on the continuity of the map ϑ ϑ from the predual of a w algebra abstract let be elements in the predual of a w algebra for their absolute value parts the estimate 2 1 2 is obtained

disturbance localization with dead beat control for linear periodic discrete time systems in this paper the classical notion of a controlled invariant subspace together with its main properties is extended to the case of linear periodic discrete time systems and used for deriving necessary and sufficient solvability conditions for the disturbance localization problem and a synthesis procedure for the solution moreover the notions of outer and inner controllable subspaces are introduced and studied for the same class of system thus allowing the derivation of necessary and sufficient solvability conditions for the disturbance localization problem with output or state dead beat control and to give synthesis procedures for the solutions

convex envelopes generated from finitely many compact convex sets we consider the problem of constructing the convex envelope of a lower semi continuous function defined over a compact convex set we formulate the envelope representation problem as a convex optimization problem for functions whose generating sets consist of finitely many compact convex sets in particular we consider nonnegative functions that are products of convex and component wise concave functions and derive closed form expressions for the convex envelopes of a wide class of such functions several examples demonstrate that these envelopes reduce significantly the relaxation gaps of widely used factorable relaxation techniques

convex envelopes for edge concave functions deterministic global optimization algorithms frequently rely on the convex underestimation of nonconvex functions in this paper we describe the structure of the polyhedral convex envelopes of edge concave functions over polyhedral domains using geometric arguments an algorithm for computing the facets of the convex envelope over hyperrectangles in 3 is described sufficient conditions are described under which the convex envelope of a sum of edge concave functions may be shown to be equivalent to the sum of the convex envelopes of these functions

robust disturbance rejection problems for linear spl omega periodic discrete time systems two robust disturbance rejection problems with state feedback and with incomplete state feedback for linear spl omega periodic discrete time systems are studied in the framework of the so called geometric approach also some necessary conditions and or sufficient conditions for the problems to be solvable are presented

blum hanson type ergodic theorems in noncommutative symmetric spaces we prove a version of blum hanson ergodic theorem for l1 l l1 l contractions acting in noncommutative symmetric spaces with order continuous norm a similar result is established for arbitrary contractions on hardy subspaces of p convex schatten ideals of compact operators

orlicz spaces with convexity or concavity constant one abstract we characterize the classes of function and sequence orlicz spaces that satisfy upper or lower p estimate with constant one we also present a characterization of p convexity or p concavity with constant one in orlicz spaces

explicit convex and concave envelopes through polyhedral subdivisions in this paper we derive explicit characterizations of convex and concave envelopes of several nonlinear functions over various subsets of a hyper rectangle these envelopes are obtained by identifying polyhedral subdivisions of the hyper rectangle over which the envelopes can be constructed easily in particular we use these techniques to derive in closed form the concave envelopes of concave extendable supermodular functions and the convex envelopes of disjunctive convex functions

on convex envelopes for bivariate functions over polytopes in this paper we discuss convex envelopes for bivariate functions satisfying suitable assumptions over polytopes we first propose a technique to compute the value and a supporting hyperplane of the convex envelope over a general two dimensional polytope through the solution of a three dimensional convex subproblem with continuously differentiable constraint functions then for quadratic functions as well as for some polynomial and rational ones again satisfying suitable assumptions we show how the same computations can be carried out through the solution of a single semidefinite problem

deriving convex hulls through lifting and projection we consider convex hull descriptions for certain sets described by inequality constraints over a hypercube and propose a lifting and projection technique to construct them the general procedure obtains the convex hulls as an intersection of semi infinite families of linear inequalities each derived using lifting techniques that are interpreted using convexification tools we demonstrate that differentiability and concavity of certain perturbation functions help reduce the number of inequalities needed for this characterization each family of inequalities yields a few linear nonlinear constraints fully characterized in the space of the original problem variables when the projection problems are amenable to a closed form solution in particular we illustrate the complete procedure by constructing the convex hulls of the subsets of a compact hypercube defined by the constraints x 1 b 1 x 2 b 2 ge x 3 and x 1 x 2 b 2 le x 3 where b 1 b 2 ge 1 as a consequence we obtain a closed form description of the convex hull of the bilinear equality x 1x 2 x 3 in the presence of variable bounds as an intersection of a few linear and nonlinear inequalities this explicit characterization hitherto unknown can improve relaxation techniques for factorable functions which utilize this equality to relax products of functions with known relaxations

on power bounded operators in finite von neumann algebras abstract it is proved that any power bounded operator of class c 1 in a finite von neumann algebra is conjugate to a unitary this solves a conjecture stated by i kovacs in 1970 an important ingredient of the proof is the study of completely positive projections on some operator space

