two c functions the minimum and the fischer burmeister functions and 2 a new inexact newton method based on the jacobian smoothing method for a smooth version of the fischer burmeister function we show that the new method is robust and efficient for standard benchmark problems as well as for realistic examples with highly heterogeneous media such as the spe10 benchmark

convergence theory for nonconvex stochastic programming with an application to mixed logit monte carlo methods have extensively been used and studied in the area of stochastic programming their convergence properties typically consider global minimizers or first order critical points of the sample average approximation saa problems and minimizers of the true problem and show that the former converge to the latter for increasing sample size however the assumption of global minimization essentially restricts the scope of these results to convex problems we review and extend these results in two directions we allow for local saa minimizers of possibly nonconvex problems and prove under suitable conditions almost sure convergence of local second order solutions of the saa problem to second order critical points of the true problem we also apply this new theory to the estimation of mixed logit models for discrete choice analysis new useful convergence properties are derived in this context both for the constrained and unconstrained cases and associated estimates of the simulation bias and variance are proposed

newton and quasi newton methods for a class of nonsmooth equations and related problems the paper presents concrete realizations of quasi newton methods for solving several standard problems including complementarity problems special variational inequality problems and the karush kuhn tucker kkt system of nonlinear programming a new approximation idea is introduced in this paper the q superlinear convergence of the newton method and the quasi newton method are established under suitable assumptions in which the existence of f x is not assumed the new algorithms only need to solve a linear equation in each step for complementarity problems the qr factorization on the quasi newton method is discussed

a characterization of graphs with rank 4 the rank of a graph g is defined to be the rank of its adjacency matrix in this paper we consider the following problem what is the structure of a connected graph with rank 4 this question has not yet been fully answered in the literature and only some partial results are known in this paper we resolve this question by completely characterizing graphs g whose adjacency matrix has rank 4

computing the permanental polynomials of graphs let m be an nn matrix with entries mij i j 1 2 n the permanent of m is defined to be per m i 1nmi i where the sum is taken over all permutations of 1 2 n the permanental polynomial of m is defined by per xinm where in is the identity matrix of size n in this paper we give recursive formulas for computing permanental polynomials of the laplacian matrix and the signless laplacian matrix of a graph respectively

combined monte carlo sampling and penalty method for stochastic nonlinear complementarity problems in this paper we consider a new formulation with recourse for a class of stochastic nonlinear complementarity problems we show that the new formulation is equivalent to a smooth semi infinite program that no longer contains recourse variables we then propose a combined monte carlo sampling and penalty method for solving the problem in which the underlying sample space is assumed to be compact furthermore we suggest a compact approximation approach for the case where the sample space is unbounded two preliminary numerical examples are included as well

which graphs are determined by their spectrum for almost all graphs the answer to the question in the title is still unknown here we survey the cases for which the answer is known not only the adjacency matrix but also other types of matrices such as the laplacian matrix are considered

regularized mathematical programs with stochastic equilibrium constraints estimating structural demand models the article considers a particular class of optimization problems involving set valued stochastic equilibrium constraints we develop a solution procedure that relies on an approxima tion scheme for the equilibrium constraints based on regularization we replace the approximated equilibrium constraints by those involving only single valued lipschitz continuous functions in ad dition sampling has the further effect of replacing the simplified equilibrium constraints by more manageable ones obtained by implicitly discretizing the given probability measure so as to render the problem computationally tractable convergence is obtained by relying in particular on the graphical convergence of the approximated equilibrium constraints the problem of estimating the characteristics of a demand model a widely studied problem in microeconometrics serves both as motivation and illustration of the regularization and sampling procedure

