lower bounds for the perron root of a nonnegative matrix abstract new lower bounds frequently better than previous bounds and readily computable for the perron root r a of a nonnegative matrix a are obtained the theoretical comparison of new bounds with the known ones is supplemented with two numerical examples

functional inequalities for spectral radii of nonnegative matrices abstract let p n denote the set of all square n n nonnegative matrices for a k a k ij n i j 1 k 1 m k not a power we set f a 1 a m f a 1 ij a m ij n i j 1 for each a p n we let a denote its spectral radius this paper is concerned with the characterization of those functions f r m r satisfying either f a 1 a m f a 1 a m 1 or f a 1 a m f a 1 a m 2 for all a 1 a m p n and every n e n we totally characterize all functions satisfying 1 we delineate various classes of functions which satisfy 2 if f 0 0 and f is bounded above in some neighborhood of any eint r m then we totally characterize all f satisfying 2

bounds on the joint and generalized spectral radius of the hadamard geometric mean of bounded sets of positive kernel operators abstract let 1 m be bounded sets of positive kernel operators on a banach function space l we prove that for the generalized spectral radius and the joint spectral radius the inequalities 1 1 m m 1 m 1 2 m 1 m 1 1 m m 1 m 1 2 m 1 m hold where 1 1 m m 1 m denotes the hadamard schur geometric mean of the sets 1 m

on lipschitz optimization based on gray box piecewise linearization we address the problem of minimizing objectives from the class of piecewise differentiable functions whose nonsmoothness can be encapsulated in the absolute value function they possess local piecewise linear approximations with a discrepancy that can be bounded by a quadratic proximal term this overestimating local model is continuous but generally nonconvex it can be generated in its abs normal form by a minor extension of standard algorithmic differentiation tools here we demonstrate how the local model can be minimized by a bundle type method which benefits from the availability of additional gray box information via the abs normal form in the convex case our algorithm realizes the consistent steepest descent trajectory for which finite convergence was established earlier specifically covering counterexamples where steepest descent with exact line search famously fails the analysis of the abs normal representation and the design of the optimization algorithm are geared toward the general case whereas the convergence proof so far covers only the convex case

a constrained maximum likelihood approach to estimating switching regressions abstract it is widely known that the likelihood function for the switching regression model is unbounded if the error variances are unconstrained this paper shows that a constrained maximum likelihood formulation makes the likelihood function bounded relatively mild constraints are imposed on the parameters and if the true parameters satisfy the constraints there is a global maximizer of the likelihood function on the constrained parameter space which is consistent asymptotically normal and efficient a well known em algorithm is modified in order to compute constrained maximizers of the likelihood function

limit eigenvalues of nonnegative matrices abstract in this paper we give a simple closed formula for a certain limit eigenvalue of a nonnegative matrix we bound the spectral radius of a nonnegative matrix in terms of this limit eigenvalue

probability inequalities for empirical processes and a law of the iterated logarithm given a class f of functions on x we can view vn as a stochastic process indexed by f and consider limit theorems for this process to prove such theorems it is often helpful to have bounds on the tail of the r v sup w i n f i see for example dudley 1978 kuelbs and dudley 1980 dudley and phillip 1983 or gine and zinn 1983 our main question of interest is for what classes f can best possible bounds be obtained so first we must ask what are these best possible bounds of course

on nesterov s nonsmooth chebyshev rosenbrock functions abstract we discuss two nonsmooth functions on r n introduced by nesterov we show that the first variant is partly smooth in the sense of lewis and that its only stationary point is the global minimizer in contrast we show that the second variant has 2 n 1 clarke stationary points none of them local minimizers except the global minimizer but also that its only mordukhovich stationary point is the global minimizer nonsmooth optimization algorithms from multiple starting points generate iterates that approximate all 2 n 1 clarke stationary points not only the global minimizer but it remains an open question as to whether the nonminimizing clarke stationary points are actually points of attraction for optimization algorithms

functional inequalities for max eigenvalues of nonnegative matrices let pn denote the set of all n n nonnegative matrices for a function f r m r and matrices ak aijk i j 1n k 1 m define f a1 am f aij1 aijm ij 1n for each a pn we denote its spectral radius by a and its max eigenvalue by a in a previous paper all functions f which satisfy f a1 am f a1 am n n a1 am pn and some functions which satisfy f a1 am f a1 am n n a1 am pn were characterized here for an interval i in r we characterize those functions f satisfying f a1 am f a1 am n n a1 am inn as well as the functions satisfying f 0 0 and f a1 am f a1 am n n a1 am inn

