on the generalized lanczos trust region method the so called trust region subproblem gets its name in the trust region method in optimization and also plays a vital role in various other applications several numerical algorithms have been proposed in the literature for solving small to medium size dense problems as well as for large scale sparse problems the generalized lanczos trust region gltr method proposed by n i m gould s lucidi m roma and p l toint siam j optim 9 1999 pp 561 580 is a natural extension of the classical lanczos method for the linear system to the trust region subproblem in this paper we first analyze the convergence of gltr to reveal its convergence behavior in theory and then propose new stopping criteria that can be integrated into gltr for better numerical performance specifically we develop a priori upper bounds for the convergence to both the optimal objective value as well as the optimal solution and argue that these bounds can be efficiently estimated numerically and serve as stopping criteria f

on copositive matrices abstract a criterion for copositive matrices is given and for n 3 the set of all copositive matrices is determined in terms of matrix elements copositive matrices are applied to the problem of excluding periodic solutions of certain algebraic differential equations

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

nonlinear semidefinite programming sensitivity convergence and an application in passive reduced order modeling we consider the solution of nonlinear programs with nonlinear semidefiniteness constraints the need for an efficient exploitation of the cone of positive semidefinite matrices makes the solution of such nonlinear semidefinite programs more complicated than the solution of standard nonlinear programs this paper studies a sequential semidefinite programming ssp method which is a generalization of the well known sequential quadratic programming method for standard nonlinear programs we present a sensitivity result for nonlinear semidefinite programs and then based on this result we give a self contained proof of local quadratic convergence of the ssp method we also describe a class of nonlinear semidefinite programs that arise in passive reduced order modeling and we report results of some numerical experiments with the ssp method applied to problems in that class

criteria for copositive matrices abstract some finite criteria for copositive copositive plus and strictly copositive matrices are proposed and compared with existing determinantal tests the basic mathematical tool is principal pivoting

a new matrix free algorithm for the large scale trust region subproblem we present a new method for the large scale trust region subproblem the method is matrix free in the sense that only matrix vector products are required we recast the trust region subproblem as a parameterized eigenvalue problem and compute an optimal value for the parameter we then find the solution of the trust region subproblem from the eigenvectors associated with two of the smallest eigenvalues of the parameterized eigenvalue problem corresponding to the optimal parameter the new algorithm uses a different interpolating scheme than existing methods and introduces a unified iteration that naturally includes the so called hard case we show that the new iteration is well defined and convergent at a superlinear rate we present computational results to illustrate convergence properties and robustness of the method

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

on groups that have normal forms computable in logspace abstract we consider the class of finitely generated groups which have a normal form computable in logspace we prove that the class of such groups is closed under passing to finite index subgroups direct products wreath products and certain free products and infinite extensions and includes the solvable baumslag solitar groups as well as non residually finite and hence non linear examples we define a group to be logspace embeddable if it embeds in a group with normal forms computable in logspace we prove that finitely generated nilpotent groups are logspace embeddable it follows that all groups of polynomial growth are logspace embeddable

the word and geodesic problems in free solvable groups we study the computational complexity of the word problem wp in free solvable groups s r d where r 2 is the rank and d 2 is the solvability class of the group it is known that the magnus embedding of s r d into matrices provides a polynomial time decision algorithm for wp in a fixed group s r d unfortunately the degree of the polynomial grows together with d so the uniform algorithm is not polynomial in d in this paper we show that wp has time complexity o rn log 2 n in s r 2 and o n 3 rd in s r d for d 3 however it turns out that a seemingly close problem of computing the geodesic length of elements in s r 2 is np complete we prove also that one can compute fox derivatives of elements from s r d in time o n 3 rd in particular one can use efficiently the magnus embedding in computations with free solvable groups our approach is based on such classical tools as the magnus embedding and fox calculus as well as on relatively new geometric ideas in particular we establish a direct link between fox derivatives and geometric flows on cayley graphs

solving the trust region subproblem by a generalized eigenvalue problem the state of the art algorithms for solving the trust region subproblem trs are based on an iterative process involving solutions of many linear systems eigenvalue problems subspace optimization or line search steps a relatively underappreciated fact due to gander golub and von matt linear algebra appl 114 1989 pp 815 839 is that trss can be solved by one generalized eigenvalue problem with no outer iterations in this paper we rediscover this fact and discover its great practicality which exhibits good performance both in accuracy and efficiency moreover we generalize the approach in various directions namely by allowing for an ellipsoidal constraint dealing with the so called hard case and obtaining approximate solutions efficiently when high accuracy is unnecessary we demonstrate that the resulting algorithm is a general purpose trs solver effective both for dense and large sparse problems including the so called hard case our algorithm is easy to implement its essence i

