nonconcavity of the spectral radius in levinger s theorem abstract let a r n n be a nonnegative irreducible square matrix and let r a be its spectral radius and perron frobenius eigenvalue levinger asserted and several have proven that r t r 1 t a t a increases over t 0 1 2 and decreases over t 1 2 1 it has further been stated that r t is concave over t 0 1 here we show that the latter claim is false in general through a number of counterexamples but prove it is true for a r 2 2 weighted shift matrices but not cyclic weighted shift matrices tridiagonal toeplitz matrices and the 3 parameter toeplitz matrices from fiedler but not toeplitz matrices in general a general characterization of the range of t or the class of matrices for which the spectral radius is concave in levinger s homotopy remains an open problem

iterative reweighted algorithms for matrix rank minimization the problem of minimizing the rank of a matrix subject to affine constraints has applications in several areas including machine learning and is known to be np hard a tractable relaxation for this problem is nuclear norm or trace norm minimization which is guaranteed to find the minimum rank matrix under suitable assumptions in this paper we propose a family of iterative reweighted least squares algorithms irls p with 0 p 1 as a computationally efficient way to improve over the performance of nuclear norm minimization the algorithms can be viewed as locally minimizing certain smooth approximations to the rank function when p 1 we give theoretical guarantees similar to those for nuclear norm minimization that is recovery of low rank matrices under certain assumptions on the operator defining the constraints for p 1 irls p shows better empirical performance in terms of recovering low rank matrices than nuclear norm minimization we provide an efficient implementation for irls p and also present a related family of algorithms sirls p these algorithms exhibit competitive run times and improved recovery when compared to existing algorithms for random instances of the matrix completion problem as well as on the movielens movie recommendation data set

the mathematics of eigenvalue optimization optimization problems involving the eigenvalues of symmetric and nonsymmetric matrices present a fascinating mathematical challenge such problems arise often in theory and practice particularly in engineering design and are amenable to a rich blend of classical mathematical techniques and contemporary optimization theory this essay presents a personal choice of some central mathematical ideas outlined for the broad optimization community i discuss the convex analysis of spectral functions and invariant matrix norms touching briefly on semidefinite representability and then outlining two broader algebraic viewpoints based on hyperbolic polynomials and lie algebra analogous nonconvex notions lead into eigenvalue perturbation theory the last third of the article concerns stability for polynomials matrices and associated dynamical systems ending with a section on robustness the powerful and elegant language of nonsmooth analysis appears throughout as a unifying narrative thread

analysis of nonsmooth symmetric matrix valued functions with applications to semidefinite complementarity problems for any function f from mathbb r to mathbb r one can define a corresponding function on the space of n n block diagonal real symmetric matrices by applying f to the eigenvalues of the spectral decomposition we show that this matrix valued function inherits from f the properties of continuity local lipschitz continuity directional differentiability frechet differentiability continuous differentiability as well as rho order semismoothness our analysis uses results from nonsmooth analysis as well as perturbation theory for the spectral decomposition of symmetric matrices we also apply our results to the semidefinite complementarity problem addressing some basic issues in the analysis of smoothing semismooth newton methods for solving this problem

extremal problems on the set of nonnegative definite matrices abstract an optimization problem of minimizing a real valued function of certain elements of a symmetric matrix subject to this matrix being nonnegative definite is considered optimality conditions are proposed the duality result of olkin and pukelsheim 1982 is extended to a wide class of such problems applications are discussed

second order directional derivatives of spectral functions a spectral function of a symmetric matrix x is a function which depends only on the eigenvalues of x l 1 x l 2 x l n x and may be written as l 1 x l 2 x l n x for some symmetric function in this paper we assume that is a c 1 1 function and discuss second order directional derivatives of such a spectral function we obtain an explicit expression of second order directional derivative for the spectral function

on max k sums the max k sum of a set of real scalars is the maximum sum of a subset of size k or alternatively the sum of the k largest elements we study two extensions first we show how to obtain smooth approximations to functions that are pointwise max k sums of smooth functions second we discuss how the max k sum can be defined on vectors in a finite dimensional real vector space ordered by a closed convex cone

convexity and differentiability properties of spectral functions and spectral mappings on euclidean jordan algebras abstract we study in this paper several properties of the eigenvalues function of a euclidean jordan algebra extending several known results in the framework of symmetric matrices in particular we give a concise form for the directional differential of a single eigenvalue we especially focus on spectral functions f on euclidean jordan algebras which are the composition of a symmetric real valued function f with the eigenvalues function we explore several properties of f that are transferred to f in particular convexity strong convexity and differentiability spectral mappings are also considered a special case of which is the gradient mapping of a spectral function answering a problem proposed by h sendov we give a formula for the jacobian of these functions

an approximation scheme for distributionally robust nonlinear optimization we consider distributionally robust optimization problems drops with nonlinear and nonconcave dependence on uncertain parameters the drop can be written as a nonsmooth nonlinear program with a

