least squares solution of f pg over positive semidefinite symmetric p abstract a matrix optimization problem of interest is the infimization for arbitrary f g rn m of f pg with respect to positive semidefinite symmetric p rn n where denotes the frobenius norm one motivation for its study is that solutions may be used as estimates of the inverse hessian of a nonlinear differentiable function f rn r1 which is to be minimized with respect to a parameter vector x rn by a quasi newton type algorithm another is where the compliance matrix of an elastic structure is to be estimated from experimental measurements of the displacements resulting from some static loading necessary and sufficient conditions for the existence of the minimum are given a sufficient condition is that rank g n which is also necessary and sufficient for uniqueness of minimizers arbitrarily accurate infimization can be achieved via solution of an instance of the minimization problem where rank g n however no general solution procedure for the latter is known which motivates algorithmic solution an algorithm is given which for the case rank g n arbitrarily accurately approximates the unique minimizer the algorithm is globally convergent requiring the computation of inner products and solutions of symmetric eigenvalue problems

a convex analytic framework for ergodic control of semi markov processes the ergodic control problem for semi markov processes is reformulated as an optimization problem over the set of suitably defined ergodic occupation measures this set is shown to be closed and convex with its extreme points corresponding to stationary strategies this leads to the existence of optimal stationary strategies under additional hypotheses a pathwise analysis of the joint empirical occupation measures of the state and control processes shows that this optimality is in the strong i e almost sure sense

some new results on eigenvectors via dimension diameter and ricci curvature abstract we generalise for a general symmetric elliptic operator the different notions of dimension diameter and ricci curvature which coincide with the usual notions in the case of the laplace beltrami operators on riemannian manifolds if 1 denotes the spectral gap that is the first nonzero eigenvalue we investigate in this paper the best lower bound on 1 one can obtain under an upper bound on the dimension an upper bound on the diameter and a lower bound of the ricci curvature two cases are known namely if the ricci curvature is bounded below by a constant r 0 then 1 nr n 1 and this estimate is sharp for the n dimensional spheres lichnerowicz s bound if the ricci curvature is bounded below by zero then zhong yang s estimate asserts that 1 2 d 2 where d is an upper bound on the diameter this estimate is sharp for the 1 dimensional torus in the general case many interesting estimates have been obtained this paper provides a general optimal comparison result for 1 which unifies and sharpens lichnerowicz and zhong yang s estimates together with other comparison results concerning the range of the associated eigenfunctions and their derivatives

elliptic gradient estimates for a nonlinear heat equation and applications abstract in this paper we study elliptic gradient estimates for a nonlinear f heat equation which is related to the gradient ricci soliton and the weighted log sobolev constant of smooth metric measure spaces precisely we obtain hamilton s and souplet zhang s gradient estimates for positive solutions to the nonlinear f heat equation only assuming the bakry emery ricci tensor is bounded below as applications we prove parabolic liouville properties for some kind of ancient solutions to the nonlinear f heat equation some special cases are also discussed

gradient estimates and liouville theorems for nonlinear parabolic equations on noncompact riemannian manifolds abstract let m be a complete noncompact riemannian manifold in this paper we derive a local gradient estimate for positive solutions of the nonlinear parabolic equation t u x t x t u x t 0 on m 0 we also obtain a theorem of liouville type for positive solutions of this nonlinear equation this paper extends the result of souplet and zhang p souplet qi s zhang sharp gradient estimate and yau s liouville theorem for the heat equation on noncompact manifolds bull lond math soc 38 2006 1045 1053

positive semidefinite matrices characterization via conical hulls and least squares solution of a matrix equation any real symmetric n times n matrix a can be described by an n n 1 2 component vector positive semidefiniteness of a is characterized by the associated vector belonging to the conical hull of a suitable convex set this characterization is used to facilitate least squared error solution with respect to such a of f ag where f and g are given matrices the solution method involves finding the point in the conical hull of a convex set which is nearest to a vector an algorithm is given for solving that proximal point problem

on the parabolic kernel of the schrodinger operator etude des equations paraboliques du type q x t t u x t 0 sur une variete riemannienne generale introduction estimations de gradients inegalites de harnack majorations et minorations des solutions fondamentales equation de la chaleur et noyau de green operateur de schrodinger

hamilton s gradient estimates and liouville theorems for porous medium equations on noncompact riemannian manifolds abstract let m be a complete noncompact riemannian manifold of dimension n in this paper we derive a local gradient estimate for positive solutions of porous medium equation t u u 1 1 1 2 n 1 on m 0 we also obtain a theorem of liouville type for positive solutions of the porous medium equation