spectral analysis of coupled linear complementarity problems abstract this note deals with the so called cone constrained bivariate eigenvalue problem the equilibrium model under consideration is a system of linear complementarity problems p x ax by x p q y cx dy y q involving two closed convex cones and their corresponding duals we study the set of pairs r 2 for which this system has a nontrivial solution x y r n m we discuss also the link between the cone constrained version and the unconstrained one

growth series of some wreath products the growth series of certain finitely generated groups which are wreath products are investigated these growth series are intimately related to the traveling salesman problem on certain graphs a large class of these growth series is shown to consist of irrational algebraic functions n 0 since only one generating set will be associated with each group below the generating set associated with fr x will be obvious let h be a group with a finite generating set sh these will be fixed for the rest of the paper the cayley graph of the pair h sh is as usual the directed graph whose vertices are the elements of h and there is an edge from a vertex hx xoa vertex h2 if and only if h2 hxh for some h in sh us 1 in particular the edge from hx to h2 has an opposite edge from h2 to hx let c be the graph gotten from the cayley graph of h sh simply by identifying opposite edges in other words c might be called the undirected cayley graph of h sh hex k also be a group with a finite generating set sk these will also be fixed for the rest of the paper it is possible to form what might be called a restricted direct product group p which consists of all functions p from the vertices of c to k such that there are only finitely many vertices v in c with p v t 1 this is a subgroup of the direct product group whose elements consist of all functions from the vertex set of c to k the group p admits h as a group of automorphisms by means of the action of h on c the resulting semidirect product p x h is the restricted wreath product k h

a subspace minimization method for the trust region step we consider methods for large scale unconstrained minimization based on finding an approximate minimizer of a quadratic function subject to a two norm trust region constraint the steihaug toint method uses the conjugate gradient method to minimize the quadratic over a sequence of expanding subspaces until the iterates either converge to an interior point or cross the constraint boundary however if the conjugate gradient method is used with a preconditioner the steihaug toint method requires that the trust region norm be defined in terms of the preconditioning matrix if a different preconditioner is used for each subproblem the shape of the trust region can change substantially from one subproblem to the next which invalidates many of the assumptions on which standard methods for adjusting the trust region radius are based in this paper we propose a method that allows the trust region norm to be defined independently of the preconditioner the method solves the inequality constrained trust region subproblem over a sequence of evolving low dimensional subspaces each subspace includes an accelerator direction defined by a regularized newton method for satisfying the optimality conditions of a primal dual interior method a crucial property of this direction is that it can be computed by applying the preconditioned conjugate gradient method to a positive definite system in both the primal and dual variables of the trust region subproblem numerical experiments on problems from the cuter test collection indicate that the method can require significantly fewer function evaluations than other methods in addition experiments with general purpose preconditioners show that it is possible to significantly reduce the number of matrix vector products relative to those required without preconditioning

criteria for copositive matrices of order four abstract it is proved that the copositivity of a symmetric matrix of order n is equivalent to the copositivity of two symmetric matrices of order n 1 if the matrix has a row whose off diagonal elements are all nonpositive this result is used for deriving criteria for copositive matrices of order 3 and 4

on unique existence theorem and exact solution formula of the inverse black body radiation problem the unique existence theorem of the inverse black body radiation problem is proved its exact solution formula of closed form and some general laurent expansions are derived the gaussian spectrum problem has been investigated numerically by many scientists but its exact solution has not yet been obtained the exact solution for the gaussian spectrum problem is obtained by the derived exact solution formula and the criterion canceling divergence is described

on a subproblem of trust region algorithms for constrained optimization we study a subproblem that arises in some trust region algorithms for equality constrained optimization it is the minimization of a general quadratic function with two special quadratic constraints properties of such subproblems are given it is proved that the hessian of the lagrangian has at most one negative eigenvalue and an example is presented to show that the hessian may have a negative eigenvalue when one constraint is inactive at the solution

