lifts of convex sets and cone factorizations in this paper we address the basic geometric question of when a given convex set is the image under a linear map of an affine slice of a given closed convex cone such a representation or lift of the convex set is especially useful if the cone admits an efficient algorithm for linear optimization over its affine slices we show that the existence of a lift of a convex set to a cone is equivalent to the existence of a factorization of an operator associated to the set and its polar via elements in the cone and its dual this generalizes a theorem of yannakakis that established a connection between polyhedral lifts of a polytope and nonnegative factorizations of its slack matrix symmetric lifts of convex sets can also be characterized similarly when the cones live in a family our results lead to the definition of the rank of a convex set with respect to this family we present results about this rank in the context of cones of positive semidefinite matrices our methods provide new tools for understanding cone lifts of convex sets

completely positive reformulations for polynomial optimization polynomial optimization encompasses a very rich class of problems in which both the objective and constraints can be written in terms of polynomials on the decision variables there is a well established body of research on quadratic polynomial optimization problems based on reformulations of the original problem as a conic program over the cone of completely positive matrices or its conic dual the cone of copositive matrices as a result of this reformulation approach novel solution schemes for quadratic polynomial optimization problems have been designed by drawing on conic programming tools and the extensively studied cones of completely positive and of copositive matrices in particular this approach has been applied to solve key combinatorial optimization problems along this line of research we consider polynomial optimization problems that are not necessarily quadratic for this purpose we use a natural extension of the cone of completely positive matrices namely the cone of completely positive tensors we provide a general characterization of the class of polynomial optimization problems that can be formulated as a conic program over the cone of completely positive tensors as a consequence of this characterization it follows that recent related results for quadratic problems can be further strengthened and generalized to higher order polynomial optimization problems also we show that the conditions underlying the characterization are conceptually the same regardless of the degree of the polynomials defining the problem to illustrate our results we discuss in further detail special and relevant instances of polynomial optimization problems

computing m dfts over gf q with one dft over gf q sup m over the field of complex numbers it is well known that if the input is real then it is possible to compute two real dfts with one complex dft the authors extend the result to finite fields and show how to compute m dfts over gf q with one dft over gf q sup m

global optimization of rational functions a semidefinite programming approach we consider the problem of global minimization of rational functions on unconstrained case and on an open connected semi algebraic subset of or the partial closure of such a set constrained case we show that in the univariate case n 1 these problems have exact reformulations as semidefinite programming sdp problems by using reformulations introduced in the phd thesis of jibetean 16 this extends the analogous results by nesterov 13 for global minimization of univariate polynomials for the bivariate case n 2 we obtain a fully polynomial time approximation scheme fptas for the unconstrained problem if an a priori lower bound on the infimum is known by using results by de klerk and pasechnik 1 for the np hard multivariate case we discuss semidefinite programming based relaxations for obtaining lower bounds on the infimum by using results by parrilo 15 and lasserre 12

a new approach to subquadratic space complexity parallel multipliers for extended binary fields based on toeplitz matrix vector products and coordinate transformation techniques we present a new scheme for subquadratic space complexity parallel multiplication in gf 2n using the shifted polynomial basis both the space complexity and the asymptotic gate delay of the proposed multiplier are better than those of the best existing subquadratic space complexity parallel multipliers for example with n being a power of 2 the space complexity is about 8 percent better while the asymptotic gate delay is about 33 percent better respectively another advantage of the proposed matrix vector product approach is that it can also be used to design subquadratic space complexity polynomial dual weakly dual and triangular basis parallel multipliers to the best of our knowledge this is the first time that subquadratic space complexity parallel multipliers are proposed for dual weakly dual and triangular bases a recursive design algorithm is also proposed for efficient construction of the proposed subquadratic space complexity multipliers this design algorithm can be modified for the construction of most of the subquadratic space complexity multipliers previously reported in the literature

a quadratically constrained quadratic optimization model for completely positive cone programming we propose a class of quadratic optimization problems which can be reformulated by completely positive cone programming with the same optimal values the objective function can be any quadratic form the constraints of each problem are described in terms of quadratic forms with no linear terms and all constraints are homogeneous equalities except one inhomogeneous equality where a quadratic form is set to be a positive constant for the equality constraints a hierarchy of copositvity condition is assumed this model is a generalization of the standard quadratic optimization problem of minimizing a quadratic form over the standard simplex and covers many of the existing quadratic optimization problems studied for exact copositive cone and completely positive cone programming relaxations in particular it generalizes the recent results on quadratic optimization problems by burer and the set semidefinite representation by eichfelder and povh optim lett 7 2013 pp 1373 1386

fft algorithm for binary extension finite fields and its application to reed solomon codes recently a new polynomial basis over binary extension fields was proposed such that the fast fourier transform fft over such fields can be computed in the complexity of order mathcal o n lg n where n is the number of points evaluated in fft in this paper we reformulate this fft algorithm such that it can be easier understood and be extended to develop frequency domain decoding algorithms for n 2 m k systematic reed solomon rs codes over mathbb f 2 m m in mathbb z with n k a power of two first the basis of syndrome polynomials is reformulated in the decoding procedure so that the new transforms can be applied to the decoding procedure a fast extended euclidean algorithm is developed to determine the error locator polynomial the computational complexity of the proposed decoding algorithm is mathcal o n lg n k n k lg 2 n k improving upon the best currently available decoding complexity mathcal o n lg 2 n lg lg n and reaching the best known complexity bound that was established by justesen in 1976 however justesen s approach is only for the codes over some specific fields which can apply cooley tukey ffts as revealed by the computer simulations the proposed decoding algorithm is 50 times faster than the conventional one for the 2 16 2 15 rs code over mathbb f 2 16