universality for the parameter mismatching effect on weak synchronization in coupled chaotic systems to examine the universality for the parameter mismatching effect on weak chaotic synchronization we study coupled multidimensional invertible systems such as the coupled henon maps and coupled pendula by generalizing the method proposed in coupled one dimensional 1d noninvertible maps we introduce the parameter sensitivity exponent to measure the degree of the parameter sensitivity of a weakly stable synchronous chaotic attractor in terms of the parameter sensitivity exponents we characterize the effect of the parameter mismatch on the intermittent bursting and the basin riddling occurring in the regime of weak synchronization it is thus found that the scaling exponent for the average characteristic time i e the average interburst time and the average chaotic transient lifetime for both the bubbling and riddling cases is given by the reciprocal of the parameter sensitivity exponent as in the simple system of coupled 1d maps hence the reciprocal relation i e 1 seems to be universal in the sense that it holds in typical coupled chaotic systems of different nature

a new levenberg marquardt type algorithm for solving nonsmooth constrained equations we consider the problem of finding a solution of a nonsmooth constrained not necessarily square system of equations based upon the smoothing reformulation of the original problem we present a levenberg marquardt l m type algorithm for solving nonsmooth constrained system of equations which solves a linear system of equations at each iteration this algorithm has global convergence property moreover this algorithm is shown to converge locally quadratically under an error bound condition which is much weaker than the standard nonsingularity condition some numerical results for the presented method indicate that the algorithm works quite well in practice

sub quadratic convergence of a smoothing newton algorithm for the p 0 and monotone lcp given equation here the linear complementarity problem lcp is to find equation here such that x s 0 s mx q xts 0 by using the chen harker kanzow smale chks smoothing function the lcp is reformulated as a system of parameterized smooth nonsmooth equations as a result a smoothing newton algorithm which is a modified version of the qi sun zhou algorithm mathematical programming vol 87 2000 pp 1 35 is proposed to solve the lcp with m being assumed to be a p0 matrix p0 lcp the proposed algorithm needs only to solve one system of linear equations and to do one line search at each iteration it is proved in this paper that the proposed algorithm has the following convergence properties i it is well defined and any accumulation point of the iteration sequence is a solution of the p0 lcp ii it generates a bounded sequence if the p0 lcp has a nonempty and bounded solution set iii if an accumulation point of the iteration sequence satisfies a nonsingularity condition which implies the p0 lcp has a unique solution then the whole iteration sequence converges to this accumulation point sub quadratically with a q rate 2 t where t 0 1 is a parameter and iv if m is positive semidefinite and an accumulation point of the iteration sequence satisfies a strict complementarity condition then the whole sequence converges to the accumulation point quadratically

smooth sample average approximation of stationary points in nonsmooth stochastic optimization and applications inspired by a recent work by alexander et al j bank finance 30 583 605 2006 which proposes a smoothing method to deal with nonsmoothness in a conditional value at risk problem we consider a smoothing scheme for a general class of nonsmooth stochastic problems assuming that a smoothed problem is solved by a sample average approximation method we investigate the convergence of stationary points of the smoothed sample average approximation problem as sample size increases and show that w p 1 accumulation points of the stationary points of the approximation problem are weak stationary points of their counterparts of the true problem moreover under some metric regularity conditions we obtain an error bound on approximate stationary points the convergence result is applied to a conditional value at risk problem and an inventory control problem

scaling laws for noise induced superpersistent chaotic transients a super persistent chaotic transient is characterized by the following scaling law for its average lifetime tau approximately exp c p p c alpha where c 0 and alpha 0 are constants p or p c is a bifurcation parameter and p c is its critical value as p approaches p c from above the exponent in the exponential dependence diverges leading to an extremely long transient lifetime historically the possibility of such transient raised the question of whether asymptotic attractors are relevant to turbulence here we investigate the phenomenon of noise induced super persistent chaotic transients in particular we construct a prototype model based on random maps to illustrate this phenomenon we then approximate the model by stochastic differential equations and derive the scaling laws for the transient lifetime versus the noise amplitude epsilon for both the subcritical p p c cases our results are the following in the subcritical case where a chaotic attractor exists in the absence of noise noise induced transients can be more persistent in the following sense of double exponential and algebraic scaling tau approximately exp k0 exp k1epsilon gamma for small noise amplitude epsilon where k0 0 k1 0 and gamma 0 are constants the longevity of the transient lifetime in this case is striking for the supercritical case where there is already a super persistent chaotic transient noise can significantly reduce the transient lifetime these results add to the understanding of the interplay between random and deterministic chaotic dynamics with surprising physical implications

