eigenvalue theorems for discrete multipoint conjugate boundary value problems abstract we consider the following boundary value problem n y k p k y y n 1 y k 0 m j y k i 0 j 0 n i 1 i 1 r where r 2 ni 1 for i 1 r i 1 r n i n and 0 k1

a semi smooth newton method for projection equations and linear complementarity problems with respect to the second order cone abstract in this paper a special semi smooth equation associated to the second order cone is studied it is shown that under mild assumptions the semi smooth newton method applied to this equation is well defined and the generated sequence is globally and q linearly convergent to a solution as an application the obtained results are used to study the linear second order cone complementarity problem with special emphasis on the particular case of positive definite matrices moreover some computational experiments designed to investigate the practical viability of the method are presented

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 primal dual interior point algorithm for second order cone optimization with full nesterov todd step in this paper we propose a primal dual path following interior point algorithm for second order cone optimization the algorithm is based on a new technique for finding the search directions and the strategy of the central path at each iteration we use only full nesterov todd step moreover we derive the currently best known iteration bound for the algorithm with small update method namely email protected where n denotes the number of second order cones in the problem formulation and e the desired accuracy

a new smoothing newton type method for second order cone programming problems a new smoothing function of the well known fischer burmeister function is given based on this new function a smoothing newton type method is proposed for solving second order cone programming at each iteration the proposed algorithm solves only one system of linear equations and performs only one line search this algorithm can start from an arbitrary point and it is q quadratically convergent under a mild assumption numerical results demonstrate the effectiveness of the algorithm

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 regularized smoothing newton method for solving soccps based on a new smoothing c function abstract in this paper an soc complementarity function is constructed based on this function a regularized smoothing newton method is proposed for solving monotone second order cone complementarity problems denoted by soccps the proposed algorithm is proved to be globally and quadratically convergent under mild conditions unlike some existing smoothing newton type methods we take the regularization parameter the same as the smoothing parameter and treat the parameter as independent variables in our algorithm some numerical results are reported and indicate that the proposed method is quite effective

a non monotone regularization newton method for the second order cone complementarity problem based on the smoothing newton method and the tikhonov regularization method we construct a regularization newton method for the second order cone complementarity problem the method uses a non monotone line search scheme which contains the usual monotone line search as a special case by using the theory of euclidean jordan algebras we prove that the proposed method is globally and locally quadratically convergent under suitable assumptions some numerical results are reported which indicate the effectiveness of the method

on the quadratic eigenvalue complementarity problem over a general convex cone the solution of the conic quadratic eigenvalue complementarity problem cqeicp is first investigated without assuming symmetry on the matrices defining the problem a new sufficient condition for existence of solutions of cqeicp is presented extending to arbitrary pointed closed and convex cones a condition known to hold when the cone is the nonnegative orthant we also address the symmetric cqeicp where all its defining matrices are symmetric we show that assuming that two of its defining matrices are positive definite this symmetric cqeicp reduces to the computation of a stationary point of an appropriate merit function on a convex set furthermore we discuss the use of the so called spectral projected gradient spg algorithm for solving cqeicp when the cone of interest is the second order cone socqeicp a new algorithm is designed for the computation of the projections required by the spg method to deal with socqeicp numerical results are included to illustrate the efficiency of the spg method and the new projection technique in practice

a proximal gradient descent method for the extended second order cone linear complementarity problem we consider an extended second order cone linear complementarity problem soclcp including the generalized soclcp the horizontal soclcp the vertical soclcp and the mixed soclcp as special cases in this paper we present some simple second order cone constrained and unconstrained reformulation problems and under mild conditions prove the equivalence between the stationary points of these optimization problems and the solutions of the extended soclcp particularly we develop a proximal gradient descent method for solving the second order cone constrained problems this method is very simple and at each iteration makes only one euclidean projection onto second order cones we establish global convergence and under a local lipschitzian error bound assumption linear rate of convergence numerical comparisons are made with the limited memory bfgs method for the unconstrained reformulations which verify the effectiveness of the proposed method