an explicit equivalent positive semidefinite program for nonlinear 0 1 programs we consider the general nonlinear optimization problem in 0 1 variables and provide an explicit equivalent positive semidefinite program in 2n 1 variables the optimal values of both problems are identical from every optimal solution of the former one easily finds an optimal solution of the latter and conversely from every solution of the latter one may construct an optimal solution of the former for illustration the equivalent positive semidefinite program is explicated for quadratic 0 1 programs and max cut in mathbb r 3 for unconstrained 0 1 programs a special representation in terms of a weighted sum of squares is provided

doubly nonnegative relaxations for quadratic and polynomial optimization problems with binary and box constraints we propose a doubly nonnegative dnn relaxation for polynomial optimization problems pops with binary and box constraints this work is an extension of the work by kim kojima and toh in 2016 from quadratic optimization problems to pops the dense and sparse dnn relaxations are reduced to a simple conic optimization problem cop to which an accelerated bisection and projection bp algorithm is applied the cop involves a single equality constraint in a matrix variable which is restricted to the intersection of the positive semidefinite cone and a polyhedral cone representing the algebraic properties of binary and box constraints as well as the sparsity structure of the objective polynomial our dnn relaxation serves as a variant of the lasserre moment sos hierarchy of relaxations for binary and box constrained pops when the size of the cones is gradually increased to compute tighter lower bounds for their optimal values a significant part of the paper is devoted to deriving and analyzing a class of polyhedral cones to strengthen the dnn relaxation and the design of efficient computation of the metric projections onto these cones in the accelerated bp algorithm numerical results on various pops are available in ito et al acm trans math softw 45 article 34 2019

fast algorithm for computing the roots of error locator polynomials up to degree 11 in reed solomon decoders the central problem in the implementation of a reed solomon code is finding the roots of the error locator polynomial in 1967 berlekamp et al found an algorithm for finding the roots of an affine polynomial in gf 2 sup m that can be used to solve this problem in this paper it is shown that this berlekamp rumsey solomon 1967 algorithm together with the chien 1964 search method makes possible a fast decoding algorithm in the standard basis representation that is naturally suitable in a software implementation finally simulation results for this fast algorithm are given

semidefinite programming in combinatorial optimization we discuss the use of semidefinite programming for combinatorial optimization problems the main topics covered include i the lovasz theta function and its applications to stable sets perfect graphs and coding theory ii the automatic generation of strong valid inequalities iii the maximum cut problem and related problems and iv the embedding of finite metric spaces and its relationship to the sparsest cut problem

reduced complexity decoders of long reed solomon codes based on composite cyclotomic fourier transforms long reed solomon rs codes are desirable for digital communication and storage systems due to their improved error performance but the high computational complexity of their decoders is a key obstacle to their adoption in practice as discrete fourier transforms dfts can evaluate a polynomial at multiple points efficient dft algorithms are promising in reducing the computational complexities of syndrome based decoders for long rs codes in this correspondence we first propose partial composite cyclotomic fourier transforms ccfts and then devise syndrome based decoders for long rs codes over large finite fields based on partial ccfts the new decoders based on partial ccfts achieve a significant saving of computational complexities for long rs codes in comparison to previous results based on horner s rule our hardware implementation for a 2720 2550 shortened rs code over gf 212 achieves much higher throughputs and better area time complexity

convex relaxations for nonconvex quadratically constrained quadratic programming matrix cone decomposition and polyhedral approximation we present a decomposition approximation method for generating convex relaxations for nonconvex quadratically constrained quadratic programming qcqp we first develop a general conic program relaxation for qcqp based on a matrix decomposition scheme and polyhedral piecewise linear underestimation by employing suitable matrix cones we then show that the convex conic relaxation can be reduced to a semidefinite programming sdp problem in particular we investigate polyhedral underestimations for several classes of matrix cones including the cones of rank 1 and rank 2 matrices the cone generated by the coefficient matrices the cone of positive semidefinite matrices and the cones induced by rank 2 semidefinite inequalities we demonstrate that in general the new sdp relaxations can generate lower bounds at least as tight as the best known sdp relaxations for qcqp moreover we give examples for which tighter lower bounds can be generated by the new sdp relaxations we also report comparison results of different convex relaxation schemes for nonconvex qcqp with convex quadratic linear constraints nonconvex quadratic constraints and 0 1 constraints

generalized lagrangian duals and sums of squares relaxations of sparse polynomial optimization problems sequences of generalized lagrangian duals and their sums of squares sos of polynomials relaxations for a polynomial optimization problem pop are introduced the sparsity of polynomials in the pop is used to reduce the sizes of the lagrangian duals and their sos relaxations it is proved that the optimal values of the lagrangian duals in the sequence converge to the optimal value of the pop using a method from the penalty function approach the sequence of sos relaxations is transformed into a sequence of semidefinite programing sdp relaxations of the pop which correspond to duals of modification and generalization of sdp relaxations given by lasserre for the pop

