design of capacity approaching irregular low density parity check codes we design low density parity check ldpc codes that perform at rates extremely close to the shannon capacity the codes are built from highly irregular bipartite graphs with carefully chosen degree patterns on both sides our theoretical analysis of the codes is based on the work of richardson and urbanke see ibid vol 47 no 2 p 599 618 2000 assuming that the underlying communication channel is symmetric we prove that the probability densities at the message nodes of the graph possess a certain symmetry using this symmetry property we then show that under the assumption of no cycles the message densities always converge as the number of iterations tends to infinity furthermore we prove a stability condition which implies an upper bound on the fraction of errors that a belief propagation decoder can correct when applied to a code induced from a bipartite graph with a given degree distribution our codes are found by optimizing the degree structure of the underlying graphs we develop several strategies to perform this optimization we also present some simulation results for the codes found which show that the performance of the codes is very close to the asymptotic theoretical bounds

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

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

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

high rate girth eight low density parity check codes on rectangular integer lattices this letter introduces a combinatorial construction of girth eight high rate low density parity check codes based on integer lattices the parity check matrix of a code is defined as a point line incidence matrix of a 1 configuration based on a rectangular integer lattice and the girth eight property is achieved by a judicious selection of sets of parallel lines included in a configuration a class of codes with a wide range of lengths and column weights is obtained the resulting matrix of parity checks is an array of circulant matrices

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

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

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

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

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

on finite length performance of polar codes stopping sets error floor and concatenated design this paper investigates properties of polar codes that can be potentially useful in real world applications we start with analyzing the performance of finite length polar codes over the binary erasure channel bec while assuming belief propagation as the decoding method we provide a stopping set analysis for the factor graph of polar codes where we find the size of the minimum stopping set we also find the girth of the graph for polar codes our analysis along with bit error rate ber simulations demonstrate that finite length polar codes show superior error floor performance compared to the conventional capacity approaching coding techniques in order to take advantage from this property while avoiding the shortcomings of polar codes we consider the idea of combining polar codes with other coding schemes we propose a polar code based concatenated scheme to be used in optical transport networks otns as a potential real world application comparing against conventional concatenation techniques for otns we show that the proposed scheme outperforms the existing methods by closing the gap to the capacity while avoiding error floor and maintaining a low complexity at the same time

a recursive approach to low complexity codes a method is described for constructing long error correcting codes from one or more shorter error correcting codes referred to as subcodes and a bipartite graph a graph is shown which specifies carefully chosen subsets of the digits of the new codes that must be codewords in one of the shorter subcodes lower bounds to the rate and the minimum distance of the new code are derived in terms of the parameters of the graph and the subeodes both the encoders and decoders proposed are shown to take advantage of the code s explicit decomposition into subcodes to decompose and simplify the associated computational processes bounds on the performance of two specific decoding algorithms are established and the asymptotic growth of the complexity of decoding for two types of codes and decoders is analyzed the proposed decoders are able to make effective use of probabilistic information supplied by the channel receiver e g reliability information without greatly increasing the number of computations required it is shown that choosing a transmission order for the digits that is appropriate for the graph and the subcodes can give the code excellent burst error correction abilities the construction principles

lagrangian subcategories and braided tensor equivalences of twisted quantum doubles of finite groups we classify lagrangian subcategories of the representation category of a twisted quantum double d g where g is a finite group and is a 3 cocycle on it in view of results of dgno this gives a complete description of all braided tensor equivalent pairs of twisted quantum doubles of finite groups we also establish a canonical bijection between lagrangian subcategories of rep d g and module categories over the category rm vec g omega of twisted g graded vector spaces such that the dual tensor category is pointed this can be viewed as a quantum version of v drinfeld s characterization of homogeneous spaces of a poisson lie group in terms of lagrangian subalgebras of the double of its lie bialgebra d as a consequence we obtain that two group theoretical fusion categories are weakly morita equivalent if and only if their centers are equivalent as braided tensor categories

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

maximum girth cylinder type block circulant ldpc codes in this paper a particular class of block circulant low density parity check bc ldpc codes referred to as cylinder type bc ldpc ctbc ldpc codes is studied we represent a cylinder type parity check matrix h by a graph called the block structure graph of h and denoted by bsg h using the properties of bsg h we show that ctbc matrices with column weight two and girth an arbitrary multiple of 8 can be constructed while for a ctbc matrix h with column weight l 3 this girth cannot exceed 12 an algorithm generating ctbc ldpc codes of arbitrary possible girth is given the algorithm produces a large group of codes with flexible rate length and girth from performance perspective over awgn channel the maximum girth 12 ctbc ldpc codes with column weight at least three generated by the given algorithm are at least as good as their random like counterpart codes and outperform the codes of the same size constructed by successive level growth progressive edge growth and partition and shift methods