extending the akaike information criterion to mixture regression models we examine the problem of jointly selecting the number of components and variables in finite mixture regression models we find that the akaike information criterion is unsatisfactory for this purpose because it overestimates the number of components which in turn results in incorrect variables being retained in the model therefore we derive a new information criterion the mixture regression criterion mrc that yields marked improvement in model selection due to what we call the clustering penalty function moreover we prove the asymptotic efficiency of the mrc we show that it performs well in monte carlo studies for the same or different covariates across components with equal or unequal sample sizes we also present an empirical example on sales territory management to illustrate the application and efficacy of the mrc finally we generalize the mrc to mixture quasi likelihood and mixture autoregressive models thus extending its applicability to non gaussian models discrete responses and d

a constrained formulation of maximum likelihood estimation for normal mixture distributions in this context the normal densities are sometimes referred to as component densities the log likelihood function l zy corresponding to a random sample xl xnj is defined by l ty ek 1 10g pm xk y the first problem associated with maximum likelihood estimation arises from the unboundedness of l qy on r day 1969 a global maximum likelihood estimate always fails to exist in addition the unboundedness of l y causes failures of optimization algorithms of both the em redner and walker 1984 and quasi newton fowlkes 1979 types in spite of the unboundedness of l 4y statistical theory kiefer 1978 guarantees that a particular local maximizer of l y is strongly consistent and asymptotically efficient several local maximizers can exist for a given sample and the other major maximum likelihood difficulty is in determining when the correct one has been found day 1969 noted that spurious maximizers corresponding to parameter points having some component standard deviations very small relative to others are generated by any small number of sample points grouped sufficiently close together the spurious maximizers like the unboundedness of l zy can create difflculties when using the em or quasi newton algorithms the constrained maximum likelihood formulation presented in the next section avoids the unconstrained problems of singularities and spurious maximizers of the type noted by day section 3 contains the statement and proof of a consistency result for the corresponding constrained estimator

some new inequalities for the hadamard product of nonnegative matrices abstract in this paper we propose some sharper upper bounds for the spectral radius of the hadamard product of two nonnegative matrices these bounds improve the existing results in addition examples are also given to show that the new upper bounds can be achieved

variable metric bundle methods from conceptual to implementable forms to minimize a convex function we combine moreau yosida regularizations quasi newton matrices and bundling mechanisms first we develop conceptual forms using reversal quasi newton formulae and we state their global and local convergence then to produce implementable versions we incorporate a bundle strategy together with a curve search no convergence results are given for the implementable versions however some numerical illustrations show their good behaviour even for large scale problems

lexicographic differentiation of nonsmooth functions we present a survey on the results related to the theory of lexicographic differentiation this theory ensures an efficient computation of generalized lexicographic derivative of a nonsmooth function belonging to a special class of lexicographically smooth functions this class is a linear space which contains all differentiable functions all convex functions and which is closed with respect to component wise composition of the members in order to define lexicographic derivative in a unique way it is enough to fix a basis in the space of variables lexicographic derivatives can be used in black box optimization methods we give some examples of applications of these derivatives in analysis of nonsmooth functions it is shown that the system of lexicographic derivatives along a fixed basis correctly represents corresponding nonsmooth function newton leibnitz formula we present nonsmooth versions of standard theorems on potentiality of nonlinear operators on differentiation of parametric integrals and on differentiation of functional sequences finally we show that an appropriately defined lexicographic subdifferential ensures a more rigorous selection of a candidate optimal solution than the subdifferential of clarke

relaxing kink qualifications and proving convergence rates in piecewise smooth optimization in the paper optim methods softw 31 2016 pp 904 930 we derived first order kkt and second order second order sufficiency condition sosc optimality conditions for functions defined b

