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

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

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

free idempotent generated semigroups and endomorphism monoids of free g acts abstract the study of the free idempotent generated semigroup ig e over a biordered set e began with the seminal work of nambooripad in the 1970s and has seen a recent revival with a number of new approaches both geometric and combinatorial here we study ig e in the case e is the biordered set of a wreath product g t n where g is a group and t n is the full transformation monoid on n elements this wreath product is isomorphic to the endomorphism monoid of the free g act end f n g on n generators and this provides us with a convenient approach we say that the rank of an element of end f n g is the minimal number of free generators in its image let e e 2 end f n g for rather straightforward reasons it is known that if rank e n 1 respectively n then the maximal subgroup of ig e containing e is free respectively trivial we show that if rank e r where 1 r n 2 then the maximal subgroup of ig e containing e is isomorphic to that in end f n g and hence to g s r where s r is the symmetric group on r elements we have previously shown this result in the case r 1 however for higher rank a more sophisticated approach is needed our current proof subsumes the case r 1 and thus provides another approach to showing that any group occurs as the maximal subgroup of some ig e on the other hand varying r again and taking g to be trivial we obtain an alternative proof of the recent result of gray and ruskuc for the biordered set of idempotents of t n

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 continuation method for nonlinear complementarity problems over symmetric cones in this paper we introduce a new p type property for nonlinear functions defined over euclidean jordan algebras and study a continuation method for nonlinear complementarity problems over symmetric cones this new p type property represents a new class of nonmonotone nonlinear complementarity problems that can be solved numerically

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

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

the jacobian consistency of a smoothed fischer burmeister function associated with second order cones abstract this paper deals with the second order cone complementarity problem soccp which is an important class of problems containing various optimization problems the soccp can be reformulated as a system of nonsmooth equations for solving this system of nonsmooth equations smoothing newton methods are widely used the jacobian consistency property plays an important role for achieving a rapid convergence of the methods in this paper we show the jacobian consistency of a smoothed fischer burmeister function moreover we estimate the distance between the subgradient of the fischer burmeister function and the gradient of its smoothing function

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

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

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

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

on the coerciveness of some merit functions for complementarity problems over symmetric cones abstract one of the popular solution methods for the complementarity problem over symmetric cones is to reformulate it as the global minimization of a certain merit function an important question to be answered for this class of methods is under what conditions the level sets of the merit function are bounded the coerciveness of the merit function in this paper we introduce the generalized weak coerciveness of a continuous transformation under this condition we prove the coerciveness of some merit functions such as the natural residual function the normal map and the fukushima yamashita function for complementarity problems over symmetric cones we note that this is a much milder condition than strong monotonicity used in the current literature

merit functions for semi definite complementarity problems merit functions such as the gap function the regularized gap function the implicit lagrangian and the norm squared of the fischer burmeister function have played an important role in the solution of complementarity problems defined over the cone of nonnegative real vectors we study the extension of these merit functions to complementarity problems defined over the cone of block diagonal symmetric positive semi definite real matrices the extension suggests new solution methods for the latter problems

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

convexity and differentiability properties of spectral functions and spectral mappings on euclidean jordan algebras abstract we study in this paper several properties of the eigenvalues function of a euclidean jordan algebra extending several known results in the framework of symmetric matrices in particular we give a concise form for the directional differential of a single eigenvalue we especially focus on spectral functions f on euclidean jordan algebras which are the composition of a symmetric real valued function f with the eigenvalues function we explore several properties of f that are transferred to f in particular convexity strong convexity and differentiability spectral mappings are also considered a special case of which is the gradient mapping of a spectral function answering a problem proposed by h sendov we give a formula for the jacobian of these functions

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 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

differentiable exact penalty functions for nonlinear second order cone programs we propose a method for solving nonlinear second order cone programs socps based on a continuously differentiable exact penalty function the construction of the penalty function is given by incorporating a multipliers estimate in the augmented lagrangian for socps under the nondegeneracy assumption and the strong second order sufficient condition we show that a generalized newton method has global and superlinear convergence we also present some preliminary numerical experiments