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

some p properties for linear transformations on euclidean jordan algebras a real square matrix is said to be a p matrix if all its principal minors are positive it is well known that this property is equivalent to the nonsign reversal property based on the componentwise product of vectors the order p property based on the minimum and maximum of vectors uniqueness property in the standard linear complementarity problem lipschitzian homeomorphism property of the normal map corresponding to the nonnegative orthant in this article we extend these notions to a linear transformation defined on a euclidean jordan algebra we study some interconnections between these extended concepts and specialize them to the space s n of all n n real symmetric matrices with the semidefinite cone s n and to the space r n with the lorentz cone

a one step smoothing newton method for second order cone programming a new smoothing function for the second order cone programming is given by smoothing the symmetric perturbed fischer burmeister function based on this new function a one step smoothing newton method is presented for solving the second order cone programming the proposed algorithm solves only one linear system of equations and performs only one line search at each iteration this algorithm does not have restrictions regarding its starting point and is q quadratically convergent numerical results suggest the effectiveness of our algorithm

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

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

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

a note on treating a second order cone program as a special case of a semidefinite program it is well known that a vector is in a second order cone if and only if its arrow matrix is positive semidefinite but much less well known is about the relation between a second order cone program socp and its corresponding semidefinite program sdp the correspondence between the dual problem of socp and sdp is quite direct and the correspondence between the primal problems is much more complicated given a sdp primal optimal solution which is not necessarily arrow shaped we can construct a socp primal optimal solution the mapping from the primal optimal solution of sdp to the primal optimal solution of socp can be shown to be unique conversely given a socp primal optimal solution we can construct a sdp primal optimal solution which is not an arrow matrix indeed in general no primal optimal solutions of the socp related sdp can be an arrow matrix

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

a multisplitting method for symmetrical affine second order cone complementarity problem the affine second order cone complementarity problem soccp is a wide class of problems that contains the linear complementarity problem lcp as a special case in this paper a multisplitting method for the symmetrical affine soccp is developed and analysed this method makes use of a set of different splittings of the coefficient matrix convergence is established for the proposed method moreover an maor like method is considered to solve the subproblems we also report some simple numerical results for the proposed method

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

iterative solution of linear systems in the 20th century this paper sketches the main research developments in the area of iterative methods for solving linear systems during the 20th century although iterative methods for solving linear systems nd their origin in the early 19th century work by gauss the eld has seen an explosion of activity spurred by demand due to extraordinary technological advances in engineering and sciences the past ve decades have been particularly rich in new developments ending with the availability of large toolbox of specialized algorithms for solving the very large problems which arise in scientic and industrial computational models as in any other scientic area research in iterative methods has been a journey characterized by a chain of contributions building on each other it is the aim of this paper not only to sketch the most signicant of these contributions during the past century but also to relate them to one another c 2000 elsevier science b v all rights reserved

interior proximal methods and central paths for convex second order cone programming we make a unified analysis of interior proximal methods of solving convex second order cone programming problems these methods use a proximal distance with respect to second order cones which can be produced with an appropriate closed proper univariate function in three ways under some mild conditions the sequence generated is bounded with each limit point being a solution and global rates of convergence estimates are obtained in terms of objective values a class of regularized proximal distances is also constructed which can guarantee the global convergence of the sequence to an optimal solution these results are illustrated with some examples in addition we also study the central paths associated with these distance like functions and for the linear socp we discuss their relations with the sequence generated by the interior proximal methods from this we obtain improved convergence results for the sequence for the interior proximal methods using a proximal distance continuous at the boundary of second order cones

analysis of nonsmooth vector valued functions associated with infinite dimensional second order cones abstract given a hilbert space h the infinite dimensional lorentz second order cone k is introduced for any x h a spectral decomposition is introduced and for any function f r r we define a corresponding vector valued function f h x on hilbert space h by applying f to the spectral values of the spectral decomposition of x h with respect to k we show that this vector valued function inherits from f the properties of continuity lipschitz continuity differentiability smoothness as well as s semismoothness these results can be helpful for designing and analyzing solution methods for solving infinite dimensional second order cone programs and complementarity problems

the gus property of second order cone linear complementarity problems the globally uniquely solvable gus property of the linear transformation of the linear complementarity problems over symmetric cones has been studied recently by gowda et al via the approach of euclidean jordan algebra in this paper we contribute a new approach to characterizing the gus property of the linear transformation of the second order cone linear complementarity problems soclcp via some basic linear algebra properties of the involved matrix of soclcp some more concrete and checkable sufficient and necessary conditions for the gus property are thus derived

a merit function method for infinite dimensional soccps abstract we introduce the jordan product associated with the second order cone k into the real hilbert space h and then define a one parametric class of complementarity functions t on h h with the parameter t 0 2 we show that the squared norm of t with t 0 2 is a continuously f rechet differentiable merit function by this the second order cone complementarity problem soccp in h can be converted into an unconstrained smooth minimization problem involving this class of merit functions and furthermore under the monotonicity assumption every stationary point of this minimization problem is shown to be a solution of the soccp