a lagrangian dnn relaxation a fast method for computing tight lower bounds for a class of quadratic optimization problems we propose an efficient computational method for linearly constrained quadratic optimization problems qops with complementarity constraints based on their lagrangian and doubly nonnegative dnn relaxation and first order algorithms the simplified lagrangian completely positive programming cpp relaxation of such qops proposed by arima kim and kojima in 2012 takes one of the simplest forms an unconstrained conic linear optimization problem with a single lagrangian parameter in a cpp matrix variable with its upper left element fixed to 1 replacing the cpp matrix variable by a dnn matrix variable we derive the lagrangian dnn relaxation and establish the equivalence between the optimal value of the dnn relaxation of the original qop and that of the lagrangian dnn relaxation we then propose an efficient numerical method for the lagrangian dnn relaxation using a bisection method combined with the proximal alternating direction multiplier and the accelerated proximal gradient methods numerical results on binary qops quadratic multiple knapsack problems maximum stable set problems and quadratic assignment problems illustrate the superior performance of the proposed method for attaining tight lower bounds in shorter computational time

duhamel hollmann like discrete fourier transform algorithm with the smallest multiplicative complexity over a finite field the new method for the discrete fourier transform computation over a finite field is introduced this method is a nontrivial generalization of the duhamel hollmann algorithm with replacement of the toeplitz convolution calculation by the normalized cyclic convolution calculation both algorithms have the smallest multiplicative complexity

quaternionic differential operators motivated by a quaternionic formulation of quantum mechanics we discuss quaternionic and complex linear differential equations we touch only a few aspects of the mathematical theory namely the resolution of the second order differential equations with constant coefficients we overcome the problems coming out from the loss of the fundamental theorem of the algebra for quaternions and propose a practical method to solve quaternionic and complex linear second order differential equations with constant coefficients the resolution of the complex linear schrodinger equation in the presence of quaternionic potentials represents an interesting application of the mathematical material discussed in this paper

quaternionic symmetry groups and particle multiplets we consider the quaternionic complete symmetry group of a massive physical system obtained extending the connected poincare group and the internal symmetry group by means of the cpt and the generalized parity operators we classify the irreducible q representations of this group crossing the generalized wigner and frobenius schur classifications and obtain 14 different cases some novelties arise in this context such as the failure of the statement that only irreducible representations must be associated with particle multiplets and a suggestion on the possible forms of a parity violating hamiltonian

pseudoanti hermitian operators in quaternionic quantum mechanics we introduce the concept of pseudoanti hermitian operators in quaternionic quantum mechanics and give a complete characterization of their spectra we highlight some physical properties related to time reversal symmetry of the pseudoanti hermitian quaternionic hamiltonians

some complexity issues in digital signal processing over the past decade a large class of problems called np complete 5 have been shown to be equivalent in the sense that if a fast algorithm can be found for one fast algorithms can be found for all at the same time despite much effort no fast algorithms have been found for any and these problems are widely regarded as intractable this class includes such notoriously difficult problems as the traveling salesman problem graph coloring and satisfiability of boolean expressions using fir filter implementation as an illustration we describe some problems in digital signal processing that are np complete these include 1 minimize the number of additions needed to implement a fixed fir filter 2 minimize the number of registers needed to implement a fixed fir filter and 3 minimize the time to perform the additions of such an fir filter using p adders large seale instances of such problems may become important with the use of programmable chips to implement signal processing our main purpose in this paper is to illustrate the usefulness of asymptotic complexity theory in the field of digital signal processing the theory discriminates between tractable and intractable problems sometimes identifies fast algorithms for the former and justifies heuristics for the latter

alternative descriptions in quaternionic quantum mechanics we characterize the quasianti hermitian quaternionic operators in quaternionic quantum mechanics by means of their spectra moreover we state a necessary and sufficient condition for a set of quasianti hermitian quaternionic operators to be anti hermitian with respect to a uniquely defined positive scalar product in a infinite dimensional right quaternionic hilbert space according to such results we obtain two alternative descriptions of a quantum optical physical system in the realm of quaternionic quantum mechanics while no alternative can exist in complex quantum mechanics and we discuss some differences between them

a lift and project cutting plane algorithm for mixed 0 1 programs we propose a cutting plane algorithm for mixed 0 1 programs based on a family of polyhedra which strengthen the usual lp relaxation we show how to generate a facet of a polyhedron in this family which is most violated by the current fractional point this cut is found through the solution of a linear program that has about twice the size of the usual lp relaxation a lifting step is used to reduce the size of the lp s needed to generate the cuts an additional strengthening step suggested by balas and jeroslow is then applied we report our computational experience with a preliminary version of the algorithm this approach is related to the work of balas on disjunctive programming the matrix cone relaxations of lovasz and schrijver and the hierarchy of relaxations of sherali and adams

an encoding algorithm of triply extended reed solomon codes with asymptotically optimal complexities in this paper we devise a fast encoding algorithm for triply extended reed solomon codes the proposed approach requires approximately two xors per bit which improves the prior result of three xors per bit established by certain maximum distance separable mds array codes we also prove that for mds codes with two and three parities the scheduling algorithms require at least two xors per bit to the best of our knowledge this is the first provable scheduling algorithm for the triple parity mds codes to approach the theoretical lower bounds the implementation with simd instructions is provided the simulations show that the proposed approach is competitive as compared with other cutting edge implementations

