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

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

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

aperiodic virasoro algebra an aperiodic analog to the virasoro algebra is introduced and its representation theory is investigated in particular highest and lowest weight representations are constructed an analog to the kac determinant formula is derived and the implications for unitarity are discussed

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

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

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

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

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

principal type scheme and unification for intersection type discipline abstract the intersection type discipline for the calculus itd is an extension of the classical functionality theory of curry in the itd a term satisfying a given property has a principal type scheme in an extended meaning i e there is a type scheme deducible for it from which all and only the type schemes deducible for it are reachable by means of suitable operations the problem of finding the principal type scheme for a term if it exists is semidecidable in the paper a procedure is shown building the principal type scheme of a term through the construction of the most general unifier for intersection type schemes

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

intersection types for λ trees we introduce a type assignment system which is parametric with respect to five families of trees obtained by evaluating terms bhm trees lvy longo trees etc then we prove in an almost uniform way that each type assignment system fully describes the observational equivalences induced by the corresponding tree representation of terms more precisely for each family of trees two terms have the same tree if and only if they get assigned the same types in the corresponding type assignment system copyright 2002 elsevier science b v

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

intersection types for combinatory logic abstract two different translations of the usual formulation of intersection types for calculus into combinatory logic are proposed in the first one the rule is unchanged while in the second one the rule is replaced by three new rules and four axiom schemes which seem to be simpler than rule itself

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

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

on quasicrystal lie algebras we realize the aperiodic witt and virasoro algebras as well as other quasicrystal lie algebras as factor algebras of some subalgebras of the higher rank virasoro algebras this realization allows us to generalize the notion of quasicrystal lie algebras in the case when the constructed algebra admits a conjugation we compute the kac determinant for the shapovalov form on the corresponding verma modules in the case of the aperiodic virasoro algebra this proves the conjecture of r twarock

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

characterization theorems for a filter lambda model the filter model is a model of the calculus based on a system of type assignment which extends the basic functionality theory of curry invented in order to give a completeness proof for scott s semantics of curry s type assignment in this paper the local structure of this model i e the syntactical characterization of the equality and inclusion relation is given

symmetries of quasicrystals we consider tiling models of round quasicrystals which would have diffraction patterns which are fully rotation invariant rings instead of bragg peaks they can be distinguished from glasses by self similarity of the pattern of radii of the rings

quasicrystals atomic coverings and windows are dual projects in the window approach to quasicrystals the atomic position space e is embedded into a space en e e windows are attached to points of a lattice en for standard fivefold and icosahedral tiling models the windows are perpendicular projections of dual voronoi and delone cells from their cuts by the position space e mark tiles and atomic positions in the alternative covering approach the position space is covered by overlapping copies of a quasi unit cell which carries a fixed atomic configuration the covering and window approach to quasicrystals are shown to be dual projects d and v clusters are defined as projections to position space e of delone or voronoi cells decagonal v clusters in the penrose tiling related to the decagon covering and two types of pentagonal d clusters in the triangle tiling of fivefold point symmetry with their windows are analysed they are linked cover position space and have definite windows for functions compatible with the tilings they form domains of definition for icosahedral tilings the v clusters are kepler triacontahedra the d clusters are two icosahedra and one dodecahedron

virasoro type algebras associated with a penrose tiling a family of infinite dimensional lie algebras with generators in a one to one correspondence with the points of a penrose tiling is introduced central extensions leading to virasoro type algebras are constructed and highest weight representations for these algebras are considered furthermore extensions to a super symmetric setting and thus aperiodic analogues to virasoro super algebras are discussed

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

infinite l calculus and types recent work on infinitary versions of the lambda calculus has shown that the infinite lambda calculus can be a useful tool to study the unsolvable terms of the classical lambda calculus working in the framework of the intersection type disciplines we devise a type assignment system such that two terms are equal in the infinite lambda calculus iff they can be assigned the same types in any basis a novel feature of the system is the presence of a type constant to denote the set of all terms of order zero and the possibility of applying a type to another type we prove a completeness and an approximation theorem for our system our results can be considered as a first step towards the goal of giving a denotational semantics for the lambda calculus which is suited for the study of the unsolvable terms however some noncontinuity phenomena of the infinite lambda calculus make a full realization of this idea namely the construction of a filter model a quite difficult task

intersection and union types type assignment systems with intersection and union types are introduced although the subject reduction property with respect to s reduction does not hold for a natural deduction like system we manage to overcome this problem in two different ways the first is to adopt a notion of parallel reduction which is a refinement of gross knuth reduction the second is to introduce type theories to refine the system among which is the theory called that induces an assignment system preserving s reduction this type assignment system further clarifies the relation with the intersection discipline through the decomposition first of a disjunctive type into a set of conjunctive types and second of a derivation in the new type assignment system into a set of derivations in the intersection type assignment system for this system we propose three semantics and prove soundness and completeness theorems

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

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

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