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

an inexact smoothing method for soccps based on a one parametric class of smoothing function in this paper we introduce a one parametric class of smoothing functions which contains the fischer burmeister smoothing function as special case based on this class of smoothing functions an inexact smoothing method for solving second order cone complementarity problems soccps is proposed in each iteration the corresponding linear system is solved only approximately under mild assumptions it is proved that the proposed method has global convergence and local superlinear convergence properties preliminary numerical results indicate that the method is effective for large scale problems

on the convergence of the central path in semidefinite optimization the central path in linear optimization always converges to the analytic center of the optimal set this result was extended to semidefinite optimization in d goldfarb and k scheinberg siam j optim 8 1998 pp 871 886 in this paper we show that this latter result is not correct in the absence of strict complementarity we provide a counterexample where the central path converges to a different optimal solution this unexpected result raises many questions we also give a short proof that the central path always converges in semidefinite optimization by using ideas from algebraic geometry

convergence analysis of the augmented lagrangian method for nonlinear second order cone optimization problems abstract the paper focuses on the convergence rate of the augmented lagrangian method for nonlinear second order cone optimization problems under a set of assumptions of sufficient conditions including the componentwise strict complementarity condition the constraint nondegeneracy condition and the second order sufficient condition we first study some properties of the augmented lagrangian and then show that the rate of local convergence of the augmented lagrangian method is proportional to 1 where the penalty parameter is not less than a threshold 0

a multiplicity result for second order impulsive differential equations via the leggett williams fixed point theorem using the leggett williams fixed point theorem we present in this paper criteria which guarantee the existence of three nonnegative solutions to second order impulsive equations

contact analysis of cable networks by using second order cone programming a method based on mathematical programming is proposed for large deformation and contact analysis of cable networks by explicitly considering these nonsmooth behaviors we formulate the linear complementarity problems over symmetric cones under some practically acceptable assumptions we also present the equivalent second order cone programming socp problems which can be regarded as the minimization problem of total potential energy and complementary energy with the subsidiary constraints on the displacements and contact forces respectively by solving the presented socp problems by using the primal dual interior point method the equilibrium configurations and internal forces of several cable networks are obtained without any assumptions on stress states and contact conditions

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

two unconstrained optimization approaches for the euclidean κ centrum location problem abstract consider the single facility euclidean centrum location problem in r n this problem is a generalization of the classical euclidean 1 median problem and 1 center problem in this paper we develop two efficient algorithms that are particularly suitable for problems where n is large by using unconstrained optimization techniques the first algorithm is based on the neural networks smooth approximation for the plus function and reduces the problem to an unconstrained smooth convex minimization problem the second algorithm is based on the fischer burmeister merit function for the second order cone complementarity problem and transforms the kkt system of the second order cone programming reformulation for the problem into an unconstrained smooth minimization problem our computational experiments indicate that both methods are extremely efficient for large problems and the first algorithm is able to solve problems of dimension n up to 10 000 efficiently

a power penalty method for second order cone nonlinear complementarity problems a power penalty method for solving nonlinear second order cone complementarity problems soccps is proposed by using this method the nonlinear soccp is converted to asymptotic nonlinear equations the merit of this method shows that the solution sequence of the asymptotic nonlinear equations converges to the solution of the nonlinear soccp at an exponential rate when the penalty parameter tends to positive infinity under mild assumptions an algorithm is constructed and numerical examples indicate the feasibility of our method