robustness in the presence of mixed parametric uncertainty and unmodeled dynamics continuing the development of the structured singular value approach to robust control design the authors investigate the problem of computing mu in the case of mixed real parametric and complex uncertainty the problem is shown to be equivalent to a smooth constrained finite dimensional optimization problem in view of the fact that the functional to be maximized may have several local extrema an upper bound on mu whose computation is numerically tractable is established this leads to a sufficient condition of robust stability and performance a historical perspective on the development of the mu theory is included

the rate of convergence of the augmented lagrangian method for nonlinear semidefinite programming we analyze the rate of local convergence of the augmented lagrangian method in nonlinear semidefinite optimization the presence of the positive semidefinite cone constraint requires extensive tools such as the singular value decomposition of matrices an implicit function theorem for semismooth functions and variational analysis on the projection operator in the symmetric matrix space without requiring strict complementarity we prove that under the constraint nondegeneracy condition and the strong second order sufficient condition the rate of convergence is linear and the ratio constant is proportional to 1 c where c is the penalty parameter that exceeds a threshold overline c 0

solving maximum entropy sampling problems using factored masks we present a practical approach to anstreicher and lee s masked spectral bound for maximum entropy sampling and we describe favorable results that we have obtained with a branch and bound algorithm based on our approach by representing masks in factored form we are able to easily satisfy a semidefiniteness constraint moreover this representation allows us to restrict the rank of the mask as a means for attempting to practically incorporate second order information

solving large scale semidefinite programs in parallel we describe an approach to the parallel and distributed solution of large scale block structured semidefinite programs using the spectral bundle method various elements of this approach such as data distribution an implicitly restarted lanczos method tailored to handle block diagonal structure a mixed polyhedral semidefinite subdifferential model and other aspects related to parallelism are combined in an implementation called lambda which delivers faster solution times than previously possible and acceptable parallel scalability on sufficiently large problems

lowner s operator and spectral functions in euclidean jordan algebras we study analyticity differentiability and semismoothness of lowner s operator and spectral functions under the framework of euclidean jordan algebras in particular we show that many optimization related classical results in the symmetric matrix space can be generalized within this framework for example the metric projection operator over any symmetric cone defined in a euclidean jordan algebra is shown to be strongly semismooth the research also raises several open questions whose answers would be of strong interest for optimization research

derivatives of spectral functions a spectral function of a hermitian matrix x is a function which depends only on the eigenvalues of x 1x 2x nx and hence may be written f 1x 2x nx for some symmetric function f such functions appear in a wide variety of matrix optimization problems we give a simple proof that this spectral function is differentiable at x if and only if the function f is differentiable at the vector x and we give a concise formula for the derivative we then apply this formula to deduce an analogous expression for the clarke generalized gradient of the spectral function a similar result holds for real symmetric matrices

volatility and correlation based systemic risk measures in the us market this paper deals with the problem of how to use simple systemic risk measures to assess portfolio risk characteristics using three simple examples taken from previous literature one based on raw and partial correlations another based on the eigenvalue decomposition of the covariance matrix and the last one based on an eigenvalue entropy a granger causation analysis revealed some of them are not always a good measure of risk in the sp therefore in the sense of risk as volatility the indicators are not always suitable nevertheless their results towards returns are similar to previous works that accept them a deeper analysis has shown that any symmetric measure based on eigenvalue decomposition of correlation matrices however is not useful as a measure of correlation risk the empirical counterpart analysis of this proposition stated that negative correlations are usually small and therefore do not heavily distort the behavior of the indicator

on eigenvalue optimization in this paper we study optimization problems involving eigenvalues of symmetric matrices one of the difficulties with numerical analysis of such problems is that the eigenvalues considered as functions of a symmetric matrix are not differentiable at those points where they coalesce we present a general framework for a smooth differentiable approach to such problems it is based on the concept of transversality borrowed from differential geometry in that framework we discuss first and second order optimality conditions and rates of convergence of the corresponding second order algorithms finally we present some results on the sensitivity analysis of such problems

a quadratically convergent local algorithm on minimizing the largest eigenvalue of a symmetric matrix abstract optimization involving eigenvalues arise in many engineering problems we propose a new algorithm on minimizing the largest eigenvalue over an affine family of symmetric matrices under certain assumptions it is shown that if started close enough to the minimizer x the proposed algorithm converges to x quadratically the proposed algorithm can be readily extended to minimizing the largest eigenvalue over an affine family of hermitian matrices also it has been extended to minimizing sums of the largest eigenvalues of a symmetric or hermitian matrix