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

presentations of general products of monoids abstract conditions are found under which a general product of two finitely presented monoids is itself finitely presented presentations in terms of the presentations of the factors are given subject to these conditions

semismooth karush kuhn tucker equations and convergence analysis of newton and quasi newton methods for solving these equations there are several forms of systems of nonsmooth equations which are equivalent to the karush kuhn tucker kkt system of a nonlinearly constrained optimization problem nlp if the nlp is twice continuously differentiable and the hessian functions of its objective and constraint functions are locally lipschitzian then these kkt equations are strongly semismooth if furthermore the linear independence condition and the strong second order sufficiency condition are satisfied at a kkt point then the generalized jacobians of these kkt equations are nonsingular at that point and the sequence generated by the generalized newton method converges to this point q quadratically however direct application of quasi newton methods cannot guarantee q superlinear convergence we present a mixed quasi newton method which converges q superlinearly with common symmetrical updating rules under the above conditions for the generalized newton method superlinear convergence of the primal variables and global convergence are also discussed

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

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

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

a generalized newton method for absolute value equations associated with second order cones in this paper we introduce the absolute value equations associated with second order cones socave in short which is a generalization of the absolute value equations discussed recently in the literature it is proved that the socave is equivalent to a class of second order cone linear complementarity problems soclcp in short in particular we propose a generalized newton method for solving the socave and show that the proposed method is globally linearly and locally quadratically convergent under suitable assumptions we also report some preliminary numerical results of the proposed method for solving the socave and the soclcp which show the 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

weakly mixing invariant tori of hamiltonian systems we note that every finite or infinite dimensional real analytic hamiltonian system with a quasi periodic invariant kam torus of finite dimension d 2 can be perturbed in such a way that the new real analytic hamiltonian system has a weakly mixing invariant torus of the same dimension

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

numerical comparison between prediction correction methods for general variational inequalities in this paper we propose two prediction correction methods for solving general variational inequalities the first method makes use of a descent direction to produce the new iterate while the second method generates the corrector by the improved extragradient method under certain conditions the global convergence of both methods is proved it is proved theoretically that the lower bound of the progress obtained by the second method is greater than that by the first one numerical results are given to verify and compare the numerical efficiency of these two predictor corrector methods

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

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

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 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

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

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

polynomial convergence of primal dual algorithms for the second order cone program based on the mz family of directions in this paper we study primal dual path following algorithms for the second order cone programming socp based on a family of directions that is a natural extension of the monteiro zhang mz family for semidefinite programming we show that the polynomial iteration complexity bounds of two well known algorithms for linear programming namely the short step path following algorithm of kojima et al and monteiro and adler and the predictor corrector algorithm of mizuno et al carry over to the context of socp that is they have an o loge 1 iteration complexity to reduce the duality gap by a factor of e where n is the number of second order cones since the mz type family studied in this paper includes an analogue of the alizadeh haeberly and overton pure newton direction we establish for the first time the polynomial convergence of primal dual algorithms for socp based on this search direction

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

applications of second order cone programming in a second order cone program socp a linear function is minimized over the intersection of an affine set and the product of second order quadratic cones socps are nonlinear convex problems that include linear and convex quadratic programs as special cases but are less general than semidefinite programs sdps several efficient primaldual interior point methods for socp have been developed in the last few years after reviewing the basic theory of socps we describe general families of problems that tan be recast as socps these include robust linear programming and robust leastsquares problems problems involving sums or maxima of norms or with convex hyperbolic constraints we discuss a variety of engineering applications such as filter design antenna array weight design truss design and grasping forte optimization in robotics we describe an efficient primaldual interior point method for solving socps which shares many of the features of primaldual interior point methods for linear program

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

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

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

operators with singular continuous spectrum i general operators the baire category theorem implies that the family f of dense sets g in fixed metric space x is a candidate for generic sets since it is closed under countable intersections and if x is perfect has no isolated point then a f has uncountable intersections with any open ball in x

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

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

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

finite presentability of bruck reilly extensions of groups abstract the purpose of this paper is to consider finite generation and finite presentability of a bruck reilly extension s br g of a group g with respect to an endomorphism it is proved that s is finitely generated if and only if g can be generated by a set of the form i 0 a i where a g is finite the main result states that s is finitely presented if and only if g can be defined by a presentation of the form a r where a is finite and r is of the form i 0 r i for some finite set of relations r finally it is proved that s is finitely presented as an inverse monoid if and only if it is finitely presented as an ordinary monoid