robustness analysis of trusses with separable load and structural uncertainties this paper discusses evaluation techniques of the robustness function of trusses which is regarded as one of measures of structural robustness under the uncertainties of member stiffnesses and external forces by using quadratic embedding of the uncertainty and the s procedure we formulate a quasiconvex optimization problem which provides lower bounds of the robustness functions a bisection method is proposed where we solve a finite number of semidefinite programming problems in order to obtain a global optimal solution to the proposed quasiconvex optimization problem the lower bounds of the robustness functions are computed for various trusses under several uncertainty circumstances

fast transform for decoding both errors and erasures of reed solomon codes over gf 2 sup m for 8 spl les m spl les 10 in this letter it is shown that a fast prime factor discrete fourier transform dft algorithm can be modified to compute fourier like transforms of long sequences of 2 sup m 1 points over gf 2 sup m where 8 spl les m spl les 10 using these transforms together with the berlekamp massey algorithm the complexity of the transform domain decoder for correcting both errors and erasures of the reed solomon codes of block length 2 sup m 1 over gf 2 sup m for 8 spl les m spl les 10 is reduced substantially from the previous time domain decoder a computer simulation verifies these new results

on the reduction of additive complexity of cyclotomic ffts we investigate a property that we have found in many efficient bilinear cyclic convolutions in gf 2m we show that this property can reduce the additive complexity of cyclotomic ffts we explain how it arises and why the most common constructions of cyclic convolutions yield this beneficial feature

cones of matrices and set functions and 0 1 optimization it has been recognized recently that to represent a polyhedron as the projection of a higher dimensional but simpler polyhedron is a powerful tool in polyhedral combinatorics a general method is developed to construct higher dimensional polyhedra or in some cases convex sets whose projection approximates the convex hull of 0 1 valued solutions of a system of linear inequalities an important feature of these approximations is that one can optimize any linear objective function over them in polynomial time in the special case of the vertex packing polytope a sequence of systems of inequalities is obtained such that the first system already includes clique odd hole odd antihole wheel and orthogonality constraints in particular for perfect and many other graphs this first system gives the vertex packing polytope for various classes of graphs including t perfect graphs it follows that the stable set polytope is the projection of a polytope with a polynomial number of facets an extension o

generalized fs w versus dyson s classification of irreducible representations we study the characterization of the 13 cases obtained in the classification of the irreducible linear antilinear representations of semigroups in a finite dimensional vector space x over an algebraically closed field k with a conjugation j generalized frobenius schur wigner or fs w classification it has already been shown that each case can be characterized by various equivalent properties some of which can be endowed with a physical interpretation we show here that whenever k is the complex field c each case can be characterized by the structure in the sense specified by the weyl theorem on the structure of the matrix algebras and their commutators of a pair of suitable operator algebras over the real field r this characterization coincides with the one given by dyson for each case of his classification of symmetry groups thus the latter classification is recovered under more general assumptions and in a generalized framework and its one to one correspondence with the generalized fs w cla

on a representation of the matching polytope via semidefinite liftings we consider the relaxation of the matching polytope defined by the non negativity and degree constraints we prove that given an undirected graph on n nodes and the corresponding relaxation of the matching polytope n 2 iterations of the lovasz schrijver semidefinite lifting procedure are needed to obtain the matching polytope in the worst case we show that n 2 iterations of the procedure always suffice

on algorithms and complexities of cyclotomic fast fourier transforms over arbitrary finite fields discrete fourier transforms over finite fields are significant due to their widespread applications in cryptography and error control codes which in turn are used in all digital communication and storage systems cyclotomic fast fourier transforms cffts are of great interest due to their low multiplicative complexities however all existing cffts are for characteristic 2 fields and the computational complexities of cffts have not been analyzed theoretically this paper addresses both problems for cffts and has three main contributions to this end first we propose an efficient bilinear algorithm to compute toeplitz matrix vector products tmvps which has a lower computational complexity than existing algorithms and works on all finite fields as well as the real and complex fields second we propose an efficient algorithm for cyclic convolutions over arbitrary finite fields which is essential in deriving efficient cffts over arbitrary finite fields finally we derive bounds on the additive and multiplicative complexities of cffts over arbitrary finite fields our results confirm that cffts have the smallest multiplicative complexities among all known algorithms although their high additive complexities render them asymptotically suboptimal cffts remain valuable since they have the smallest overall complexities for dfts of up to thousands of symbols in most cases

on minimal valid inequalities for mixed integer conic programs we study disjunctive conic sets involving a general regular closed convex full dimensional and pointed cone x1d4a6 such as the nonnegative orthant the lorentz cone or the positive semidefinite cone in a unified framework we introduce x1d4a6 minimal inequalities and show that under mild assumptions these inequalities together with the trivial cone implied inequalities are sufficient to describe the convex hull we focus on the properties of x1d4a6 minimal inequalities by establishing algebraic necessary conditions for an inequality to be x1d4a6 minimal this characterization leads to a broader algebraically defined class of x1d4a6 sublinear inequalities we demonstrate a close connection between x1d4a6 sublinear inequalities and the support functions of convex sets with a particular structure this connection results in practical ways of verifying x1d4a6 sublinearity and or x1d4a6 minimality of inequalities our study generalizes some of the results from the mixed integer linear case it is well known that the minimal inequalities for