solution of underdetermined electromagnetic and seismic problems by the maximum entropy method many inversion problems require solution of a fredholm integral equation of the form t bar r int d bar r bar r sigma bar r dv where t is the observable d is an operator and sigma is the unknown parameter distribution examples occur in the areas of radiation and scattering tomography and geotomography we reduce the equation to matrix form and apply a maximum entropy technique based on the first principle of data reduction to obtain a most probable sigma distribution we illustrate the technique by synthetic data examples of geotomography assuming straight rays with and without noise the examples show how sharp anomalies may be identified in grossly underdetermined situations we outline the algorithm used and describe some computational properties our method suggests a way of overcoming the ill conditioned nature of fredholm integral equation inversion

on spectral maps induced by convex cones abstract let l e be the vector space of linear endomorphisms on a euclidean space e and k be a closed convex cone in e the spectral map k l e 2 r induced by k is defined as follows to each linear endomorphism a on e it associates the set k a of all r for which the complementarity problem k x a x x k has a nonzero solution x e here k is the dual cone of k and stands for orthogonality the set k a is known as the k spectrum of a the purpose of this work is twofold firstly we wish to compare algebraic properties of k and geometric properties of k secondly we wish to identify classes of linear transformations f l e l e that leave the k spectrum invariant i e such that k f a k a for all a l e the set of linear transformations preserving the k spectrum depends of course on the cone k under consideration

first and second order analysis of nonlinear semidefinite programs in this paper we study nonlinear semidefinite programming problems convexity duality and first order optimality conditions for such problems are presented a second order analysis is also given second order necessary and sufficient optimality conditions are derived finally sensitivity analysis of such programs is discussed

solution of inverse blackbody radiation problem with conjugate gradient method the inverse blackbody radiation problem for the estimation of the area temperature distribution is investigated in this paper the inverse problem is solved by minimizing the integration of the square of the difference between the calculated total radiated power spectrum and the simulated experimental data the conjugate gradient method is applied for the solution of the minimization problem the effects of the functional form of the area temperature distribution and the measurement errors on the accuracy of the estimation in the inverse analysis are examined

on the finiteness of the cone spectrum of certain linear transformations on euclidean jordan algebras abstract let l be a linear transformation on a finite dimensional real hilbert space h and k be a closed convex cone with dual k in h the cone spectrum of l relative to k is the set of all real for which the linear complementarity problem x k y l x x k and x y 0 admits a nonzero solution x in the setting of a euclidean jordan algebra h and the corresponding symmetric cone k we discuss the finiteness of the cone spectrum for z transformations and quadratic representations on h

criteria for copositive matrices using simplices and barycentric coordinates abstract we present criteria for verifying the copositivity of an n n matrix given that all its principal submatrices of order n 1 are copositive for n 4 5 necessary and sufficient conditions for copositivity are given based on the sign distribution of the off diagonal elements of a single row here it is only assumed that one principal submatrix is copositive

a test for copositive matrices abstract the paper presents necessary and sufficient conditions that a symmetric matrix be copositive or strictly copositive the conditions are given in terms of the eigenvalues and eigenvectors of the principal submatrices of the given matrix

the difference between 5 5 doubly nonnegative and completely positive matrices abstract the convex cone of n n completely positive cp matrices and its dual cone of copositive matrices arise in several areas of applied mathematics including optimization every cp matrix is doubly nonnegative dnn i e positive semidefinite and component wise nonnegative and it is known that for n 4 only every dnn matrix is cp in this paper we investigate the difference between 5 5 dnn and cp matrices defining a bad matrix to be one which is dnn but not cp we i design a finite procedure to decompose any n n dnn matrix into the sum of a cp matrix and a bad matrix which itself cannot be further decomposed ii show that every bad 5 5 dnn matrix is the sum of a cp matrix and a single bad extreme matrix and iii demonstrate how to separate bad extreme matrices from the cone of 5 5 cp matrices

finite criteria for conditional definiteness of quadratic forms abstract a real symmetric n n matrix q is a conditionally positivesemidefinite where a is a given m n matrix if x qx 0 whenever ax 0 and is a conditionally positive definite if strict inequality holds except when x 0 when a is the identity matrix these notions reduce to the well studied notions of copositivity and strict copositivity respectively this paper presents finite criteria involving only the solution of sets of linear equations constructed from the matrices q a for testing both types of conditional definiteness these criteria generalize known facts about copositive matrices and when q is invertible and all row submatrices of a have maximal rank can be very elegantly stated in terms of schur complements of the matrix aq 1 a