computable representations for convex hulls of low dimensional quadratic forms let mathcal c be the convex hull of points 1 choose x 1 choose x t x in mathcal f subset re n representing or approximating mathcal c is a fundamental problem for global optimization algorithms based on convex relaxations of products of variables we show that if n 4 and mathcal f is a simplex then mathcal c has a computable representation in terms of matrices x that are doubly nonnegative positive semidefinite and componentwise nonnegative we also prove that if n 2 and mathcal f is a box then mathcal c has a representation that combines semidefiniteness with constraints on product terms obtained from the reformulation linearization technique rlt the simplex result generalizes known representations for the convex hull of x 1 x 2 x 1x 2 x in mathcal f when mathcal f subset re 2 is a triangle while the result for box constraints generalizes the well known fact that in this case the rlt constraints generate the convex hull of x 1 x 2 x 1x 2 x in mathcal f when n 3 and mathcal f is a box we show that a representation for mathcal c can be obtained by utilizing the simplex result for n 4 in conjunction with a triangulation of the 3 cube

rademacher averages on noncommutative symmetric spaces let e be a separable or the dual of a separable symmetric function space let m be a semifinite von neumann algebra and let e m be the associated noncommutative function space let ek k 1 be a rademacher sequence on some probability space for finite sequences xk k 1 of e m we consider the rademacher averages kek xk as elements of the noncommutative function space e l m and study estimates for their norms kek xk e calculated in that space we establish general khintchine type inequalities in this context then we show that if e is 2 concave kek xk e is equivalent to the infimum of yk yk 12 zkzk 12 over all yk zk in e m such that xk yk zk for any k 1 dual estimates are given when e is 2 convex and has a nontrivial upper boyd index in this case kek xk e is equivalent to xk xk 12 xkxk 12 we also study rademacher averages i jei ej xij for doubly indexed families xij i j of e m

norm inequalities related to operator monotone functions let a b be positive semidefinite matrices and cdot any unitarily invariant norm on the space of matrices we show f a f b geq f a b for any non negative operator monotone function f t on 0 infty and g a g b leq g a b for non negative increasing function g t on 0 infty with g 0 0 and g infty infty whose inverse function is operator monotone

strong valid inequalities for orthogonal disjunctions and bilinear covering sets in this paper we derive a closed form characterization of the convex hull of a generic nonlinear set when this convex hull is completely determined by orthogonal restrictions of the original set although the tools used in this construction include disjunctive programming and convex extensions our characterization does not introduce additional variables we develop and apply a toolbox of results to check the technical assumptions under which this convexification tool can be employed we demonstrate its applicability in integer programming by providing an alternate derivation of the split cut for mixed integer polyhedral sets and finding the convex hull of certain mixed pure integer bilinear sets we then extend the utility of the convexification tool to relaxing nonconvex inequalities which are not naturally disjunctive by providing sufficient conditions for establishing the convex extension property over the non negative orthant we illustrate the utility of this result by deriving the convex hull of a continuous bilinear covering set over the non negative orthant although we illustrate our results primarily on bilinear covering sets they also apply to more general polynomial covering sets for which they yield new tight relaxations

generalized hermitian operators in this paper the concept of generalized hermitian operators defined on a complex hilbert space is introduced it is shown that the spectrums and the fredholm fields of generalized hermitian operators are both symmetric with respect to the real axis some other results on generalized hermitian operators are obtained

generalized controlled and conditioned invariances for linear ω periodic discrete time systems abstract in this paper the three notions of generalized controlled a b invariance generalized conditioned c a invariance and generalized a b c invariance for uncertain linear periodic discrete time systems are studied and then the parameter insensitive disturbance rejection problems with static output feedback and or with state feedback are formulated and their solvability conditions are presented

jointly constrained biconvex programming this paper presents a branch and bound algorithm for minimizing the sum of a convex function in x a convex function in y and a bilinear term in x and y over a closed set such an objective function is called biconvex with biconcave functions similarly defined the feasible region of this model permits joint constraints in x and y to be expressed the bilinear programming problem becomes a special case of the problem addressed in this paper we prove that the minimum of a biconcave function over a nonempty compact set occurs at a boundary point of the set and not necessarily an extreme point the algorithm is proven to converge to a global solution of the nonconvex program we discuss extensions of the general model and computational experience in solving jointly constrained bilinear programs for which the algorithm has been implemented