on the rank of extreme matrices in semidefinite programs and the multiplicity of optimal eigenvalues we derive some basic results on the geometry of semidefinite programming sdp and eigenvalue optimization i e the minimization of the sum of the k largest eigenvalues of a smooth matrix valued function we provide upper bounds on the rank of extreme matrices in sdps and the first theoretically solid explanation of a phenomenon of intrinsic interest in eigenvalue optimization in the spectrum of an optimal matrix the kth and k 1 st largest eigenvalues tend to be equal and frequently have multiplicity greater than two this clustering is intuitively plausible and has been observed as early as 1975 when the matrix valued function is affine we prove that clustering must occur at extreme points of the set of optimal solutions if the number of variables is sufficiently large we also give a lower bound on the multiplicity of the critical eigenvalue these results generalize to the case of a general matrix valued function under appropriate conditions

convex analysis on the hermitian matrices there is growing interest in optimization problems with real symmetric matrices as variables generally the matrix functions involved are spectral they depend only on the eigenvalues of the matrix it is known that convex spectral functions can be characterized exactly as symmetric convex functions of the eigenvalues a new approach to this characterization is given via a simple fenchel conjugacy formula we then apply this formula to derive expressions for subdifferentials and to study duality relationships for convex optimization problems with positive semidefinite matrices as variables analogous results hold for hermitian matrices

the u lagrangian of the maximum eigenvalue function in this paper we apply the u lagrangian theory to the maximum eigenvalue function la and to its precomposition with affine matrix valued mappings we first give geometrical interpretations of the u objects that we introduce we also show that the u lagrangian of la has a hessian which can be explicitly computed the second order development of the u lagrangian provides a second order development of la along a characteristic smooth manifold the set of symmetric matrices whose maximal eigenvalues have a fixed multiplicity the same results can be obtained when we precompose la with an affine matrix valued mapping a provided that this mapping satisfies a regularity condition em transversality condition we show that the hessian of the u lagrangian of la circ a coincides with the reduced hessian encountered in sequential quadratic programming finally we use the u lagrangian to derive second order algorithms for minimizing la circ a

a new variational result for quasi newton formulae the recent measure function of byrd and nocedal siam j numer anal 26 1989 pp 727 739 is considered and simple proofs of some of its properties are given it is then shown that the bfgs and dfp formulae satisfy a least change property with respect to this new measure

computation of the maximum likelihood estimator in low rank factor analysis factor analysis is a classical multivariate dimensionality reduction technique popularly used in statistics econometrics and data science estimation for factor analysis is often carried out via the maximum likelihood principle which seeks to maximize the gaussian likelihood under the assumption that the positive definite covariance matrix can be decomposed as the sum of a low rank positive semidefinite matrix and a diagonal matrix with nonnegative entries this leads to a challenging rank constrained nonconvex optimization problem for which very few reliable computational algorithms are available we reformulate the low rank maximum likelihood factor analysis task as a nonlinear nonsmooth semidefinite optimization problem study various structural properties of this reformulation and propose fast and scalable algorithms based on difference of convex optimization our approach has computational guarantees gracefully scales to large problems is applicable to situations where the sample covariance matrix is rank deficient and adapts to variants of the maximum likelihood problem with additional constraints on the model parameters our numerical experiments validate the usefulness of our approach over existing state of the art approaches for maximum likelihood factor analysis

a new derivation of a formula by kato abstract if the m th largest eigenvalue m a of a real symmetric matrix a is simple then m is an analytic function in a neighbourhood of a in this note we provide a new derivation of the classical formulae for the coefficients in the power series expansion of t m a te for any real symmetric matrix e and t close to 0 kato s classical derivation of that formula uses a complex analytic approach involving properties of the resolvent of a te our derivation uses simple real analytic and combinatorial arguments in particular we derive and utilize a formula for the derivative of the moore penrose generalized inverse of the map x m x i x in direction e at real symmetric matrix a for any real symmetric matrix e

on analyticity of functions involving eigenvalues abstract let a z be an n n complex matrix whose elements depend analytically on z c m it is well known that any individual eigenvalue of a z may be nondifferentiable when it coalesces with others in this paper we investigate the analycity property of functions on the eigenvalues z 1 z n z of a z we first introduce the notion of functions that are symmetric with respect to partitions it is then shown that if a function c n c is analytic at a where a c m and is symmetric with respect to a certain partition induced by a then the composite function g z 1 z n z is analytic at a when z is real a z is symmetric or hermitian and the aforementioned assumptions hold so that g z is analytic at a we also derive formulae for its first and second order partial derivatives we apply the results to several problems involving eigenvalues

a unified approach to operator monotone functions abstract the notion of operator monotonicity dates back to a work by lowner in 1934 a map f s n s m is called operator monotone if a b implies f a f b here s n is the space of symmetric matrices with the semidefinite partial order often the function f is defined in terms of an underlying simpler function f of main interest is to find the properties of f that characterize operator monotonicity of f in that case it is said that f is also operator monotone classical examples are the lowner operators and the spectral scalar valued isotropic functions operator monotonicity for these two classes of functions is characterized in seemingly very different ways this work extends the notion of operator monotonicity to symmetric functions f on k arguments the latter is used to define generated k isotropic maps f s n s n k for any n k necessary and sufficient conditions are given for f to generate an operator monotone k isotropic map f when k 1 the k isotropic map becomes a lowner operator and when k n it becomes a spectral function this allows us to reconcile and explain the differences between the conditions for monotonicity for the lowner operators and the spectral functions