a note on computing the smallest conic singular value abstract the goal of this note is to study the smallest conic singular value of a matrix from a lagrangian duality viewpoint and provide an efficient method for its computation

inverse black body radiation at microwave frequencies the inverse black body radiation which is the problem of determining the area temperature distribution of a black body from its measured radiated power spectrum was recently solved by bojarski his solution employs the inverse laplace transform to reconstruct the area coldness distribution from an integral equation based on planck s law our solution is similar but restricted to microwave frequencies and employs the rayleigh jeans approximation to yield the corresponding portion of the area coldness distribution using the inverse laplace transform but avoiding the iterative procedure of bojarski

algorithmic copositivity detection by simplicial partition we present new criteria for copositivity of a matrix i e conditions which ensure that the quadratic form induced by the matrix is nonnegative over the nonnegative orthant these criteria arise from the representation of the quadratic form in barycentric coordinates with respect to the standard simplex and simplicial partitions thereof we show that as the partition gets finer and finer the conditions eventually capture all strictly copositive matrices we propose an algorithmic implementation which considers several numerical aspects as an application we present results on the maximum clique problem we also briefly discuss extensions of our approach to copositivity with respect to arbitrary polyhedral cones

globally solving the trust region subproblem using simple first order methods we consider the trust region subproblem which is given by a minimization of a quadratic not necessarily convex function over the euclidean ball based on the well known second order necessary and

on eigenvalues induced by a cone constraint abstract let a be an n n real matrix and k r n be a closed convex cone the spectrum of a relative to k denoted by a k is the set of all r for which the linear complementarity problem x k ax x k x ax x 0 admitsa nonzero solution x r n the notation k stands for the positive dual cone of k the purpose of this work is to study the main properties of the mapping k and discuss some structural differences existing between the polyhedral case i e k is finitely generated and the nonpolyhedral case

higher rank lamplighter groups are graph automatic abstract we show that the higher rank lamplighter groups or diestel leader groups d q for d 3 are graph automatic this introduces a new family of graph automatic groups which are not automatic

a quasi newton approach to nonsmooth convex optimization problems in machine learning we extend the well known bfgs quasi newton method and its memory limited variant lbfgs to the optimization of nonsmooth convex objectives this is done in a rigorous fashion by generalizing three components of bfgs to subdifferentials the local quadratic model the identification of a descent direction and the wolfe line search conditions we prove that under some technical conditions the resulting subbfgs algorithm is globally convergent in objective function value we apply its memory limited variant sublbfgs to l2 regularized risk minimization with the binary hinge loss to extend our algorithm to the multiclass and multilabel settings we develop a new efficient exact line search algorithm we prove its worst case time complexity bounds and show that our line search can also be used to extend a recently developed bundle method to the multiclass and multilabel settings we also apply the direction finding component of our algorithm to l1 regularized risk minimization with logistic loss in all these contexts our methods perform comparable to or better than specialized state of the art solvers on a number of publicly available data sets an open source implementation of our algorithms is freely available

inverse black body radiation an extant closed form solution the inverse black body radiation problem concerned with the determination of the temperature distribution of a block body source from spectral measurements has recently been formulated here an exact closed form reconstruction algorithm is given in terms of on infinite series this algorithm which avoids iteration can be implemented either numerically or analytically the method is applied to several analytical and numerical examples with excellent results

theory of cones abstract this survey deals with the aspects of archimedian partially ordered finite dimensional real vector spaces and order preserving linear maps which do not involve spectral theory the first section sketches some of the background of entrywise nonnegative matrices and of systems of inequalities which motivate much of the current investigations the study of inequalities resulted in the definition of a polyhedral cone k and its face lattice f k in section ii a the face lattice of a not necessarily polyhedral cone k in a vector space v is investigated in particular the interplay between the lattice properties of f k and geometric properties of k is emphasized section ii b turns to the cones k in the space of linear maps on v recall that k is the cone of all order preserving linear maps of particular interest are the algebraic structure of k as a semiring and the nature of the group aut k of nonsingular elements a k for which a 1 k as well in a short final section the cone p n of n n positive semidefinite matrices is discussed a characterization of the set of completely positive linear maps is stated the proofs will appear in a forthcoming paper