comparison geometry for the bakry emery ricci tensor for riemannian manifolds with a measure m g edvolg we prove mean curvature and volume comparison results when the bakry emery ricci tensor is bounded from below and f is bounded or rf is bounded from below generalizing the classical ones i e when f is constant this leads to extensions of many theorems for ricci curvature bounded below to the bakry emery ricci tensor in particular we give extensions of all of the major comparison theorems when f is bounded simple examples show the bound on f is necessary for these results

computing a nearest symmetric positive semidefinite matrix the nearest symmetric positive semidefinite matrix in the frobenius norm to an arbitrary real matrix a is shown to be b h 2 where h is the symmetric polar factor of b a at 2 in the 2 norm a nearest symmetric positive semidefinite matrix and its distance 2 a from a are given by a computationally challenging formula due to halmos we show how the bisection method can be applied to this formula to compute upper and lower bounds for 2 a differing by no more than a given amount a key ingredient is a stable and efficient test for positive definiteness based on an attempted choleski decomposition for accurate computation of 2 a we formulate the problem as one of zero finding and apply a hybrid newton bisection algorithm some numerical difficulties are discussed and illustrated by example

elliptic gradient estimates and liouville theorems for a weighted nonlinear parabolic equation abstract let m n g e f d v be a complete smooth metric measure space with bakry emery ricci tensor bounded from below we derive elliptic gradient estimates for positive solutions of a weighted nonlinear parabolic equation f t u x t q x t u x t 0 where x t m n and is an arbitrary constant as applications we prove a liouville type theorem for positive ancient solutions and harnack type inequalities for positive bounded solutions

projection like retractions on matrix manifolds this paper deals with constructing retractions a key step when applying optimization algorithms on matrix manifolds for submanifolds of euclidean spaces we show that the operation consisting of taking a tangent step in the embedding euclidean space followed by a projection onto the submanifold is a retraction we also show that the operation remains a retraction if the projection is generalized to a projection like procedure that consists of coming back to the submanifold along admissible directions and we give a sufficient condition on the admissible directions for the generated retraction to be second order this theory offers a framework in which previously proposed retractions can be analyzed as well as a toolbox for constructing new ones illustrations are given for projection like procedures on some specific manifolds for which we have an explicit easy to compute expression

a convex analytic approach to markov decision processes this paper develops a new framework for the study of markov decision processes in which the control problem is viewed as an optimization problem on the set of canonically induced measures on the trajectory space of the joint state and control process this set is shown to be compact convex one then associates with each of the usual cost criteria infinite horizon discounted cost finite horizon control up to an exit time a naturally defined occupation measure such that the cost is an integral of some function with respect to this measure these measures are shown to form a compact convex set whose extreme points are characterized classical results about existence of optimal strategies are recovered from this and several applications to multicriteria and constrained optimization problems are briefly indicated

positive solutions of yamabe type equations on complete manifolds and applications we study the semilinear equation u a x u b x u 1 on a complete riemannian manifold we determine conditions on the coefficients that guarantee existence and nonexistence of positive solutions a very general uniqueness result is also established our main results are valid without explicit curvature assumptions and appear to be new even in rm

gradient estimates for the heat equation under the ricci flow the paper considers a manifold m evolving under the ricci flow and establishes a series of gradient estimates for positive solutions of the heat equation on m

average cost markov decision processes optimality conditions we are concerned in this paper with discrete time markov decision processes mdps with borel state and action spaces x and a respectively and the long run expected average cost criterion when x is a denumerable set man y necessary and or sufficient conditions for the existence of optimal control policies are known however when x is a borel space i e a borel subset of a complete separable metric space most of the available results impose on the mdp very restrictive topological conditions e g compactness and or strong recurrence assumptions such as doeblin s condition see e g 4 9 12 and their references another related work is 7 where we have studied mdps from the viewpoint of the recurrence or ergodicity properties of the state process ln the present paper however we are concerned with the existence of average optimal policies by looking at static optimization problems see condition cs in section 4 related in sorne cases equivalent to the existence of a bounded solution to the so called optimality equation see c4 in section 4 these optimization problems are dual in the sense that under appropriate conditions the existence of an optimal solution to one of the problems implies existence of an optimal

elliptic gradient estimates for a weighted heat equation and applications we obtain two elliptic gradient estimates for positive solutions to the f heat equation on a complete smooth metric measure space with only bakry emery ricci tensor bounded below one is a local sharp souplet zhang s type and the other is a global hamilton s type as applications we prove parabolic liouville theorems for ancient solutions satisfying some growth restriction near infinity in particular the liouville results are suitable for the gradient shrinking or steady ricci solitons the estimates of derivation of the f heat kernel are also obtained