stabilization of solutions of a degenerate nonlinear diffusion problem i 4 u f u in l l x r u tl t o in r u x 0 uo x in l l where m 1 is a parameter f is locally lipschitz continuous f 0 0 and u is bounded problems of this form arise in a number of areas of science for instance in models for gas or fluid flow in porous media 2 and for the spread of certain biological populations 13 161 this paper is divided into two parts in part i we consider what may be called the motivating example problem i which consists of problem i with the special choice f u u 1 u u a 1 for suitably restricted parameters a we begin by describing in detail the set l of nonnegative equilibrium solutions of problem i clearly 8 l contains the trivial solution u 0 for all l 0 write 8 l e l o in the description of z l there are two critical parameter values lo and l1 with 0 lo l1 j we show that i g l 4 for 0 l lo

a regularized explicit exchange method for semi infinite programs with an infinite number of conic constraints the semi infinite program sip is normally represented with infinitely many inequality constraints and has been studied extensively so far however there have been very few studies on the sip involving conic constraints even though it has important applications such as chebyshev like approximation filter design and so on in this paper we focus on the sip with infinitely many conic constraints called an sicp for short we show that under the robinson constraint qualification a local optimum of the sicp satisfies the kkt conditions that can be represented only with a finite subset of the conic constraints we also introduce two exchange type algorithms for solving the convex sicp we first provide an explicit exchange method and show that it has global convergence under the strict convexity assumption on the objective function we then propose an algorithm combining a regularization method with the explicit exchange method and establish global convergence of the hybrid algorithm without the strict c

preconditioning techniques for large linear systems a survey this article surveys preconditioning techniques for the iterative solution of large linear systems with a focus on algebraic methods suitable for general sparse matrices covered topics include progress in incomplete factorization methods sparse approximate inverses reorderings parallelization issues and block and multilevel extensions some of the challenges ahead are also discussed an extensive bibliography completes the paper

the existence of multiple positive solutions to boundary value problems of nonlinear delay differential equations with countably many singularities on infinite interval in this paper we consider the existence of countably many positive solutions to a boundary value problem of a nonlinear delay differential equation with countably many singularities on infinite interval f x t a t f t x t x t 0 0 x t 0 where f r r is an increasing homeomorphism and a positive homomorphism with f 0 0 x t is a function in c r 0 r defined by x t s x t s for r s 0 and x c r 0 r by using the fixed point index theory and a new fixed point theorem in a cone we provide sufficient conditions for the existence of multiple positive solutions to the above boundary value problem the conclusions in this paper essentially extend and improve the known results

a one parametric class of merit functions for the symmetric cone complementarity problem abstract in this paper we extend the one parametric class of merit functions proposed by kanzow and kleinmichel c kanzow h kleinmichel a new class of semismooth newton type methods for nonlinear complementarity problems comput optim appl 11 1998 227 251 for the nonnegative orthant complementarity problem to the general symmetric cone complementarity problem sccp we show that the class of merit functions is continuously differentiable everywhere and has a globally lipschitz continuous gradient mapping from this we particularly obtain the smoothness of the fischer burmeister merit function associated with symmetric cones and the lipschitz continuity of its gradient in addition we also consider a regularized formulation for the class of merit functions which is actually an extension of one of the ncp function classes studied by c kanzow y yamashita m fukushima new ncp functions and their properties j optim theory appl 97 1997 115 135 to the sccp by exploiting the cartesian p properties for a nonlinear transformation we show that the class of regularized merit functions provides a global error bound for the solution of the sccp and moreover has bounded level sets under a rather weak condition which can be satisfied by the monotone sccp with a strictly feasible point or the sccp with the joint cartesian r 02 property all of these results generalize some recent important works in j s chen p tseng an unconstrained smooth minimization reformulation of the second order cone complementarity problem math program 104 2005 293 327 c k sim j sun d ralph a note on the lipschitz continuity of the gradient of the squared norm of the matrix valued fischer burmeister function math program 107 2006 547 553 p tseng merit function for semidefinite complementarity problems math program 83 1998 159 185 under a unified framework

high performance finite element approximate inverse preconditioning abstract a new parallel normalized optimized approximate inverse algorithm based on the concept of the fish bone computational approach satisfying an antidiagonal data dependency for computing classes of explicit approximate inverses is introduced for symmetric multiprocessor systems the parallel normalized explicit approximate inverses are used in conjunction with parallel normalized explicit preconditioned conjugate gradient square schemes for the efficient solution of finite element sparse linear systems the parallel design and implementation issues of the new proposed algorithms are discussed and the parallel performance is presented using openmp

eigenvalues and eigenfunctions of discrete conjugate boundary value problems abstract we consider the following boundary value problem 1 n p n y f k y y n 1 y n 2 0 k m i y 0 0 0 i p 1 i y m n i 0 0 i n p 1 where 1 p n 1 is fixed and 0 a characterization of the values of is carried out so that the boundary value problem has a positive solution next for 1 criteria are developed for the existence of two positive solutions of the boundary value problem in addition for particular cases we also offer upper and lower bounds for these positive solutions several examples are included to dwell upon the importance of the results obtained