stability of linear quadratic minimization over euclidean balls stability properties of the problem of minimizing a nonconvex linear quadratic function over a euclidean ball known as the trust region subproblem are studied in this paper we investigate in detail the case where the linear part of the objective function is perturbed and obtain necessary and sufficient conditions for the upper lower semicontinuity of the karush kuhn tucker kkt point set map and the global solution map explicit formulas for computing the directional derivative and the frechet derivative of the optimal value function stability of the karush kuhn tucker point set under the perturbation of the quadratic form is also studied

solving inverse cone constrained eigenvalue problems we compare various algorithms for constructing a matrix of order n whose pareto spectrum contains a prescribed set lambda lambda 1 ldots lambda p of reals in order to avoid overdetermination one assumes that p does not exceed n 2 the inverse pareto eigenvalue problem under consideration is formulated as an underdetermined system of nonlinear equations we also address the issue of computing lorentz spectra and solving inverse lorentz eigenvalue problems

theoretical investigation on the inverse black body radiation problem after reviewing previous work on the inverse black body radiation problem a completely new approach to solve the problem not only theoretically but also numerically is presented the first exact and concise expression for the solution is shown in this work based on the fourier transform theory its existence uniqueness and stability are discussed in detail the problem of convergence has been completely eliminated

asymptotic properties of maximum likelihood estimators and likelihood ratio tests under nonstandard conditions abstract large sample properties of the likelihood function when the true parameter value may be on the boundary of the parameter space are described specifically the asymptotic distribution of maximum likelihood estimators and likelihood ratio statistics are derived these results generalize the work of moran 1971 chant 1974 and chernoff 1954 some of chant s results are shown to be incorrect the approach used in deriving these results follows from comments made by moran and chant the problem is shown to be asymptotically equivalent to the problem of estimating the restricted mean of a multivariate gaussian distribution from a sample of size 1 in this representation the gaussian random variable corresponds to the limit of the normalized score statistic and the estimate of the mean corresponds to the limit of the normalized maximum likelihood estimator thus the limiting distribution of the maximum likelihood estimator is the same as the distribution of the projection of the gaussian random v

a variational approach to copositive matrices this work surveys essential properties of the so called copositive matrices the study of which has been spread over more than fifty five years special emphasis is given to variational aspects related to the concept of copositivity in addition some new results on the geometry of the cone of copositive matrices are presented here for the first time

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

perron frobenius property of copositive matrices and a block copositivity criterion abstract haynsworth and hoffman proved in 1969 that the spectral radius of a symmetric copositive matrix is an eigenvalue of this matrix this note investigates conditions which guarantee that an eigenvector corresponding to this dominant eigenvalue has no negative coordinates i e whether the perron frobenius property holds also a block copositivity criterion using the schur complement is specified which may be helpful to reduce dimension in copositivity checks and which generalizes results proposed by andersson et al in 1995 and johnson and reams in 2005 apparently the latter five researchers were unaware of the more general results by the author precedingly published in 1987 and 1996 respectively

mobius transform pairs a continuous version of a theorem from classical number theory due to mobius has recently been derived and used to solve some inverse problems in physics that are otherwise difficult to solve explicitly using standard results a table of mobius transform pairs is presented that may be useful in identifying other applications of the transform to physical problems

on the complexity of steepest descent newton s and regularized newton s methods for nonconvex unconstrained optimization problems it is shown that the steepest descent and newton s methods for unconstrained nonconvex optimization under standard assumptions may both require a number of iterations and function evaluations arbitrarily close to o epsilon 2 to drive the norm of the gradient below epsilon this shows that the upper bound of o epsilon 2 evaluations known for the steepest descent is tight and that newton s method may be as slow as the steepest descent method in the worst case the improved evaluation complexity bound of o epsilon 3 2 evaluations known for cubically regularized newton s methods is also shown to be tight

some np complete problems in quadratic and nonlinear programming in continuous variable smooth nonconvex nonlinear programming we analyze the complexity of checking whether a a given feasible solution is not a local minimum and b the objective function is not bounded below on the set of feasible solutions we construct a special class of indefinite quadratic programs with simple constraints and integer data and show that checking a or b on this class is np complete as a corollary we show that checking whether a given integer square matrix is not copositive is np complete