on the finite convergence of successive sdp relaxation methods abstract let f be a subset of the n dimensional euclidean space r n represented in terms of a compact convex subset of r n and a set of finitely or infinitely many quadratic inequalities this paper investigates some fundamental properties related to the finite convergence of the successive semidefinite programming relaxation method proposed by the authors for approximating the convex hull of f

approximation of the stability number of a graph via copositive programming lovasz and schrijver siam j optim 1 1991 pp 166 190 showed how to formulate increasingly tight approximations of the stable set polytope of a graph by solving semidefinite programs sdps of increasing size lift and project method in this paper we present a similar idea we show how the stability number can be computed as the solution of a conic linear program lp over the cone of copositive matrices subsequently we show how to approximate the copositive cone ever more closely via a hierarchy of linear or semidefinite programs of increasing size liftings the latter idea is based on recent work by parrilo structured semidefinite programs and semi algebraic geometry methods in robustness and optimization ph d thesis california institute of technology pasadena ca 2000 in this way we can compute the stability number alpha g of any graph g v e after at most alpha g 2 successive liftings for the lp based approximations one can compare this to the n alpha g 1 bound for the lp based lift and project scheme of lovasz and schrijver our approach therefore requires fewer liftings for families of graphs where alpha g o sqrt n we show that the first sdp based approximation for alpha g in our series of increasingly tight approximations coincides with the vartheta function of schrijver ieee trans inform theory 25 1979 pp 425 429 we further show that the second approximation is tight for complements of triangle free graphs and for odd cycles

normalized cyclic convolution the case of even length a novel method for computation of the discrete fourier transform over a finite field with reduced multiplicative complexity is described if the number of multiplications is to be minimized then the novel method for the finite field of even extension degree is the best known method of the discrete fourier transform computation a constructive method of constructing for a cyclic convolution over a finite field is introduced

intersection cuts for nonlinear integer programming convexification techniques for structured sets we study the generalization of split k branch split and intersection cuts from mixed integer linear programming to the realm of mixed integer nonlinear programming constructing such cuts requires calculating the convex hull of the difference between a convex set and an open set with a simple geometric structure we introduce two techniques to give precise characterizations of such convex hulls and use them to construct split k branch split and intersection cuts for several classes of non polyhedral sets in particular we give simple formulas for split cuts for essentially all convex sets described by a single conic quadratic inequality we also give simple formulas for k branch split cuts and some general intersection cuts for a wide variety of convex quadratic sets

t violation and quaternionic state oscillations we derive using the quaternionic group representation theory the possible forms of the t violating hamiltonians and consider time evolution in a two level system when a t violating term is added to a t preserving unperturbed hamiltonian transition probabilities are calculated moreover the time evolution of a complex state is determined analytically showing that a quaternionic component arises naturally in its evolution finally some properties of the time reversal observable are studied and a set of selection rules is obtained

irreducible linear antilinear representations and internal symmetries we reconsider a well known classification due to wigner of the unitary antiunitary finite dimensional irreducible group representations within a somewhat generalized mathematical framework where in particular any algebraically closed field k with an involutory automorphism j is considered in place of the complex field c we show that each case of the classification can be characterized by the set u l of the linear mappings that commute with the given set u which is now assumed to be an irreducible semigroup of linear and antilinear i e j semilinear mappings and explicitly exhibit u l then this classification is crossed with the classification of the sets of linear and antilinear mappings that has been obtained in some previous work and that generalizes the old classification of the unitary representations introduced by frobenius and schur we obtain a new classification in which every case can be characterized by the group uc of the invertible linear and antilinear mappings which commute with

fast prime factor discrete fourier transform algorithms over gf 2m for 8 m 10 in this paper it is shown that winograd s algorithm for computing convolutions and a fast prime factor discrete fourier transform dft algorithm can be modified to compute fourier like transforms of long sequences of 2 m 1 points over gf 2 m for 8

on the copositive representation of binary and continuous nonconvex quadratic programs in this paper we model any nonconvex quadratic program having a mix of binary and continuous variables as a linear program over the dual of the cone of copositive matrices this result can be viewed as an extension of earlier separate results which have established the copositive representation of a small collection of np hard problems a simplification which reduces the dimension of the linear conic program and an extension to complementarity constraints are established and computational issues are discussed

polynomials nonnegative on a grid and discrete optimization we characterize the real valued polynomials on r n that are nonnegative not necessarily strictly positive on a grid k of points of r n in terms of a weighted sum of squares whose degree is bounded and known in advance we also show that the mimimization of an arbitrary polynomial on k a discrete optimization problem reduces to a convex continuous optimization problem of fixed size the case of concave polynomials is also investigated the proof is based on a recent result of curto and fialkow on the k moment problem

composite cyclotomic fourier transforms with reduced complexities discrete fourier transforms dfts over finite fields have widespread applications in digital communication and storage systems hence reducing the computational complexities of dfts is of great significance recently proposed cyclotomic fast fourier transforms cffts are promising due to their low multiplicative complexities unfortunately there are two issues with cffts 1 they rely on efficient short cyclic convolution algorithms which have not been sufficiently investigated in the literature and 2 they have very high additive complexities when directly implemented to address both issues we make three main contributions in this paper first for any odd prime p we reformulate a p point cyclic convolution as the product of a p 1 p 1 toeplitz matrix and a vector which has well known efficient algorithms leading to efficient bilinear algorithms for p point cyclic convolutions second to address the high additive complexities of cffts we propose composite cyclotomic fourier transforms ccfts in comparison to previously proposed fast fourier transforms our ccfts achieve lower overall complexities for moderate to long lengths and the improvement significantly increases as the length grows third our efficient algorithms for p point cyclic convolutions and ccfts allow us to obtain longer dfts over larger fields e g the 2047 point dft over gf 211 and 4095 point dft over gf 212 which are first efficient dfts of such lengths to the best of our knowledge finally our ccfts are also advantageous for hardware implementations due to their modular structure