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

log sigmoid nonlinear lagrange method for nonlinear optimization problems over second order cones this paper analyzes the rate of local convergence of the log sigmoid nonlinear lagrange method for nonconvex nonlinear second order cone programming under the componentwise strict complementarity condition the constraint nondegeneracy condition and the second order sufficient condition we show that the sequence of iteration points generated by the proposed method locally converges to a local solution when the penalty parameter is less than a threshold and the error bound of solution is proportional to the penalty parameter finally we report numerical results to show the efficiency of the method

soc monotone and soc convex functions vs matrix monotone and matrix convex functions abstract the soc monotone function respectively soc convex function is a scalar valued function that induces a map to preserve the monotone order respectively the convex order when imposed on the spectral factorization of vectors associated with second order cones socs in general hilbert spaces in this paper we provide the sufficient and necessary characterizations for the two classes of functions and particularly establish that the set of continuous soc monotone respectively soc convex functions coincides with that of continuous matrix monotone respectively matrix convex functions of order 2

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

an explicit exchange algorithm for linear semi infinite programming problems with second order cone constraints in this paper we propose an explicit exchange algorithm for solving the semi infinite programming problem sip with second order cone soc constraints we prove by using the complementarity slackness conditions that the algorithm terminates in a finite number of iterations and the obtained solution sufficiently approximates the original sip solution in existing studies on sips only the nonnegative constraints were considered and hence the complementarity slackness conditions were separable to each scalar component however in our study the existing componentwise analyses are no longer applicable since the complementarity slackness conditions are associated with socs in order to overcome such a difficulty we introduce a certain coordinate system based on the spectral factorization in the numerical experiments we solve some test problems to see the effectiveness of the proposed algorithm first we compare our algorithm with two other algorithms based on the linear sip lsip reformulation and observe that by exploiting the sdpt3 solver for subproblems our algorithm finds the solution much faster than lsip reformulation approaches when the size of the problem is large also we apply the algorithm to some filter design problems and see that those problems can be solved efficiently

solvability of newton equations in smoothing type algorithms for the soccp in this paper we first investigate the invertibility of a class of matrices based on the obtained results we then discuss the solvability of newton equations appearing in the smoothing type algorithm for solving the second order cone complementarity problem soccp a condition ensuring the solvability of such a system of newton equations is given in addition our results also show that the assumption that the jacobian matrix of the function involved in the soccp is a p 0 matrix is not enough for ensuring the solvability of such a system of newton equations which is different from the one of smoothing type algorithms for solving many traditional optimization problems in n

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

perturbation analysis of second order cone programming problems we discuss first and second order optimality conditions for nonlinear second order cone programming problems and their relation with semidefinite programming problems for doing this we extend in an abstract setting the notion of optimal partition then we state a characterization of strong regularity in terms of second order optimality conditions this is the first time such a characterization is given for a nonpolyhedral conic problem

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

new second order cone linear complementarity formulation and semi smooth newton algorithm for finite element analysis of 3d frictional contact problem we propose a new second order cone linear complementarity problem soclcp formulation for the numerical finite element analysis of three dimensional 3d frictional contact problems by the parametric variational principle specifically we develop a regularization technique to resolve the multi valued difficulty involved in the frictional contact law and use a second order cone complementarity condition to handle the regularized coulomb friction law in contact analysis the governing equations of the 3d frictional contact problem is represented by an soclcp via the parametric variational principle and the finite element method which avoids the polyhedral approximation to the coulomb friction cone so that the problem to be solved has much smaller size and the solution has better accuracy in this paper we reformulate the soclcp as a semi smooth system of equations via a one parametric class of second order cone complementarity functions and then apply the non smooth newton method for solving this system numerical results confirm the effectiveness and robustness of the soclcp approach developed