some inertia theorems in euclidean jordan algebras abstract this paper deals with some inertia theorems in euclidean jordan algebras first based on the continuity of eigenvalues we give an alternate proof of kaneyuki s generalization of sylvester s law of inertia in simple euclidean jordan algebras as a consequence we show that the cone spectrum of any quadratic representation with respect to a symmetric cone is finite second we present ostrowski schneider type inertia results in euclidean jordan algebras in particular we relate the inertias of objects a and x in a euclidean jordan algebra when l a x 0 or s a x 0 where l a and s a denote lyapunov and stein transformations respectively

word problems recognisable by deterministic blind monoid automata we consider blind deterministic finite automata equipped with a register which stores an element of a given monoid and which is modified by right multiplication by monoid elements we show that for monoids m drawn from a large class including groups such an automaton accepts the word problem of a group h if and only if h has a finite index subgroup which embeds in the group of units of m in the case that m is a group this answers a question of elston and ostheimer

inverse black body radiation the total power spectrum radiated by a black body with a given area temperature distribution is given by planck s law the inverse black body radiation problem is thus the problem of determining the area temperature distribution of a black body given its measured total radiated power spectrum the concept of a normalized absolute coldness as the proportional to the reciprocal of the absolute temperature is introduced yielding an integral equation for the inverse black body radiation problem this integral equation is solved analytically with the aid of the inverse laplace transform and a very simple iterative process the method of successive approximations that converges in two iterations numerico experimental results for the reconstruction of a gaussian area coldness distribution are presented

minimizing a quadratic over a sphere a new method the sequential subspace method ssm is developed for the problem of minimizing a quadratic over a sphere in our scheme the quadratic is minimized over a subspace which is adjusted in successive iterations to ensure convergence to an optimum when a sequential quadratic programming iterate is included in the subspace convergence is locally quadratic numerical comparisons with other recent methods are given

iterative methods for nearly singular linear systems iterative methods are developed and studied for near singular linear systems cx b our approach called the transformed minimal residual algorithm tmres is derived from any convergent iterative scheme sxk 1 txk b associated with a splitting c s t in each step of tmres the transformed residual s 1 b cx is minimized over a krylov space generated by s 1 t the original iterative scheme typically converges slowly when c is nearly singular while a krylov space generated by s 1 t often contains a much better approximation to a solution tmres is algebraically equivalent to the generalized minimal residual algorithm gmres preconditioned by s 1 although there are numerical differences since a different matrix s 1 c is used to generate the krylov space in preconditioned gmres special attention is given to sparsity and convergence issues related to linear systems of the form bf aa tr sigma bf i bf x bf b where sigma ge 0

conjugate cone characterization of positive definite and semidefinite matrices abstract positive definite and semidefinite matrices are characterized in terms of positive definiteness and semidefiniteness on arbitrary closed convex cones in rn these results are obtained by generalizing moreau s polar decomposition to a conjugate decomposition some typical results are the matrix a is positive definite if and only if for some closed convex cone k a is positive definite on k and a at 1 exists and is semidefinite on the polar cone k the matrix a is positive semidefinite if and only if for some closed convex cone k such that either k is polyhedral or a at k is closed a is positive semidefinite on both k and the conjugate cone ka s xt a at s 0 x k and a at x 0 for all x in k such that xtax 0

a block active set algorithm with spectral choice line search for the symmetric eigenvalue complementarity problem in this paper we address the solution of the symmetric eigenvalue complementarity problem eicp by treating an equivalent reformulation of finding a stationary point of a fractional quadratic program on the unit simplex the spectral projected gradient spg method has been recommended to this optimization problem when the dimension of the symmetric eicp is large and the accuracy of the solution is not a very important issue we suggest a new algorithm which combines elements from the spg method and the block active set method where the latter was originally designed for box constrained quadratic programs in the new algorithm the projection onto the unit simplex in the spg method is replaced by the much cheaper projection onto a box this can be of particular advantage for large and sparse symmetric eicps global convergence to a solution of the symmetric eicp is established computational experience with medium and large symmetric eicps is reported to illustrate the efficacy and efficiency of the new algorithm

global optimization with polynomials and the problem of moments we consider the problem of finding the unconstrained global minimum of a real valued polynomial p x mathbb r n to mathbb r as well as the global minimum of p x in a compact set k defined by polynomial inequalities it is shown that this problem reduces to solving an often finite sequence of convex linear matrix inequality lmi problems a notion of karush kuhn tucker polynomials is introduced in a global optimality condition some illustrative examples are provided