a benders decomposition based matheuristic for the cardinality constrained shift design problem abstract the shift design problem is an important optimization problem which arises when scheduling personnel in industries that require continuous operation based on the forecast required staffing levels for a set of time periods a set of shift types that best covers the demand must be determined a shift type is a consecutive sequence of time periods that adheres to legal and union rules and can be assigned to an employee on any day in this paper we introduce the cardinality constrained shift design problem a variant of the shift design problem in which the number of permitted shift types is bounded by an upper limit we present an integer programming model for this problem and show that its structure lends itself very naturally to benders decomposition due to convergence issues with a conventional implementation we propose a matheuristic based on benders decomposition for solving the problem furthermore we argue that an important step in this approach is finding dual alternative optimal solutions to the benders subproblems and describe an approach to obtain a diverse set of these numerical tests show that the described methodology significantly outperforms a commercial mixed integer programming solver on instances with 1241 different shift types and remains competitive for larger cases with 2145 shift types on all classes of problems the heuristic is able to quickly find good solutions

optimal shift scheduling with multiple break windows this paper presents a new integer programming model for optimal shift scheduling with multiple rest and lunch breaks and break windows a set covering approach for this problem was originally developed by dantzig 1954 since then a number of set covering based formulations have been proposed in the literature these formulations require an integer variable for every shift type shift start time and rest lunch break placement combination unfortunately the number of integer variables required is rather large making them impractical to solve for an optimal solution in most applications we present a new approach in which a set of break variables is introduced for every shift break type combination to determine the break placements this approach leads to a significantly improved integer programming model requiring substantially smaller number of variables and computer memory we tested the proposed approach with 40 test problems involving between 1 728 and 8 640 shift variations and five demand patterns our results showed that the proposed formulation is very useful in solving large shift scheduling problems optimally

convex analysis of spectrally defined matrix functions the purpose of this work is to carry out a systematic study of a special class of convex functions defined over the space s n of symmetric matrices of order n ti n the functions under consideration phi s n ra r cup infty are spectrally defined in the sense that the value phi a depends only on the spectrum lam 1 a ldots lam n a of the matrix a in s n fenchel legendre conjugation first and second order subdifferentiability asymptotic behavior and other concepts of convex analysis are the main ingredients of our exposition

the higher order derivatives of spectral functions in this paper we are interested in the higher order derivatives of functions of the eigenvalues of symmetric matrices with respect to the matrix argument we describe the formula for the k th derivative of such functions in two general cases the flrst case concerns the derivatives of the composition of an arbitrary not necessarily symmetric k times difierentiable function with the eigenvalues of symmetric matrices at a symmetric matrix with distinct eigenvalues the second case describes the derivatives of the composition of a k times difierentiable separable symmetric function with the eigenvalues of symmetric matrices at an arbitrary symmetric matrix we show that the formula signiflcantly simplifles when the separable symmetric function is k times continuously difierentiable as an application of the developed techniques we re derive the formula for the hessian of a general spectral function at an arbitrary symmetric matrix the new tools lead to a shorter cleaner derivation than the original one in 16 to make the exposition as self contained as possible we have included the necessary background results and deflnitions the proofs of the intermediate technical results are collected in the appendices

accidental degeneracy and symmetry lie algebra the authors consider a very elementary example of accidental degeneracy and show that it is not explained by the symmetry lie algebra of the corresponding classical problem

first order sensitivity analysis of symplectic eigenvalues abstract for every 2 n 2 n positive definite matrix a there are n positive numbers d 1 a d n a associated with a called the symplectic eigenvalues of a it is known that d m are continuous functions of a but are not differentiable in general in this paper we show that the directional derivative of d m exists and derive its expression we also discuss various subdifferential properties of d m such as clarke and michel penot subdifferentials

secure key rate of the bb84 protocol using finite sample bits we improve the non asymptotic key rate shown by scarani and renner by proposing several methods to construct tighter conservative confidence intervals of the phase error rate than one previously shown in addition we show that the accurate channel estimation method non asymptotically increases the key rate over the amplitude damping channel as well as the asymptotic case in the bb84 protocol

solving log determinant optimization problems by a newton cg primal proximal point algorithm we propose a newton cg primal proximal point algorithm ppa for solving large scale log determinant optimization problems our algorithm employs the essential ideas of ppa the newton method and the preconditioned cg solver when applying the newton method to solve the inner subproblem we find that the log determinant term plays the role of a smoothing term as in the traditional smoothing newton technique focusing on the problem of maximum likelihood sparse estimation of a gaussian graphical model we demonstrate that our algorithm performs favorably compared to existing state of the art algorithms and is much preferred when a high quality solution is required for problems with many equality constraints