control of markov chains with long run average cost criterion the dynamic programming equations the long run average cost control problem for discrete time markov chains on a countable state space is studied in a very general framework necessary and sufficient conditions for optimality in terms of the dynamic programming equations are given when an optimal stable stationary strategy is known to exist e g for the situations studied in stochastic differential systems stochastic control theory and applications ima vol math app 10 springer verlag new york berlin 1988 pp 57 77 a characterization of the desired solution of the dynamic programming equations is given in a special case also included is a novel convex analytic argument for deducing the existence of an optimal stable stationary strategy when that of a randomized one is known

ergodic control of markov chains with constraints the general case the problem of controlling a markov chain on a countable state space with ergodic or long run average cost is studied in the presence of additional constraints requiring finitely many say m other ergodic costs to satisfy prescribed bounds under extremely general conditions it is proved that an optimal stationary randomized strategy can be found that requires at most m randomizations this generalizes a result of ross

solving semidefinite quadratic linear programs using sdpt3 this paper discusses computational experiments with linear optimization problems involving semidefinite quadratic and linear cone constraints sqlps many test problems of this type are solved using a new release of sdpt3 a matlab implementation of infeasible primal dual path following algorithms the software developed by the authors uses mehrotra type predictor corrector variants of interior point methods and two types of search directions the hkm and nt directions a discussion of implementation details is provided and computational results on problems from the sdplib and dimacs challenge collections are reported

linear matrix equations from an inverse problem of vibration theory abstract the symmetric positive semidefinite and positive definite real solutions of matrix equations a t xa d and a t xa xa y ad d 0 are considered necessary and sufficient conditions for the existence of such solutions and their general forms are derived using the singular value decomposition the theory is motivated and illustrated with a problem of vibration theory

a semi analytical approach for the positive semidefinite procrustes problem abstract the positive semidefinite procrustes psdp problem is the following given rectangular matrices x and b find the symmetric positive semidefinite matrix a that minimizes the frobenius norm of a x b no general procedure is known that gives an exact solution in this paper we present a semi analytical approach to solve the psdp problem first we characterize a family of positive semidefinite matrices that either solve the psdp problem when the infimum is attained or give arbitrary accurate approximations to the infimum when it is not attained this characterization requires the unique optimal solution of a smaller psdp problem where b is square and x is diagonal with positive diagonal elements second we propose a very efficient strategy to solve the psdp problem combining the semi analytical approach a new initialization strategy and the fast gradient method we illustrate the effectiveness of the new approach which is guaranteed to converge linearly compared to state of the art methods

markov decision problems and state action frequencies consider a controlled markov chain with countable state and action spaces basic quantities that determine the values of average cost functionals are identified under some regularity conditions these turn out to be a collection of numbers one for each state action pair describing for each state the relative number of uses of each action these conditional frequencies which are defined pathwise are shown to determine the state action frequencies that in the finite case are known to determine the costs this is extended to the countable case allowing for unbounded costs the space of frequencies is shown to be compact and convex and the extreme points are identified with stationary deterministic policies conditions under which the search for optimality in several optimization problems may be restricted to stationary policies are given these problems include the standard markov decision process as well as constrained optimization both in terms of average cost functionals and variability sensitive optimization an application to a queueing problem is given where these results imply the existence and explicit computation of optimal policies in constrained optimization problems the pathwise definition of the conditional frequencies implies that their values can be controlled directly moreover they depend only on the limiting behavior of the control this has immediate application to adaptive control of markov chains including adaptive control under constraints

zero sum average semi markov games fixed point solutions of the shapley equation this paper deals with zero sum average semi markov games with borel state and action spaces unbounded payoffs and mean holding times a solution to the shapley equation is obtained via the banach fixed point theorem assuming that the model satisfies a lyapunov like condition a growth hypothesis on the payoff function and the mean holding times besides standard continuity and compactness requirements

sample path average optimality for markov control processes the authors consider a markov control process with borel state and actions spaces unbounded costs and under the long run sample path average cost criterion they prove that under very weak assumptions on the transition law and a moment assumption for the one step cost there exists a stationary policy with invariant probability distribution v that is sample path average cost optimal for v almost all initial states in addition every expected average cost optimal stationary policy is in fact liminf sample path average cost optimal and strongly expected average cost optimal

continuity of mean recurrence times in denumerable semi markov processes for a family of semi markov processes where the transition matrices for the embedded markov chains and the mean sojourn times depend continuously on a parameter we give equivalent as well as sufficient conditions for the continuity of the mean recurrence times the results will be used in a subsequent paper on average costs in a dynamic programming model