right eigenvalue equation in quaternionic quantum mechanics we study the right eigenvalue equation for quaternionic and complex linear matrix operators defined in n dimensional quaternionic vector spaces for quaternionic linear operators the eigenvalue spectrum consists of n complex values for these operators we give a necessary and sufficient condition for the diagonalization of their quaternionic matrix representations our discussion is also extended to complex linear operators whose spectrum is characterized by 2n complex eigenvalues we show that a consistent analysis of the eigenvalue problem for complex linear operators requires the choice of a complex geometry in defining inner products finally we introduce some examples of the left eigenvalue equations and highlight the main difficulties in their solution

a cone programming approach to the bilinear matrix inequality problem and its geometry we discuss an approach for solving the bilinear matrix inequality bmi based on its connections with certain problems defined over matrix cones these problems are among others the cone generalization of the linear programming lp and the linear complementarity problem lcp referred to as the cone lp and the cone lcp respectively specifically we show that solving a given bmi is equivalent to examining the solution set of a suitably constructed cone lp or cone lcp this approach facilitates our understanding of the geometry of the bmi and opens up new avenues for the development of the computational procedures for its solution

strong duality for a trust region type relaxation of the quadratic assignment problem abstract lagrangian duality underlies many efficient algorithms for convex minimization problems a key ingredient is strong duality lagrangian relaxation also provides lower bounds for non convex problems where the quality of the lower bound depends on the duality gap quadratically constrained quadratic programs qqps provide important examples of non convex programs for the simple case of one quadratic constraint the trust region subproblem strong duality holds in addition necessary and sufficient strengthened second order optimality conditions exist however these duality results already fail for the two trust region sub problem surprisingly there are classes of more complex non convex qqps where strong duality holds one example is the special case of orthogonality constraints which arise naturally in relaxations for the quadratic assignment problem qap in this paper we show that strong duality also holds for a relaxation of qap where the orthogonality constraint is replaced by a semidefinite inequality constraint using this strong duality result and semidefinite duality we develop new trust region type necessary and sufficient optimality conditions for these problems our proof of strong duality introduces and uses a generalization of the hoffman wielandt inequality

geometry of the copositive and completely positive cones the copositive cone and its dual the completely positive cone have useful applications in optimisation however telling if a general matrix is in the copositive cone is a co np complete problem in this paper we analyse some of the geometry of these cones we discuss a way of representing all the maximal faces of the copositive cone along with a simple equation for the dimension of each one in doing this we show that the copositive cone has faces which are isomorphic to positive semidefinite cones we also look at some maximal faces of the completely positive cone and find their dimensions additionally we consider extreme rays of the copositive and completely positive cones and show that every extreme ray of the completely positive cone is also an exposed ray but the copositive cone has extreme rays which are not exposed rays

a finite branch and bound algorithm for nonconvex quadratic programming via semidefinite relaxations existing global optimization techniques for nonconvex quadratic programming qp branch by recursively partitioning the convex feasible set and thus generate an infinite number of branch and bound nodes an open question of theoretical interest is how to develop a finite branch and bound algorithm for nonconvex qp one idea which guarantees a finite number of branching decisions is to enforce the first order karush kuhn tucker kkt conditions through branching in addition such an approach naturally yields linear programming lp relaxations at each node however the lp relaxations are unbounded a fact that precludes their use in this paper we propose and study semidefinite programming relaxations which are bounded and hence suitable for use with finite kkt branching computational results demonstrate the practical effectiveness of the method with a particular highlight being that only a small number of nodes are required

a new structured design method for convolutions over finite fields part i the structure of bilinear cyclic convolution algorithms is explored over finite fields the algorithms derived are valid for any length not divisible by the field characteristic and are based upon the small length polynomial multiplication algorithms the multiplicative complexity of these algorithms is small and depends on the field of constants the linear transformation matrices a b premultiplication and c postmultiplication defining the algorithm have block structures which are related to one another the rows of a and b and the columns of c are maximal length recurrent sequences because of the highly regular structure of a b and c the algorithms can be very easily designed even for large lengths the application of these algorithms to the decoding of reed solomon codes is also examined

exploiting symmetry in copositive programs via semidefinite hierarchies copositive programming is a relative young field which has evolved into a highly active research area in mathematical optimization an important line of research is to use semidefinite programming to approximate conic programming over the copositive cone two major drawbacks of this approach are the rapid growth in size of the resulting semidefinite programs and the lack of information about the quality of the semidefinite programming approximations these drawbacks are an inevitable consequence of the intractability of the generic problems that such approaches attempt to solve to address such drawbacks we develop customized solution approaches for highly symmetric copositive programs which arise naturally in several contexts for instance symmetry properties of combinatorial problems are typically inherited when they are addressed via copositive programming as a result we are able to compute new bounds for crossing number instances in complete bipartite graphs