the simplest semidefinite programs are trivial we consider optimization problems of the following type min left mbox tr cx colon ax b x quad mbox positive semidefinite right here tr denotes the trace operator c and x are symmetric n n matrices b is a symmetric m m matrix and a denotes a linear operator such problems are called semidefinite programs and have recently become the object of considerable interest due to important connections with max min eigenvalue problems and with new bounds for integer programming in the context of symmetric matrices the simplest linear operators have the following form ax mxm t where m is an arbitrary m n matrix in this paper we show that for such linear operators the optimization problem is trivial in the sense that an explicit solution can be given

on multiple eigenvalues of matrices depending on several parameters the eigenvalues and eigenvectors as well as their derivatives of matrices depending on one or several parameters have been studied by sun j comput math 3 1985 pp 351 364 using the implicit function theorems and others but not for the more general multiple eigenvalues in this paper the mean of the multiple eigenvalues in terms of the trace operator tr the corresponding invariant subspaces and their derivatives are studied using a generalization of sun s approach the numerical aspects concerning the computation of such derivatives by direct and iterative methods e g simultaneous and inverse simultaneous iterations will be discussed briefly the main results in this paper apply to clusters of nonmultiple eigenvalues as well the implications e g on simultaneous iteration will be explored

first and second order epi differentiability in eigenvalue optimization abstract the subject of this work is the first and second order sensitivity analysis of some spectral functions which are essential in eigenvalue optimization by the way of epi differentiability we show that the sum of the m largest eigenvalues of a real symmetric matrix is twice epi differentiable and we derive an explicit expression of its second order epi derivative we also prove that the mth largest eigenvalue function is twice epi differentiable if and only if it ranks first in a group of equal eigenvalues finally we derive chain rules and then we obtain optimality conditions for an important class of eigenvalue optimization problems

sensitivity analysis of all eigenvalues of a symmetric matrix given a x a ij x a n by n symmetric matrix depending smoothly on a parameter x we study the first order sensitivity of all the eigenvalues lambda m x of a x 1 le m le n under some smoothness assumption like the a ij be c 1 we prove that the directional derivatives d mapsto lambda prime m x d lim t to o lambda m x td lambda m x t do exist and give an explicit expression of them in terms of the data of the parametrized matrix the key idea to circumvent the difficulties inherent to the study of each lambda m taken separately is to consider the functions f m x 1 le m le n defined as the sums of the m largest eigenvalues of a x based on ky fan s variational formulation of f m and some chain rule from nonsmooth analysis we derive an explicit formula for the generalized gradient of f m and a computationally useful formula for the directional derivative of f m using these formulas and the relation lambda m f m f m 1 we then derive the directional derivative of lambda m some properties of this directional derivative as well as an illustrative example are presented

variational analysis of non lipschitz spectral functions we consider spectral functions f where f is any permutation invariant mapping from cn to r and is the eigenvalue map from the set of n n complex matrices to cn ordering the eigenvalues lexicographically for example if f is the function maximum real part then f is the spectral abscissa while if f is maximum modulus then f is the spectral radius both these spectral functions are continuous but they are neither convex nor lipschitz for our analysis we use the notion of subgradient extensively analyzed in variational analysis r t rockafellar and r j b wets springer 1998 we show that a necessary condition for y to be a subgradient of an eigenvalue function f at x is that y commutes with x we also give a number of other necessary conditions for y based on the schur form and the jordan form of x in the case of the spectral abscissa we refine these conditions and we precisely identify the case where subdifferential regularity holds we conclude by introducing the notion of a semistable program maximize a linear function on the set of square matrices subject to linear equality constraints together with the constraint that the real parts of the eigenvalues of the solution matrix are non positive semistable programming is a nonconvex generalization of semidefinite programming using our analysis we derive a necessary condition for a local maximizer of a semistable program and we give a generalization of the complementarity condition familiar from semidefinite programming

a sequential semismooth newton method for the nearest low rank correlation matrix problem based on the well known result that the sum of the largest eigenvalues of a symmetric matrix can be represented as a semidefinite programming problem sdp we formulate the nearest low rank correlation matrix problem as a nonconvex sdp and propose a numerical method that solves a sequence of least square problems each of the least square problems can be solved by a specifically designed semismooth newton method which is shown to be quadratically convergent the sequential method is guaranteed to produce a stationary point of the nonconvex sdp our numerical results demonstrate the high efficiency of the proposed method on large scale problems