a globally convergent method based on fischer burmeister operators for solving second order cone constrained variational inequality problems the karush kuhn tucker system of a second order cone constrained variational inequality problem is transformed into a semismooth system of equations with the help of fischer burmeister operators over second order cones the clarke generalized differential of the semismooth mapping is presented a modified newton method with armijo line search is proved to have global convergence with local superlinear rate of convergence under certain assumptions on the variational inequality problem an illustrative example is given to show how the globally convergent method works

exceptional family and solvability of the second order cone complementarity problems abstract in this paper we introduce a concept of exceptional family for second order cone complementarity problems denoted by soccp which is the particular case of the concept of exceptional family of elements introduced by isac and carbone 1999 25 for the nonlinear complementarity problems and the exceptional family for second order cone complementarity problems has some pretty unique properties furthermore we propose a new sufficient existence condition of a solution to soccp and give a particular example to show that the new condition is not stronger than known isac carbone s condition

a smoothing method for second order cone complementarity problem in this paper the second order cone complementarity problem is studied based on a perturbed symmetrically smoothing function which has coerciveness under proper conditions we present a smoothing newton method for this problem the boundedness of the level set can be obtained from the coerciveness which plays an important role in the convergence analysis furthermore the proposed algorithm for the reformulation has no restrictions on the starting point and solves only one system of equations preliminary numerical results indicate that the algorithm is effective

some p properties for nonlinear transformations on euclidean jordan algebras in this article we introduce the concepts of p and p0 properties for a nonlinear transformation defined on a euclidean jordan algebra and study existence of solution in the associated complementarity problems in particular we show in this general setting that if a transformation has the p0 and r0 properties then all associated complementarity problems have solutions we also describe a necessary condition for a transformation to have the global uniqueness of solution property

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

on the gpgpu parallelization issues of finite element approximate inverse preconditioning during the past decades explicit finite element approximate inverse preconditioning methods have been extensively used for efficiently solving sparse linear systems on multiprocessor systems the effectiveness of explicit approximate inverse preconditioning schemes relies on the use of efficient preconditioners that are close approximants to the coefficient matrix and are fast to compute in parallel new parallel computational techniques are proposed for the parallelization of the optimized banded generalized approximate inverse finite element matrix obgaifem algorithm based on the concept of the fish bone computational approach and for the explicit preconditioned conjugate gradient type methods on a general purpose graphics processing unit gpgpu the proposed parallel methods have been implemented using compute unified device architecture cuda developed by nvidia finally numerical results for the performance of the finite element explicit approximate inverse preconditioning for solving characteristic two dimensional boundary value problems on a massive multiprocessor interface on a gpu are presented the cuda implementation issues of the proposed methods are also discussed

a regularization method for the second order cone complementarity problem with the cartesian p0 property abstract we consider the tikhonov regularization method for the second order cone complementarity problem soccp with the cartesian p 0 property we show that many results of the regularization method for the p 0 nonlinear complementarity problem still hold for this important class of nonmonotone soccp for example under the more general setting every regularized problem has the unique solution and the solution trajectory generated is bounded if the original soccp has a nonempty and bounded solution set we also propose an inexact regularization algorithm by solving the sequence of regularized problems approximately with the merit function approach based on fischer burmeister merit function and establish the convergence result of the algorithm preliminary numerical results are also reported which verify the favorable theoretical properties of the proposed method

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

a class of interior proximal like algorithms for convex second order cone programming we propose a class of interior proximal like algorithms for the second order cone program which is to minimize a closed proper convex function subject to general second order cone constraints the class of methods uses a distance measure generated by a twice continuously differentiable strictly convex function on 0 infty and includes as a special case the entropy like proximal algorithm eggermont linear algebra appl 130 1990 pp 25 42 which was originally proposed for minimizing a convex function subject to nonnegative constraints particularly we consider an approximate version of these methods allowing the inexact solution of subproblems like the entropy like proximal algorithm for convex programming with nonnegative constraints we under some mild assumptions establish the global convergence expressed in terms of the objective values for the proposed algorithm and we show that the sequence generated is bounded and every accumulation point is a solution of the considered problem preliminary numerical results are reported for two approximate entropy like proximal algorithms and numerical comparisons are also made with the merit function approach chen and tseng math program 104 2005 pp 293 327 which verify the effectiveness of the proposed method