mastrovito multiplier for all trinomials an efficient algorithm for the multiplication in gf 2 sup m was introduced by mastrovito the space complexity of the mastrovito multiplier for the irreducible trinomial x sup m x 1 was given as m sup 2 1 xor and m sup 2 and gales in this paper we describe an architecture based on a new formulation of the multiplication matrix and show that the mastrovito multiplier for the generating trinomial x sup m x sup n 1 where m spl ne 2n also requires m sup 2 1 xor and m sup 2 and gates however m sup 2 x sup m 2 xor gates are sufficient when the generating trinomial is of the form x sup m x sup m 2 1 for an even m we also calculate the time complexity of the proposed mastrovito multiplier and give design examples for the irreducible trinomials x sup 7 x sup 4 1 and x sup 6 x sup 3 1

complexity analysis of successive convex relaxation methods for nonconvex sets this paper discusses computational complexity of conceptual successive convex relaxation methods proposed by kojima and tuna el for approximating a convex relaxation of a compact subsetof the n dimensional euclidean spacer n here c0 denotes a nonempty compact convex subset ofr n anda set of finitely or infinitely many quadratic functions we evaluate the number of iterations which the successive convex relaxation methods require to attain a convex relaxation off with a given accuracy e in terms of e the diameter ofc0 the diameter off and some other quantities characterizing the lipschitz continuity the nonlinearity and the nonconvexity of the setof quadratic functions

quaternionic representations of magnetic groups we study magnetic groups which contain a time inversion operator besides spatial symmetries in the framework of quaternionic group representation theory we obtain a classification of these groups depending on the reducibility of their spatial part and then we cross it with the generalized frobenius schur classification that has been obtained by ourselves elsewhere ten distinct cases arise but only five of them apply to factorizable groups i e groups in which the time inversion operator appears alone and not multiplied by a spatial symmetry we supply examples for these five cases and determine which of them apply to bosonic or fermionic systems respectively finally we discuss the degeneracy of energy levels in the presence of a time reversal symmetry kramers degeneracy in quaternionic quantum mechanics obtaining right values in all cases

finding roots of polynomials over finite fields we propose an improved algorithm for finding roots of polynomials over finite fields this makes possible significant speedup of the decoding process of bose chaudhuri hocquenghem reed solomon and some other error correcting codes

central projective quaternionic representations we obtain all the irreducible not similar central quaternionic projective representations of a finite group g from the knowledge of the vector complex irreducible representations of suitable extensions of g and classify them according to a generalized frobenius schur classification finally a definition of equivalence of such representations is proposed

closed form orthogonal number theoretic transform eigenvectors and the fast fractional ntt in this paper we propose a new method to find the closed form solution of number theoretic transform ntt eigenvectors we construct the complete generalized legendre sequence over the finite field cglsf and use it to solve the ntt eigenvector problem we derive the cglsf like ntt eigenvectors successfully including the case where the operation field is defined over the fermat and mersenne numbers the derived ntt eigenvector set is orthogonal and has a closed form it is suitable for constructing sub ntt building blocks for ntt implementation in addition with different eigenvalue assignment rule we can construct the fractional number theoretic transform fntt including the fractional fermat number transform ffnt the fractional complex mersenne number transform fcmnt and the fractional new mersenne number transform fnmnt they are the generalizations of the original transforms and all have the complexities of o nlog2n

a new structure preserving method for quaternion hermitian eigenvalue problems in this paper we propose a novel structure preserving algorithm for solving the right eigenvalue problem of quaternion hermitian matrices the algorithm is based on the structure preserving tridiagonalization of the real counterpart for quaternion hermitian matrices by applying orthogonal jrs symplectic matrices the algorithm is numerically stable because we use orthogonal transformations the algorithm is very efficient it costs about a quarter arithmetical operations and a quarter to one eighth cpu times comparing with standard general purpose algorithms numerical experiments are provided to demonstrate the efficiency of the structure preserving algorithm

cones of matrices and successive convex relaxations of nonconvex sets let f be a compact subset of the n dimensional euclidean space rn represented by finitely or infinitely many quadratic inequalities we propose two methods one based on successive semidefinite programming sdp relaxations and the other on successive linear programming lp relaxations each of our methods generates a sequence of compact convex subsets ck k 1 2 of rn such that a the convex hull of f subseteq c k 1 subseteq c k monotonicity b cap k 1 infty c k text the convex hull of f asymptotic convergence our methods are extensions of the corresponding lovasz schrijver lift and project procedures with the use of sdp or lp relaxation applied to general quadratic optimization problems qops with infinitely many quadratic inequality constraints utilizing descriptions of sets based on cones of matrices and their duals we establish the exact equivalence of the sdp relaxation and the semi infinite convex qop relaxation proposed originally by fujie and kojima using this equivalence we investigate some fundamental features of the two methods including a and b above

cyclotomic ffts with reduced additive complexities based on a novel common subexpression elimination algorithm in this paper we first propose a novel common subexpression elimination cse algorithm for matrix vector multiplications over characteristic 2 fields as opposed to previously proposed cse algorithms which usually focus on complexity savings due to recurrences of subexpressions our cse algorithm achieves two types of complexity reductions differential savings and recurrence savings by taking advantage of the cancellation property of characteristic 2 fields using our cse algorithm we reduce the additive complexities of cyclotomic fast fourier transforms cffts using a weighted sum of the numbers of multiplications and additions as a metric our cffts achieve smaller total complexities than previously proposed cffts and other ffts requiring both fewer multiplications and fewer additions in many cases