convexity and the quantum many body problem we recall some properties of convex functions and in particular of the sum of the largest eigenvalues of a hermitian matrix from these properties a new estimate of an arbitrary eigenvalue of a sum of hermitian matrices is derived which in turn is used to compute an approximate associated spectral projector these estimates are applied for the first time to explain the generic spectral features of quantum systems as an application of the formalism we explain the preponderance of certain ground state angular momenta as observed in the vibron model with random interactions we show that the evolution of eigenstates can be predicted from the knowledge of a limited number of spectra and investigate the effect of a three body interaction in the vibron model on eigenenergies and eigenvectors

nearest southeast submatrix that makes multiple a prescribed eigenvalue part 1 abstract given four complex matrices a b c and d where a c n n and d c m m and given a complex number z 0 what is the spectral norm distance from d to the set of matrices x c m m such that z 0 is a multiple eigenvalue of the matrix a b c x this problem is solved when z 0 is not an eigenvalue of a we also give a conjecture when z 0 is an eigenvalue of a

large scale optimization of eigenvalues optimization problems involving eigenvalues arise in many applications let x be a vector of real parameters and let a x be a continuously differentiable symmetric matrix function of x we consi

nearest common root of a set of polynomials a structured singular value approach abstract the paper considers the problem of calculating the nearest common root of a polynomial set under perturbations in their coefficients in particular we seek the minimum magnitude perturbation in the coefficients of the polynomial set such that the perturbed polynomials have a common root it is shown that the problem is equivalent to the solution of a structured singular value problem arising in robust control for which numerous techniques are available it is also shown that the method can be extended to the calculation of an approximate gcd of fixed degree by introducing the notion of the generalized structured singular value of a matrix the work generalizes previous results by the authors involving the calculation of the approximate gcd of two polynomials although the general case considered here is considerably harder and relies on a matrix dilation approach and several preliminary transformations

an interior point method for minimizing the maximum eigenvalue of a linear combination of matrices we consider the problem of finding lambda opt mathop inf limits x in r m lambda max a 0 sum limits i 1 m x i a i 1 and an optimal solution xopt if it exists here the a i are given symmetric matrices in ir n n for 0 i m and max a is the largest eigenvalue of a matrix a problem 1 can be rewritten as a convex differentiable problem with a positive definite constraint positive definiteness can easily be verified numerically via cholesky decomposition moreover the cone of positive definite matrices a is characterized by the domain of the smooth convex barrier function log det a

diabolical points in the spectra of triangles accidental degeneracies between energy levels e j and e j 1 of a real hamiltonian can occur generically in a family of hamiltonians labelled by at least two parameters x y energy level surfaces in e x y space have locally a double cone diabolo connection and we refer to the degeneracies themselves as diabolical points we studied the family of systems in which a particle moves freely within hard walled triangles vibrations of triangular membranes with x and y labelling two of the angles using an efficient green function technique to compute the levels we found several diabolical points for low lying levels as well as some symmetry degeneracies the lowest diabolical point occurred for levels 5 and 6 of the triangle 130 57 30 73 18 70 the conical structure was confirmed by noting that the normal derivative u of the wavefunction at a boundary point changed sign during a small circuit of the diabolical point the form of the variation of u around a circuit and the changing pattern of nodal lines of agreed with theoretical expectations an estimate of the total number of degeneracies n d j involving levels 1 through j based on the energy scaling of cone angles and the level spacing distribution gave n d j j 2 5 and our limited data support this prediction analytical theory confirmed that for thin triangles where our computational method is slow there are no degeneracies in the energy range studied

double and triple points in the spectra of the cubic symmetry hamiltonian accidental degeneracy of levels is studied in the spectrum of the hamiltonian used frequently in applications which is a linear combination of spherical tensors rank n or 6 invariant with respect to transformations of a cubic group the degeneracy arises only at definite values of the hamiltonian parameters and seems to be irrelevant with respect to increasing hamiltonian symmetry the basis of irreducible representations dgamma of the cubic group is constructed in the space of irreducible representations dj of the so 3 group

spectral operators of matrices the class of matrix optimization problems mops has been recognized in recent years to be a powerful tool by researchers far beyond the optimization community to model many important ap plications involving structured low rank matrices this trend can be credited to some extent to the exciting developments in the emerging field of compressed sensing the lowner operator which gen erates a matrix valued function by applying a single variable function to each of the singular values of a matrix has played an important role for a long time in solving matrix optimization problems however the classical theory developed for lowner operators has become inadequate in these recent applications the main objective of this paper is to provide some necessary theoretical foundations for designing numerical methods for solving the mop this goal is achieved by introducing and conducting a thorough study on a new class of matrix valued functions coined as spectral operators of matrices several fundamental properties of spectral operators including the well definedness continuity di rectional differentiability frechet differentiability locally lipschitzian continuity order b ouligand differentiability 0 1 order g semismooth 0 1 and the characterization of clarke s generalized jacobian are systematically studied