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

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

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

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

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

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

on the brauer picard groups of fusion categories we develop methods of computation of the brauer picard groups of fusion categories and apply them to compute such groups for several classes of fusion categories of prime power dimension representation categories of elementary abelian groups with twisted associativity extra special p groups and the kac paljutkin hopf algebra we conclude that many finite groups of lie type occur as composition factors of the brauer picard groups of pointed fusion categories

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

design of regular 2 d sub c ldpc codes over gf q using their binary images in this paper a method to design regular 2 dc ldpc codes over gf q with both good waterfall and error floor properties is presented based on the algebraic properties of their binary image first the algebraic properties of rows of the parity check matrix h associated with a code are characterized and optimized to improve the waterfall then the algebraic properties of cycles and stopping sets associated with the underlying tanner graph are studied and linked to the global binary minimum distance of the code finally simulations are presented to illustrate the excellent performance of the designed codes

ldpc block and convolutional codes based on circulant matrices a class of algebraically structured quasi cyclic qc low density parity check ldpc codes and their convolutional counterparts is presented the qc codes are described by sparse parity check matrices comprised of blocks of circulant matrices the sparse parity check representation allows for practical graph based iterative message passing decoding based on the algebraic structure bounds on the girth and minimum distance of the codes are found and several possible encoding techniques are described the performance of the qc ldpc block codes compares favorably with that of randomly constructed ldpc codes for short to moderate block lengths the performance of the ldpc convolutional codes is superior to that of the qc codes on which they are based this performance is the limiting performance obtained by increasing the circulant size of the base qc code finally a continuous decoding procedure for the ldpc convolutional codes is described

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

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

on the brauer picard group of a finite symmetric tensor category let cn denote the representation category of finite supergroup kn z 2z we compute the brauer picard group brpic cn of cn this is done by identifying brpic cn with the group of braided tensor autoequivalences of the drinfeld center of cn and studying the action of the latter group on the categorical lagrangian grassmannian of cn we show that this action corresponds to the action of a projective symplectic group on a classical lagrangian grassmannian

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

the brauer group of a braided monoidal category representation theory of finite groups prompted the introduction of the brauer group of a field and also of its schur subgroups the theory of quadratic forms and spaces introduced the zr2z graded variant of the brauer group together with the typical clifford algebras and related w x representations cf the work of wall 29 the extension of the zr2zgraded theory to gradings by other cyclic groups and consequently to abelian groups replacing the usual clifford algebras by the so called generalized clifford algebras is then natural from the algebraic point of view however extension to nonabelian groups did not seem to be possible at least not by simple modifications of the abelian theory at the same time the generalization of the brauer group of graded algebras obtained w x by long 12 13 made use of hopf algebras but again cocommutativity conditions were present the recent interest in quantum groups motivated the authors to introduce the brauer group of a quantum group as the brauer group of crossed module algebras also called quantum yang w x baxter module algebras in 4 5 the use of the category of crossed w x modules originated in the case of group algebras cf whitehead 30 module algebras or coalgebras of this crossed type were introduced by w x radford in 24 and used by majid in the description of modules over drinfel d quantum doubles the noncocommutativity aspects of the theory concerning the brauer group in terms of crossed module algebras do not have a counterpart in classical theory note that a first unifying categorical theory has been w x obtained by pareigis in 23 this theory deals with the brauer group of a

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

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

a decomposition of the brauer picard group of the representation category of a finite group abstract we present an approach of calculating the group of braided autoequivalences of the category of representations of the drinfeld double of a finite dimensional hopf algebra h and thus the brauer picard group of h mod we consider two natural subgroups and a subset as candidates for generators in this article h is the group algebra of a finite group g as our main result we prove that any element of the brauer picard group fulfilling an additional cohomological condition decomposes into an ordered product of our candidates for elementary abelian groups g our decomposition reduces to the bruhat decomposition of the brauer picard group which is in this case a lie group over a finite field our results are motivated by and have applications to symmetries and defects in 3d tqft and group extensions of fusion categories

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

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 lattice based systematic recursive construction of quasi cyclic ldpc codes this paper presents a low complexity recursive and systematic method to construct good well structured low density parity check ldpc codes the method is based on a recursive application of a partial kronecker product operation on a given gamma x q q ges 3 a prime integer lattice l gamma x q the n 1 fold product of l gamma x q by itself denoted ln gamma x q represents a regular quasi cyclic qc ldpc code denoted see pdf of high rate and girth 6 the minimum distance of see pdf is equal to that of the core code see pdf introduced by l gamma x q the support of the minimum weight codewords in see pdf are characterized by the support of the same type of codewords in see pdf from performance perspective the constructed codes compete with the pseudorandom ldpc codes