exact formulas for the proximal regular limiting normal cone of the second order cone complementarity set the proximal regular and limiting normal cones to the second order cone complementarity set play important roles in studying mathematical programs with second order cone complementarity constraints second order cone programs and the second order cone complementarity problems it is needed in the first order optimality conditions for mathematical programs with second order cone complementarity constraint the second order subdifferential criteria in characterizing the full stability for second order cone programs and second order cone complementarity problems as well as in the characterizing the pseudo lipschitz continuity of the solution mapping to parametric second order cone complementarity problems in this paper we establish explicit formulas for the proximal regular and limiting normal cone of the second order cone complementarity set

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

an inexact interior point method for the large scale simulation of granular material abstract non smooth contact dynamics provides an increasingly popular simulation framework for granular material in contrast to classical discrete element methods this approach is stable for arbitrary time steps and produces visually acceptable results in very short computing time yet when it comes to the prediction of draft forces non smooth contact dynamics is typically not accurate enough we therefore propose to combine the method class with an interior point algorithm for higher accuracy our specific algorithm is based on the so called jordan algebras and exploits the relation to symmetric cones in order to tackle the conical constraints that are intrinsic to frictional contact problems in every interior point iteration a linear system has to be solved we analyze how the interior point method behaves when it is combined with krylov subspace solvers and incomplete factorizations we show that efficient preconditioners and efficient linear solvers are essential for the method to be applicable to large scale problems using cg as a linear solver and incomplete lu factorizations we substantially improve the accuracy in comparison to the projected gaus jacobi solver

a generalized newton method for absolute value equations associated with circular cones in this paper we study a class of absolute value equations associated with circular cone ccave for short which is a generalization of the absolute value equations discussed recently in the literature analogously to the fact that circular cone is the very generalization of the second order cone we show that the ccave is equivalent to a class of circular cone linear complementarity problems and hence generalize the well known equivalence between absolute value equations and linear complementarity problems useful properties of the generalized differential of the absolute value function over the circular cone are investigated which helps to propose a generalized newton method for solving the ccave the convergence of this method is established under mild conditions as well as the efficiency of which is illustrated by some preliminary numerical results

three symmetric solutions of lidstone boundary value problems for difference and partial difference equations we consider the boundary value problem 2my k m f y k 2y k 1 2iy k 1 2 m 1 y k m 1 k a 1 b 1 2iy a 1 m 2iy b 1 m 2i 0 0 i m 1 where m 1 and 1 m f rm 0 is continuous by using amann and leggett williams fixed point theorems we develop growth conditions on f so that the boundary value problem has triple positive symmetric solutions the results obtained are then applied in the investigation of radial solutions for certain partial difference equation subject to lidstone type conditions

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

on q and r0 properties of a quadratic representation in linear complementarity problems over the second order cone abstract this paper studies the linear complementarity problem lcp m q over the second order lorentz or ice cream cone denoted by n where m is a n n real square matrix and q rn this problem is denoted as solcp m q the study of second order cone programming problems and hence an independent study of solcp is motivated by a number of applications though the second order cone is a special case of the cone of squares symmetric cone in a euclidean jordan algebra the geometry of its faces is much simpler and hence an independent study of lcp over n may yield interesting results in this paper we characterize the r0 property x n m x n and x m x 0 x 0 of a quadratic representation pa x 2a a x a2 x of n for a x n where is a jordan product and show that the r0 property of pa is equivalent to stating that solcp pa q has a solution for all q n

a modified smoothing and regularized newton method for monotone second order cone complementarity problems in this paper we propose a globally and quadratically convergent newton type algorithm for solving monotone second order cone complementarity problems denoted by soccps this algorithm is based on smoothing and regularization techniques by incorporating smoothing newton s method many newton type methods with smoothing and regularization techniques have been studied for solving nonlinear complementarity problems ncps and box constrained variational inequalities bvis our algorithm is regarded as an extension of those methods to soccp however it is different from the existing methods because we solve soccp by treating both the smoothing parameter m and the regularization parameter e as independent variables in addition numerical experiments indicate that the proposed method is quite effective

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

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