quadratic expansions of spectral functions a function f on the space of n n real symmetric matrices is called spectral if it depends only on the eigenvalues of its argument that is f a f uau t for every orthogonal u and symmetric a in its domain spectral functions are in one to one correspondence with the symmetric functions on r n those that are invariant under arbitrary swapping of their arguments in this paper we show that a spectral function has a quadratic expansion around a point a if and only if its corresponding symmetric function has quadratic expansion around a the vector of eigenvalues we also give a concise and easy to use formula for the hessian of the spectral function in the case of convex functions we show that a positive definite hessian of f implies positive definiteness of the hessian of f 2002 elsevier science inc all rights reserved

on minimizing the largest eigenvalue of a symmetric matrix abstract optimization problems involving eigenvalues arise in many engineering problems in this paper we consider the problem of minimizing the largest eigenvalue over an affine family of symmetric matrices this problem has a variety of applications such as the stability analysis of dynamic systems or the computation of structured singular values given 0 we give an optimality condition which ensures that the largest eigenvalue is within error bound of the solution also a new line search rule is proposed and it is shown to have good descent properties when the multiplicity of the largest eigenvalue the solution is known a new algorithm for the optimization problem under consideration is proposed some numerical experiments on the proposed algorithm are presented

approximating functions on stratified sets we investigate smooth approximations of functions with prescribed gradient behavior on a distinguished stratified subset of the domain as an application we outline how our results yield important consequences for a recently introduced class of stochastic processes called the matrix valued bessel processes

quantifying the informativeness of similarity measurements in this paper we describe an unsupervised measure for quantifying the informativeness of correlation matrices formed from the pairwise similarities or relationships among data instances the measure quantifies the heterogeneity of the correlations and is defined as the distance between a correlation matrix and the nearest correlation matrix with constant off diagonal entries this non parametric notion generalizes existing test statistics for equality of correlation coefficients by allowing for alternative distance metrics such as the bures and other distances from quantum information theory for several distance and dissimilarity metrics we derive closed form expressions of informativeness which can be applied as objective functions for machine learning applications empirically we demonstrate that informativeness is a useful criterion for selecting kernel parameters choosing the dimension for kernel based nonlinear dimensionality reduction and identifying structured graphs we also consider the problem of finding a maximally informative correlation matrix around a target matrix and explore parameterizing the optimization in terms of the coordinates of the sample or through a lower dimensional embedding in the latter case we find that maximizing the bures based informativeness measure which is maximal for centered rank 1 correlation matrices is equivalent to minimizing a specific matrix norm and present an algorithm to solve the minimization problem using the norm s proximal operator the proposed correlation denoising algorithm consistently improves spectral clustering overall we find informativeness to be a novel and useful criterion for identifying non trivial correlation structure

nonsmooth analysis of eigenvalues the eigenvalues of a symmetric matrix depend on the matrix nonsmoothly this paper describes the nonsmooth analysis of these eigenvalues in particular i present a simple formula for the approximate limiting frechet subdifferential of an arbitrary function of the eigenvalues subsuming earlier results on convex and clarke subgradients as an example i compute the subdifferential of the k th largest eigenvalue

semi definite matrix constraints in optimization positive semi definite matrix constraints arise in a number of optimization problems in which some or all of the elements of a matrix are variables such as the educational testing and matrix modification problems the structure of such constraints is developed including expressions for the normal cone feasible directions and their application to optimality conditions a computational framework is given within which these concepts can be exploited and which permits the quantification of second order effects the matrix of lagrange multipliers in this formulation is shown to have an important relationship to the characterization of the normal cone modifications of various iterative schemes in nonlinear programming are considered in order to develop an effective algorithm for the educational testing problem and comparative numerical experiments are described it is shown that a particular choice of the penalty parameter for an l 1 exact penalty function is appropriate for this type of problem the be

constrained maximum entropy sampling a fundamental experimental design problem is to select a most informative subset having prespecified size from a set of correlated random variables instances of this problem arise in many applied domains such as meteorology environmental statistics and statistical geology in these applications observations can be collected at different locations and possibly at different times information is measured by entropy practical situations have further restrictions on the design space for example budgetary limits geographical considerations as well as legislative and political considerations may restrict the design space in a complicated manner using techniques of linear algebra combinatorial optimization and convex optimization we develop upper and lower bounds on the optimal value for the gaussian case we describe how these bounds can be integrated into a branch and bound algorithm for the exact solution of these design problems finally we describe how we have implemented this algorithm and we present computational results for estimated covariance matrices corresponding to sets of environmental monitoring stations in the ohio valley of the united states

semismooth matrix valued functions matrix valued functions play an important role in the development of algorithms for semidefinite programming problems this paper studies generalized differential properties of such functions related to nonsmooth smoothing newton methods the first part of this paper discusses basic properties such as the generalized derivative rademacher s theorem b derivative directional derivative and semismoothness the second part shows that the matrix absolute value function the matrix semidefinite projection function and the matrix projective residual function are strongly semismooth