comments on recent solutions to the inverse black body radiation problem it is shown that both the boltzmann and the gibbs maximum entropy me methods are appropriate for the underdetermined inverse black body radiation problem the choice of either or an alternative method is subjective with a minimum of assumptions they are attractive alternatives to other formally correct but impractical solutions the me solutions illustrate by means of a synthetic data example the strong dependence of a most probable coldness distribution on measured data

on pairs of vectors achieving the maximal angle of a convex cone in this paper we explore the concept of antipodality relative to a closed convex cone the problem under consideration is that of finding a pair of unit vectors in k achieving the maximal angle of the cone we mention also a few words on the attainability of critical angles by way of application of the general theory we briefly discuss the problem of estimating the radius of pointedness of a cone

an alternative theorem for quadratic forms and extensions abstract this note concerns an alternative theorem for quadratic forms established recently by y yuan we explore some directions in which yuan s result can be extended and we indicate also some important situations where an alternative theorem a la yuan does not hold

on the spherical quasi convexity of quadratic functions abstract in this paper the spherical quasi convexity of quadratic functions on spherically convex sets is studied several conditions characterizing the spherical quasi convexity of quadratic functions are presented in particular conditions implying spherical quasi convexity of quadratic functions on the spherical positive orthant are given some examples are provided as applications of the obtained results

on copositive matrices and strong ellipticity for isotropic elastic materials on etablit les conditions necessaires et suffisantes pour une forte ellipticite des equations d un materiau elastique non lineaire isotrope compressible a l equilibre on utilise un theoreme de representation de matrices copositives du a cottle habetler et lemke 1970

searching for critical angles in a convex cone the concept of antipodality relative to a closed convex cone has been explored in detail in a recent work of ours the antipodality problem consists of finding a pair of unit vectors in k achieving the maximal angle of the cone our attention now is focused not just in the maximal angle but in the angular spectrum of the cone by definition the angular spectrum of a cone is the set of angles satisfying the stationarity or criticality condition associated to the maximization problem involved in the determination of the maximal angle in the case of a polyhedral cone the angular spectrum turns out to be a finite set among other results we obtain an upper bound for the cardinality of this set we also discuss the link between the critical angles of a cone k and the critical angles of its dual cone

copositive matrices and definiteness of quadratic forms subject to homogeneous linear inequality constraints abstract a symmetric matrix c is called copositive if the quadratic form x cx is nonnegative for all nonnegative values of the variables x 1 x 2 x n x a known sufficient condition for a quadratic form x qx to be positive unless x 0 subject to the linear inequality constraints ax 0 is that there should exist a copositive matrix c such that q a ca is positive definite the main result of this paper establishes the necessity of this condition for x qx to be merely nonnegative subject to ax 0 the situation is less straightforward the necessity of the existence of a copositive matrix c such that q a ca is positive semidefinite is proved only under various additional hypotheses regarding the size or rank of a and counterexamples are given to show that in general no such matrix may exist even when slater s constraint qualification holds our approach to these existence questions also furnishes certain tests for positivity or mere nonnegativity of x qx subject to ax 0 in which specific symmetric matrices constructed by rational operations from a and q and depending upon a single real parameter v must be tested for positive definiteness or strict copositivity for large values of v this technique is illustrated by several examples

topology trivialization and large deviations for the minimum in the simplest random optimization finding the global minimum of a cost function given by the sum of a quadratic and a linear form in n real variables over n 1 dimensional sphere is one of the simplest yet paradigmatic problems in optimization theory known as the trust region subproblem or constraint least square problem when both terms in the cost function are random this amounts to studying the ground state energy of the simplest spherical spin glass in a random magnetic field we first identify and study two distinct large n scaling regimes in which the linear term magnetic field leads to a gradual topology trivialization i e reduction in the total number mathcal n tot of critical stationary points in the cost function landscape in the first regime mathcal n tot remains of the order n and the cost function energy has generically two almost degenerate minima with the tracy widom tw statistics in the second regime the number of critical points is of the order of unity with a finite probability for a single minimum in that case the mean total number of extrema minima and maxima of the cost function is given by the laplace transform of the tw density and the distribution of the global minimum energy is expected to take a universal scaling form generalizing the tw law though the full form of that distribution is not yet known to us one of its far tails can be inferred from the large deviation theory for the global minimum in the rest of the paper we show how to use the replica method to obtain the probability density of the minimum energy in the large deviation approximation by finding both the rate function and the leading pre exponential factor