the erdos renyi strong law for pattern matching with a given proportion of mismatches 1 informal introduction the erdos bnyi law erdijs and r6nyi 1970 l for coin tossing is a strong law for the behavior of the length of long success rich runs for the length r of the longest run of pure heads in n tosses the result is that with probability l lin rjlog n 1 where p flheads 0 a more general result is quoted as formula 1 in section 2 for rz the longest head rich run in which the fraction of heads is at least a p note that r ri in many practical situations such as manufacturing or roulette observations are taken sequentially and each can be classified as success or failure for these cases it is possible to use the erdijs wnyi law to test the hypothesis that the success probability is p from another point of view the erdijs bnyi law can be used to recognize patterns of unusually long runs of succe88e8 or failures our interest is in the recognition of unusually long patterns or words common to two random sequences the patterns are unknown prior to an examination of the sequences the motivation for this work is the comparison of dna sequences which can be modeled as sequences of i i d or markov distributed letters evolution operates to conserve although imperfectly patterns important to biological function it is a task of biology to discover these patterns and their function in earlier work we generalized the erdiie bnyi law for pure head runs a 1 to exact matching patterns between two sequences in this paper we extend those results to include matchings of quality a between p and 1 the examples below illustrate the natural analogs of rz studied in this paper two words form a quality a matching if they have the same length and the

rank restricting functions abstract in this paper we characterize for given positive integers k and d the class of functions f r r such that for every n m real valued matrix a aij i 1nj 1m arbitrary n and m of rank at most k the matrix f a f aij i 1nj 1m has rank at most d as well as the class of functions g c c such that for every n m complex valued matrix a aij i 1nj 1m arbitrary n and m of rank at most k the matrix g a g aij i 1nj 1m has rank at most d for k 2 each such function f is a polynomial of an appropriate form which we shall exactly delineate while each g is a polynomial in z and z also of an explicitly delineated form for k 1 the class of such functions in each case is significantly different nonetheless it is via the study of the case k 1 that we are able to characterize such functions where k 2

a robust gradient sampling algorithm for nonsmooth nonconvex optimization let f be a continuous function on rl n and suppose f is continuously differentiable on an open dense subset such functions arise in many applications and very often minimizers are points at which f is not differentiable of particular interest is the case where f is not convex and perhaps not even locally lipschitz but is a function whose gradient is easily computed where it is defined we present a practical robust algorithm to locally minimize such functions based on gradient sampling no subgradient information is required by the algorithm when f is locally lipschitz and has bounded level sets and the sampling radius eps is fixed we show that with probability 1 the algorithm generates a sequence with a cluster point that is clarke eps stationary furthermore we show that if f has a unique clarke stationary point bar x then the set of all cluster points generated by the algorithm converges to bar x as eps is reduced to zero numerical results are presented demonstrating the robustness of the algorithm and its applicability in a wide variety of contexts including cases where f is not locally lipschitz at minimizers we report approximate local minimizers for functions in the applications literature which have not to our knowledge been obtained previously when the termination criteria of the algorithm are satisfied a precise statement about nearness to clarke eps stationarity is available a matlab implementation of the algorithm is posted at http www cs nyu edu overton papers gradsamp alg

nonsmooth optimization via quasi newton methods we investigate the behavior of quasi newton algorithms applied to minimize a nonsmooth function f not necessarily convex we introduce an inex act line search that generates a sequence of nested intervals containing a set of points of nonzero measure that satisfy the armijo and wolfe conditions if f is absolutely continuous along the line furthermore the line search is guaranteed to terminate if f is semi algebraic it seems quite difficult to establish a convergence theorem for quasi newton methods applied to such general classes of functions so we give a care ful analysis of a special but illuminating case the euclidean norm in one variable using the inexact line search and in two variables assuming that the line search is exact in practice we find that when f is locally lipschitz and semi algebraic with bounded sublevel sets the bfgs broyden fletcher goldfarb shanno method with the inexact line search almost always generates sequences whose cluster points are clarke stationary and with function values converging r linearly to a clarke station ary value we give references documenting the successful use of bfgs in a variety of nonsmooth applications particularly the design of low order controllers for linear dynamical systems we conclude with a challenging open question

a note on the height of suffix trees consider a random word in which the individual symbols are drawn from a finite or infinite alphabet with symbol probabilities p i and let h n be the height of the suffix tree constructed from the first n suffixes of this word it is shown that h n is asymptotically close to 2 log n log 1 sum i p i 2 in many respects the difference is o log log n in probability and the ratio tends to one almost surely and in the mean

a note on phillips 1991 a constrained maximum likelihood approach to estimating switching regressions abstract phillips phillips r f 1991 a constrained maximum likelihood approach to estimating switching regressions journal of econometrics 48 241 262 proposed a constrained maximum likelihood approach to estimating the parameters in a switching regression model in this note we propose a new approach which leads to a proof of a more general result than phillips s specifically we prove that the constrained mle cmle is still strongly consistent when the constant c decreases to 0 at the rate of exp n 1 2 log n as n increases to with 1 we also suggest a suitable hence c n for practice based on simulation results