explicit solutions for interval semidefinite linear programs abstract we consider the special class of semidefinite linear programs ivp maximize trace cx subject to l a x u where c x l u are symmetric matrices a is an onto linear operator and denotes the lowner positive semidefinite partial order we present explicit representations for the general primal and dual optimal solutions this extends the results for standard linear programming that appeared in ben israel and charnes 3 this work is further motivated by the explicit solutions for a different class of semidefinite problems presented recently in yang and vanderbei 15

accelerating benders decomposition algorithmic enhancement and model selection criteria this paper proposes methodology for improving the performance of benders decomposition when applied to mixed integer programs it introduces a new technique for accelerating the convergence of the algorithm and theory for distinguishing good model formulations of a problem that has distinct but equivalent mixed integer programming representations the acceleration technique is based upon selecting judiciously from the alternate optima of the benders subproblem to generate strong or pareto optimal cuts this methodology also applies to a much broader class of optimization algorithms that includes dantzig wolfe decomposition for linear and nonlinear programs and related cutting plane type algorithms that arise in resource directive and price decomposition when specialized to network location problems this cut generation technique leads to very efficient algorithms that exploit the underlying structure of these models in discussing the proper formulation of mixed integer programs we suggest criteri

asymptotic expansions of the ordered spectrum of symmetric matrices abstract in this work we build on ideas of torki 2001 6 and show that if a symmetric matrix valued map t a t has a one sided asymptotic expansion at t 0 of order k then so does t m a t where m is the m th largest eigenvalue we derive formulas for computing the coefficients a 0 a 1 a k in the asymptotic expansion as an application of the approach we give a new proof of a classical result due to kato 1976 3 about the one sided analyticity of the ordered spectrum under analytic perturbations finally as a demonstration of the derived formulas we compute the first three terms in the asymptotic expansion of m a t e for any fixed symmetric matrices a and e

an introduction to a class of matrix cone programming in this paper we define a class of linear conic programming which we call matrix cone programming or mcp involving the epigraphs of five commonly used matrix norms and the well studied symmetric cone mcp has recently been found to have many important applications for example in nuclear norm relaxations of affine rank minimization problems in order to make the defined mcp tractable and meaningful we must first understand the structure of these epigraphs so far only the epigraph of the frobenius matrix norm which can be regarded as a second order cone has been well studied here we take an initial step to study several important properties including its closed form solution calm bouligand differentiability and strong semismoothness of the metric projection operator over the epigraph of the l 1 l infty spectral or operator and nuclear matrix norm respectively these properties make it possible to apply augmented lagrangian methods which have recently received a great deal of interests due to their high efficiency in solving large scale semidefinite programming to this class of mcp problems the work done in this paper is far from comprehensive rather it is intended as a starting point to call for more insightful research on mcp so that it can serve as a basic tool to solve more challenging convex matrix optimization problems in years to come

large scale binary quadratic optimization using semidefinite relaxation and applications in computer vision many problems can be formulated as binary quadratic programs bqps which are in general np hard finding a solution when the problem is of large size to be of practical interest typically requires relaxation semidefinite relaxation usually yields tight bounds but its computational complexity is high in this work we present a semidefinite programming sdp formulation for bqps with two desirable properties first it produces similar bounds to the standard sdp formulation second compared with the conventional sdp formulation the proposed sdp formulation leads to a considerably more efficient and scalable dual optimization approach we then propose two solvers namely quasi newton and smoothing newton methods for the simplified dual problem both of them are significantly more efficient than standard interior point methods empirically the smoothing newton solver is faster than the quasi newton solver for dense or medium sized problems while the quasi newton solver is preferable for large sparse structured problems

expressing a polynomial as the characteristic polynomial of a symmetric matrix abstract we show that given a polynomial one can without knowing the roots construct a symmetric matrix whose characteristics polynomial is the given polynomial appropriately normed for a real polynomial the matrix can have purely imaginary off diagonal entries for a polynomial with only real distinct roots the matrix can be chosen real

derivatives of compound matrix valued functions abstract primary matrix functions and spectral functions are two classes of orthogonally invariant functions on a symmetric matrix argument many of their properties have been investigated thoroughly and find numerous applications both theoretical and applied in areas ranging from engineering image processing optimization and physics we propose a family of maps that provide a natural connection and generalization of these two classes of functions the family of maps also contains the well known multiplicative and additive compound matrices we explain when each member of this family is a differentiable function and exhibit a formula for its derivative

applications of parametric programming and eigenvalue maximization to the quadratic assignment problem we investigate new bounding strategies based on different relaxations of the quadratic assignment problem in particular we improve the lower bound found by using an eigenvalue decomposition of the quadratic part and by solving a linear program for the linear part the improvement is accomplished by applying a steepest ascent algorithm to the sum of the two bounds