semidefinite programming vs lp relaxations for polynomial programming we consider the global minimization of a multivariate polynomial on a semi algebraic set o defined with polynomial inequalities we then compare two hierarchies of relaxations namely lp relaxations based on products of the original constraints in the spirit of the rlt procedure of sherali and adams 1990 and recent semidefinite programming sdp relaxations introduced by the author the comparison is analyzed in light of recent results in real algebraic geometry on various representations of polynomials positive on a compact semi algebraic set

additive fast fourier transforms over finite fields we present new additive fast fourier transform fft algorithms based on taylor expansions over finite fields of characteristic two the new algorithms improve previous approaches by wang and zhu 1988 cantor 1989 and von zur gathen and gerhard 1996

on the structure of cyclotomic fourier transforms and their applications to reed solomon codes this paper is focused on cyclotomic fourier transforms in gf 2m and on their applications to algebraic decoding of reed solomon codes like the evaluation of syndromes and of error locator or evaluator polynomials cyclotomic transforms are much more efficient than straightforward evaluation in particular the number of multiplications is quite small in this paper it is shown that also the number of additions can be considerably reduced with respect to previous analyses a simple interpretation of the cyclotomic fourier transform best suited for the evaluation of syndromes allows to assemble the required matrix easily and quickly even in large fields fast construction of such matrices is important to obtain the best results since as many matrices as possible must be generated and compared it is shown that both the structure of the matrix and of bilinear convolutions need to be exploited to reduce the complexity of the costly part of cyclotomic fourier transforms which is a matrix vector product heuristic algorithms for matrix vector product are to be run as many times as possible to obtain the best transform it is shown with several examples that very good results can be obtained even with very simple algorithms

a fast algorithm for the syndrome calculation in algebraic decoding of reed solomon codes in this paper fedorenko and trifonov s procedure is applied to evaluate the syndrome of the received word in time domain reed solomon decoders this application leads to a substantial reduction of the computational complexity of the syndrome polynomial for correcting both errors and erasures moreover simulation results for this new syndrome method are given

approximating quadratic programming with bound and quadratic constraints we consider the problem of approximating the global maximum of a quadratic program qp subject to bound and simple quadratic constraints based on several early results we show that a 4 7 approximate solution can be obtained in polynomial time

a copositive programming approach to graph partitioning we consider 3 partitioning the vertices of a graph into sets s 1 s 2 and s 3 of specified cardinalities such that the total weight of all edges joining s 1 and s 2 is minimized this problem is closely related to several np hard problems like determining the bandwidth or finding a vertex separator in a graph we show that this problem can be formulated as a linear program over the cone of completely positive matrices leading in a natural way to semidefinite relaxations of the problem we show in particular that the spectral relaxation introduced by helmberg e 1995 can equivalently be formulated as a semidefinite program finally we propose a tightened version of this semidefinite program and show on some small instances that this new bound is a significant improvement over the spectral bound

discretization and localization in successive convex relaxation methods for nonconvex quadratic optimization based on the authors previous work which established theoretical foundations of two conceptual successive convex relaxation methods i e the ssdp successive semidefinite programming relaxation method and the ssilp successive semi infinite linear programming relaxation method this paper proposes their implementable variants for general quadratic optimization problems these problems have a linear objective function c t x to be maximized over a nonconvex compact feasible region f described by a finite number of quadratic inequalities we introduce two new techniques discretization and localization into the ssdp and ssilp relaxation methods the discretization technique makes it possible to approximate an infinite number of semi infinite sdps or semi infinite lps which appeared at each iteration of the original methods by a finite number of standard sdps or standard lps with a finite number of linear inequality constraints we establish given any open convex set u containing f there is an implementable discretization of the ssdp or ssilp relaxation method which generates a compact convex set c such that f c u in a finite number of iterations the localization technique is for the cases where we are only interested in upper bounds on the optimal objective value for a fixed objective function vector c but not in a global approximation of the convex hull of f this technique allows us to generate a convex relaxation of f that is accurate only in certain directions in a neighborhood of the objective direction c this cuts off redundant work to make the convex relaxation accurate in unnecessary directions we establish given any positive number e there is an implementable localization discretization of the ssdp or ssilp relaxation method which generates an upper bound of the objective value within e of its maximum in a finite number of iterations

conic mixed integer rounding cuts a conic integer program is an integer programming problem with conic constraints many problems in finance engineering statistical learning and probabilistic optimization are modeled using conic constraints here we study mixed integer sets defined by second order conic constraints we introduce general purpose cuts for conic mixed integer programming based on polyhedral conic substructures of second order conic sets these cuts can be readily incorporated in branch and bound algorithms that solve either second order conic programming or linear programming relaxations of conic integer programs at the nodes of the branch and bound tree central to our approach is a reformulation of the second order conic constraints with polyhedral second order conic constraints in a higher dimensional space in this representation the cuts we develop are linear even though they are nonlinear in the original space of variables this feature leads to a computationally efficient implementation of nonlinear cuts for conic mixed integer programming the reformulation also allows the use of polyhedral methods for conic integer programming we report computational results on solving unstructured second order conic mixed integer problems as well as mean variance capital budgeting problems and least squares estimation problems with binary inputs our computational experiments show that conic mixed integer rounding cuts are very effective in reducing the integrality gap of continuous relaxations of conic mixed integer programs and hence improving their solvability