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

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

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

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 iterative method for solving semismooth equations in this paper we combine trust region technique with line search technique to develop an iterative method for solving semismooth equations at each iteration a trust region subproblem is solved the solution of the trust region subproblem provides a descent direction for the norm of a smoothing function by using a backtracking line search a steplength is determined the proposed method shares advantages of trust region methods and line search methods under appropriate conditions the proposed method is proved to be globally and superlinearly convergent in particular we show that after finitely many iterations the unit step is always accepted and the method reduces to a smoothing newton method

vines and partial transformations abstract we introduce the partial vine monoid pv n this monoid is related to the partial transformation semigroup pt n in the same way as the braid group b n is related to the symmetric group s n and contains both the vine monoid t g lavers the theory of vines comm algebra 25 4 1997 1257 1284 and the inverse braid monoid d easdown t g lavers the inverse braid monoid adv math 186 2 2004 438 455 we give a presentation for pv n in terms of generators and relations as well as a faithful representation in a monoid of endomorphisms of a free group we also derive a new presentation for pt n

strong semismoothness of the fischer burmeister sdc and soc complementarity functions we show that the fischer burmeister complementarity functions associated to the semidefinite cone sdc and the second order cone soc respectively are strongly semismooth everywhere interestingly enough the proof relys on a relationship between the singular value decomposition of a nonsymmetric matrix and the spectral decomposition of a symmetric matrix

expected residual minimization formulation for a class of stochastic linear second order cone complementarity problems abstract this paper considers a class of stochastic linear second order cone complementarity problems slsoccp noticing that the slsoccp does not have a solution suitable to all realizations in general we present a deterministic formulation called the expected residual minimization erm formulation for it the coercive property of the erm problem and the robustness of its solutions are discussed due to the existence of expectation in the erm problem we employ the monte carlo approximation techniques to approximate the erm problem and show that under mild conditions this approximation approach possesses exponential convergence rate then we extend the above results to a general mixed slsoccp furthermore we apply the theoretical results to a stochastic optimal power flow model in radial network and report some numerical dispatching experiments for real world southern california edison 47 bus network

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

analysis of nonsmooth vector valued functions associated with second order cones let open image in new window be the lorentz second order cone in open image in new window for any function f from open image in new window to open image in new window one can define a corresponding function fsoc x on open image in new window by applying f to the spectral values of the spectral decomposition of x open image in new window with respect to open image in new window we show that this vector valued function inherits from f the properties of continuity local lipschitz continuity directional differentiability frechet differentiability continuous differentiability as well as order semismoothness these results are useful for designing and analyzing smoothing methods and nonsmooth methods for solving second order cone programs and complementarity problems

semismooth matrix valued functions matrix valued functions play an important role in the development of algorithms for semidefinite programming problems this paper studies generalized differential properties of such functions related to nonsmooth smoothing newton methods the first part of this paper discusses basic properties such as the generalized derivative rademacher s theorem b derivative directional derivative and semismoothness the second part shows that the matrix absolute value function the matrix semidefinite projection function and the matrix projective residual function are strongly semismooth

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

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

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

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 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 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

a regularized smoothing newton method for symmetric cone complementarity problems this paper extends the regularized smoothing newton method in vector complementarity problems to symmetric cone complementarity problems sccp which includes the nonlinear complementarity problem the second order cone complementarity problem and the semidefinite complementarity problem as special cases in particular we study strong semismoothness and jacobian nonsingularity of the total natural residual function for sccp we also derive the uniform approximation property and the jacobian consistency of the chen mangasarian smoothing function of the natural residual based on these properties global and quadratical convergence of the proposed algorithm is established

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

a new non interior continuation method for solving the second order cone complementarity problem abstract in this paper based on a symmetrically perturbed smoothing fischer burmeister function a non interior continuation method is proposed for solving the second order cone complementarity problem soccp the proposed algorithm solves only one linear system of equations and performs only one line search at each iteration under monotonicity it is shown that our algorithm is globally and locally superlinearly convergent without requiring strict complementarity assumption at the soccp solution furthermore the proposed algorithm has local quadratic convergence under mild conditions some numerical results are reported which indicate the effectiveness of the proposed algorithm