nonnegative matrices zero patterns and spectral inequalities we give an inequality for the spectral radius of positive linear combinations of tuples of nonnegative matrices linked in a certain way from this several observations are deduced including the perturbation inequality a t sgn a a t sgn a in which sgn denotes the signum functions applied componentwise this was motivated in part by our work on characterizing the functions ff such that f a1 ap f a1 ap in which f is applied componentwise on the left to be published elsewhere one of our observations the question of studying the combinatorial structure of the cone algebra of nonnegative matrices with given left and right perron vectors

inequalities for the perron root related to levinger s theorem abstract for the perron roots of square nonnegative matrices a b and a d 1btd where d is a diagonal matrix with positive diagonal entries the inequality a d 1btd a b is proved under the assumption that a and b have a common unordered pair of nonorthogonal right and left perron vectors the case of equality is analyzed the above inequality generalizes the inequality a 1 bt a 1 b proved under stronger assumptions by bapat and implies a generalization of levinger s theorem on the monotonicity of the perron root of a weighted arithmetic mean of a nonnegative matrix and its transpose also for the perron root a c d 1atd c c 1 0 c of a weighted entrywise geometric mean of a and d 1atd where a ij and o denotes the hadamard product the monotonicity property dual to that asserted by generalized levinger s theorem is established

markov switching model selection using kullback leibler divergence abstract in markov switching regression models we use kullback leibler kl divergence between the true and candidate models to select the number of states and variables simultaneously specifically we derive a new information criterion markov switching criterion msc which is an estimate of kl divergence msc imposes an appropriate penalty to mitigate the over retention of states in the markov chain and it performs well in monte carlo studies with single and multiple states small and large samples and low and high noise we illustrate the usefulness of msc via applications to the u s business cycle and to media advertising

solving piecewise linear systems in abs normal form abstract with the ultimate goal of iteratively solving piecewise smooth ps systems we consider the solution of piecewise linear pl equations as shown in 7 pl models can be derived in the fashion of automatic or algorithmic differentiation as local approximations of ps functions with a second order error in the distance to a given reference point the resulting pl functions are obtained quite naturally in what we call the abs normal form a variant of the state representation proposed by bokhoven in his dissertation 27 apart from the tradition of pl modelling by electrical engineers which dates back to the master thesis of thomas stern 26 in 1956 we take into account more recent results on linear complementarity problems and semi smooth equations originating in the optimization community 3 25 5 we analyze simultaneously the original pl problem opl in abs normal form and a corresponding complementary system cpl which is closely related to the absolute value equation ave studied by mangasarian and meyer 14 and a corresponding linear complementarity problem lcp we show that the cpl like kkt conditions and other simply switched systems cannot be open without being injective hence some of the intriguing pl structure described by scholtes in 25 is lost in the transformation from opl to cpl to both problems one may apply newton variants with appropriate generalized jacobians directly computable from the abs normal representation alternatively the cpl can be solved by bokhoven s modulus method and related fixed point iterations we compile the properties of the various schemes and highlight the connection to the properties of the schur complement matrix in particular its partial contractivity as analyzed by rohn and rump 23 numerical experiments and suitable combinations of the fixed point solvers and stabilized generalized newton variants remain to be realized

some monotonicity properties of schur powers of matrices and related inequalities abstract a series of inequalities are developed relating the spectral radius a b of the schur product a b of two nonnegative matrices a and b with those of a a and b b yielding a b a a b b 1 2 as a corollary it is proved that the spectral radius of the schur powers r a r a r a a a r factors satisfies 1 r log r is decreasing while 1 r 1 log r is increasing the latter provided a is a stochastic matrix the entropy of a finite stationary markov chain is identified with d r dr r 1 a number of majorization comparisons for the spectral radius of schur powers is given

an algorithm for nonsmooth optimization by successive piecewise linearization we present an optimization method for lipschitz continuous piecewise smooth ps objective functions based on successive piecewise linearization since in many realistic cases nondifferentiabilities are caused by the occurrence of abs max and min we concentrate on these nonsmooth elemental functions the method s idea is to locate an optimum of a ps objective function by explicitly handling the kink structure at the level of piecewise linear models this piecewise linearization can be generated in its abs normal form by minor extension of standard algorithmic or automatic differentiation tools in this paper it is shown that the new method when started from within a compact level set generates a sequence of iterates whose cluster points are all clarke stationary numerical results including comparisons with other nonsmooth optimization methods then illustrate the capabilities of the proposed approach