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

categorical lagrangian grassmannians and brauer picard groups of pointed fusion categories abstract we analyze the action of the brauer picard group of a pointed fusion category on the set of lagrangian subcategories of its center using this action we compute the brauer picard groups of pointed fusion categories associated to several classical finite groups as an application we construct examples of weakly group theoretical fusion categories

on fusion categories using a variety of methods developed in the literature in particular the theory of weak hopf algebras we prove a number of general results about fusion categories in characteristic zero we show that the global dimension of a fusion category is always positive and that the s matrix of any not necessarily hermitian modular category is unitary we also show that the category of module functors between two module categories over a fusion category is semisimple and that fusion categories and tensor functors between them are undeformable generalized ocneanu rigidity in particular the number of such categories functors realizing a given fusion datum is finite finally we develop the theory of frobenius perron dimensions in an arbitrary fusion category at the end of the paper we generalize some of these results to positive characteristic

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

quasicyclic low density parity check codes from circulant permutation matrices in this correspondence the construction of low density parity check ldpc codes from circulant permutation matrices is investigated it is shown that such codes cannot have a tanner graph representation with girth larger than 12 and a relatively mild necessary and sufficient condition for the code to have a girth of 6 8 10 or 12 is derived these results suggest that families of ldpc codes with such girth values are relatively easy to obtain and consequently additional parameters such as the minimum distance or the number of redundant check sums should be considered to this end a necessary condition for the codes investigated to reach their maximum possible minimum hamming distance is proposed

quasi cyclic low density parity check codes with girth larger than 12 a quasi cyclic qc low density parity check ldpc code can be viewed as the protograph code with circulant permutation matrices or circulants in this correspondence we find all the subgraph patterns of protographs of qc ldpc codes having inevitable cycles of length 2i i 6 7 8 9 10 i e the cycles that always exist regardless of the shift values of circulants it is also derived that if the girth of the protograph is 2g g 2 its protograph code cannot have the inevitable cycles of length smaller than 6g based on these subgraph patterns we propose new combinatorial construction methods of the protographs whose protograph codes can have girth larger than or equal to 14 or 18 we also propose a couple of shift value assigning rules for circulants of a qc ldpc code guaranteeing the girth 14

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

galois algebras and monoidal functors between categories of representations of finite groups abstract we use relations between galois algebras and monoidal functors to describe monoidal functors between categories of representations of finite groups we pay special attention to two kinds of these monoidal functors monoidal functors to vector spaces and monoidal equivalences between categories of representations the functors of the second kind induce isomorphisms of character tables we show that pairs of groups with the same character table obtained in this way are a generalization of the construction proposed by b fischer 1988 rend circ mat palermo 2 suppl 19 71 77

monoidal 2 structure of bimodule categories abstract we define a notion of tensor product of bimodule categories and prove that with this product the 2 category of c bimodule categories for fixed tensor c is a monoidal 2 category in the sense of kapranov and voevodsky 1991 1 we then provide a monoidal structure preserving 2 equivalence between the 2 category of c bimodule categories and z c module categories module categories over the center of c for a finite group g we show that de equivariantization is equivalent to the tensor product over rep g we derive rep g module fusion rules and show that the group of invertible rep g module categories is isomorphic to h 2 g k extending results in etingof et al 2

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

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

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

shortened array codes of large girth one approach to designing structured low density parity check ldpc codes with large girth is to shorten codes with small girth in such a manner that the deleted columns of the parity check matrix contain all the variables involved in short cycles this approach is especially effective if the parity check matrix of a code is a matrix composed of blocks of circulant permutation matrices as is the case for the class of codes known as array codes we show how to shorten array codes by deleting certain columns of their parity check matrices so as to increase their girth the shortening approach is based on the observation that for array codes and in fact for a slightly more general class of ldpc codes the cycles in the corresponding tanner graph are governed by certain homogeneous linear equations with integer coefficients consequently we can selectively eliminate cycles from an array code by only retaining those columns from the parity check matrix of the original code that are indexed by integer sequences that do not contain solutions to the equations governing those cycles we provide ramsey theoretic estimates for the maximum number of columns that can be retained from the original parity check matrix with the property that the sequence of their indices avoid solutions to various types of cycle governing equations this translates to estimates of the rate penalty incurred in shortening a code to eliminate cycles simulation results show that for the codes considered shortening them to increase the girth can lead to significant gains in signal to noise ratio snr in the case of communication over an additive white gaussian noise awgn channel

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

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

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