uniform exponential convergence of sample average random functions under general sampling with applications in stochastic programming abstract sample average approximation saa is one of the most popular methods for solving stochastic optimization and equilibrium problems research on saa has been mostly focused on the case when sampling is independent and identically distributed iid with exceptions dai et al 2000 9 homem de mello 2008 16 in this paper we study saa with general sampling including iid sampling and non iid sampling for solving nonsmooth stochastic optimization problems stochastic nash equilibrium problems and stochastic generalized equations to this end we first derive the uniform exponential convergence of the sample average of a class of lower semicontinuous random functions and then apply it to a nonsmooth stochastic minimization problem exponential convergence of estimators of both optimal solutions and m stationary points characterized by mordukhovich limiting subgradients mordukhovich 2006 23 rockafellar and wets 1998 32 are established under mild conditions we also use the unform convergence result to establish the exponential rate of convergence of statistical estimators of a stochastic nash equilibrium problem and estimators of the solutions to a stochastic generalized equation problem

quadratic one step smoothing newton method for p0 lcp without strict complementarity in this paper we propose a modified one step smoothing newton method for solving the p 0 linear complementarity problem p 0 lcp based on kanzow s smoothing function our smoothing newton method solves only one linear system of equations and performs only one line search at each iteration it is proved that our proposed algorithm has global convergence and local quadratic convergence in absence of strict complementarity assumption at the p 0 lcp solution under weaker conditions our convergence results are much stronger than many previous literatures

smoothing functions for second order cone complementarity problems smoothing functions have been much studied in the solution of optimization and complementarity problems with nonnegativity constraints in this paper we extend smoothing functions to problems in which the nonnegative orthant is replaced by the direct product of second order cones these smoothing functions include the chen mangasarian class and the smoothed fischer burmeister function we study the lipschitzian and differential properties of these functions and in particular we derive computable formulas for these functions and their jacobians these properties and formulas can then be used to develop and analyze noninterior continuation methods for solving the corresponding optimization and complementarity problems in particular we establish the existence and uniqueness of the newton direction when the underlying mapping is monotone

a fixed point method for a class of super large scale nonlinear complementarity problems we consider a class of complementarity problems involving functions which are nonlinear in this paper we reformulate this nonlinear complementarity problem as a system of absolute value equations which is nonsmooth then we propose a fixed point method to solve this nonsmooth system we prove that the proposed method is globally linearly convergent under a mild condition the proposed method is greatly effective not only for small and medium size problems but also for large and super large scale problems especially our method can efficiently solve super large scale problems with a million variables in a few tens of minutes on a pc

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

saa regularized methods for multiproduct price optimization under the pure characteristics demand model utility based choice models are often used to determine a consumer s purchase decision among a list of available products to provide an estimate of product demands and when data on purchase decisions or market shares are available to infer consumers preferences over observed product characteristics these models also serve as a building block in modeling firms pricing and assortment optimization problems we consider a firm s multiproduct pricing problem in which product demands are determined by a pure characteristics model a sample average approximation saa method is used to approximate the expected market share of products and the firm profit we propose an saa regularized method for the multiproduct price optimization problem we present convergence analysis and numerical examples to show the efficiency and the effectiveness of the proposed method

the subdifferential of measurable composite max integrands and smoothing approximation the subdifferential calculus for the expectation of nonsmooth random integrands involves many fundamental and challenging problems in stochastic optimization it is known that for clarke regular integrands the clarke subdifferential of the expectation equals the expectation of their clarke subdifferential in particular this holds for convex integrands however little is known about the calculation of clarke subgradients for the expectation of non regular integrands the focus of this contribution is to approximate clarke subgradients for the expectation of random integrands by smoothing methods applied to the integrand a framework for how to proceed along this path is developed and then applied to a class of measurable composite max integrands this class contains non regular integrands from stochastic complementarity problems as well as stochastic optimization problems arising in statistical learning

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

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