spectral theory of copositive matrices abstract let a r n n we provide a block characterization of copositive matrices with the assumption that one of the principal blocks is positive definite haynsworth and hoffman showed that if r is the largest eigenvalue of a copositive matrix then r for all other eigenvalues of a we continue their study of the spectral theory of copositive matrices and show that a copositive matrix must have a positive vector in the subspace spanned by the eigenvectors corresponding to the nonnegative eigenvalues moreover if a symmetric matrix has a positive vector in the subspace spanned by the eigenvectors corresponding to its nonnegative eigenvalues then it is possible to increase the nonnegative eigenvalues to form a copositive matrix a without changing the eigenvectors we also show that if a copositive matrix has just one positive eigenvalue and n 1 nonpositive eigenvalues then a has a nonnegative eigenvector corresponding to a nonnegative eigenvalue

global convergence of ssm for minimizing a quadratic over a sphere in an earlier paper minimizing a quadratic over a sphere siam j optim 12 2001 188 208 we presented the sequential subspace method ssm for minimizing a quadratic over a sphere this method generates approximations to a minimizer by carrying out the minimization over a sequence of subspaces that are adjusted after each iterate is computed we showed in this earlier paper that when the subspace contains a vector obtained by applying one step of newton s method to the first order optimality system ssm is locally quadratically convergent even when the original problem is degenerate with multiple solutions and with a singular jacobian in the optimality system in this paper we prove nonlocal convergence of ssm to a global minimizer whenever each ssm subspace contains the following three vectors i the current iterate ii the gradient of the cost function evaluated at the current iterate and iii an eigenvector associated with the smallest eigenvalue of the cost function hessian for nondegenerate problems the convergence rate is at least linear when vectors i iii are included in the ssm subspace

critical angles in polyhedral convex cones numerical and statistical considerations this work concerns the numerical computation of critical angles in polyhedral convex cones the set of proper critical angles is evaluated explicitly by solving a series of generalized eigenvalue problems involving the generators of the cone the local maximal angles are identified by using a necessary condition for local maximality the expected numbers of critical angles in random polyhedral convex cones are estimated experimentally

regularity versus degeneracy in dynamics games and optimization a unified approach to different aspects in this paper links are established between optimality conditions for quadratic optimization problems qualitative properties in the nonlinear selection replicator dynamics and central solution concepts of evolutionary game theory with particular emphasis on several regularity conditions that are desirable in any of the three fields mentioned above as strictness conditions for locally optimal solutions as hyperbolicity conditions for fixed points and as quasi strictness conditions for game equilibria

new results on pareto spectra abstract a pareto eigenvalue of a matrix a of order n is a scalar r for which the complementarity problem 0 x a x x 0 admits a nonzero solution x r n pareto eigenvalues are also known as complementarity eigenvalues they have found applications in graph theory cone constrained dynamical systems and mathematical modeling in general in this paper we continue our study on theoretical properties of pareto spectra special attention is paid to classification of pareto eigenvalues and cardinality issues

using copositivity for global optimality criteria in concave quadratic programming problems in this note we specify a necessary and sufficient condition for global optimality in concave quadratic minimization problems using this condition it follows that from the perspective of worst case complexity of concave quadratic problems the difference between local and global optimality conditions is not as large as in general as an essential ingredient we here use thee subdifferential calculus via an approach of hiriart urruty and lemarechal 1990

c graph automatic groups abstract we generalize the notion of a graph automatic group introduced by kharlampovich khoussainov and miasnikov by replacing the regular languages in their definition with more powerful language classes for a fixed language class c we call the resulting groups c graph automatic we prove that the class of c graph automatic groups is closed under change of generating set direct and free product for certain classes c we show that for quasi realtime counter graph automatic groups where normal forms have length that is linear in the geodesic length there is an algorithm to compute normal forms and therefore solve the word problem in polynomial time the class of quasi realtime counter graph automatic groups includes all baumslag solitar groups and the free group of countably infinite rank context sensitive graph automatic groups are shown to be a very large class which encompasses for example groups with unsolvable conjugacy problem the grigorchuk group and thompson s groups f t and v