some results on the strength of relaxations of multilinear functions we study approaches for obtaining convex relaxations of global optimization problems containing multilinear functions specifically we compare the concave and convex envelopes of these functions with the relaxations that are obtained with a standard relaxation approach due to mccormick the standard approach reformulates the problem to contain only bilinear terms and then relaxes each term independently we show that for a multilinear function having a single product term this approach yields the convex and concave envelopes if the bounds on all variables are symmetric around zero we then review and extend some results on conditions when the concave envelope of a multilinear function can be written as a sum of concave envelopes of its individual terms finally for bilinear functions we prove that the difference between the concave upper bounding and convex lower bounding functions obtained from the mccormick relaxation approach is always within a constant of the difference between the concave and convex envelopes these results along with numerical examples we provide give insight into how to construct strong relaxations of multilinear functions

rosenthal inequalities in noncommutative symmetric spaces abstract we give a direct proof of the upper khintchine inequality for a noncommutative symmetric quasi banach function space with nontrivial upper boyd index this settles an open question of c le merdy and the fourth named author le merdy and sukochev 2008 24 we apply this result to derive a version of rosenthal s theorem for sums of independent random variables in a noncommutative symmetric space as a result we obtain a new proof of rosenthal s theorem for haagerup l p spaces

noncommutative kothe duality using techniques drawn from the classical theory of rearrangement invariant banach function spaces we develop a duality theory in the sense of kothe for symmetric banach spaces of measurable operators affiliated with a semifinite von neumann algebra equipped with a distinguished trace a principal result of the paper is the identification of the kothe dual of a given banach space of measurable operators in terms of normality

quasisimilarity of power bounded operators and blum hanson property abstract we construct a power bounded operator on a hilbert space which is not quasisimilar to a contraction to this aim we solve an open problem from operator ergodic theory showing that there are power bounded hilbert space operators without the blum hanson property we also find an example of a power bounded operator quasisimilar to a unitary operator which is not similar to a contraction thus answering negatively open questions raised by kerchy and cassier on the positive side we prove that contractions on l p spaces 1 p possess the blum hanson property

the convex envelope of n 1 convex functions the question of determining strong convex underestimators for nonlinear functions is theoretically and practically of major interest unfortunately results along these lines are quite limited as very few general procedures are at hand that can be applied to general classes of functions in this paper we show how to reduce the question of determining a convex envelope to lower dimensional optimization problems when the underlying function is indefinite and n 1 convex our structural result about this reduction technique enables us to give descriptions for the convex envelope of a variety of two dimensional functions

a polynomially bounded operator on hilbert space which is not similar to a contraction let e 0 we prove that there exists an operator te e2 2 such that for any polynomial p we have iip te ii not similar to a contraction i e there does not exist an invertible operator s f2 e f2 such that jis 1tesii tributed to halmos after his well known 1970 paper ten problems in hilbert space we also give some related finite dimensional estimates department of mathematics texas a m university college station texas 77843 universite paris vi equipe d7analyse case 186 75252 paris cedex 05 france e mail address gipqccr jussieu fr this content downloaded from 157 55 39 104 on sun 19 jun 2016 06 52 47 utc all use subject to http about jstor org terms

brief performance limitations in the robust servomechanism problem for discrete time periodic systems fundamental limitations for error tracking regulation are obtained for the robust servomechanism problem rsp for discrete time periodic systems in studying this problem the rsp for a multi input multi output discrete time system is considered application of these results is then made to the periodic system robust servomechanism problem and explicit expressions for the limiting costs for error tracking regulation are obtained these limitations can be characterized completely by the number and location of the non minimum phase transmission zeros of the system s associated lifted system together with a term which depends on the order of the system the number of inputs the number of transmission zeros and the system s period

a new framework to relax composite functions in nonlinear programs in this paper we devise new relaxations for composite functions which improve the prevalent factorable relaxations without introducing additional variables by exploiting the inner function structure we outer approximate inner functions using arbitrary under and over estimators and then convexify the outer function over a polytope p which models the ordering relationships between the inner functions and their estimators and utilizes bound information on the inner functions as well as on the estimators we show that there is a subset q of p with significantly simpler combinatorial structure such that the separation problem of the graph of the outer function over p is polynomially equivalent via a fast combinatorial algorithm to that of its graph over q we specialize our study to consider the product of two inner functions with one non trivial underestimator for each inner function for the corresponding polytope p we show that there are eight valid inequalities besides the four mccormick inequalities which improve the factorable relaxation finally we show that our results generalize to simultaneous convexification of a vector of outer functions

lifting inequalities a framework for generating strong cuts for nonlinear programs in this paper we introduce the first generic lifting techniques for deriving strong globally valid cuts for nonlinear programs the theory is geometric and provides insights into lifting based cut generation procedures yielding short proofs of earlier results in mixed integer programming using convex extensions we obtain conditions that allow for sequence independent lifting in nonlinear settings paving a way for efficient cut generation procedures for nonlinear programs this sequence independent lifting framework also subsumes the superadditive lifting theory that has been used to generate many general purpose strong cuts for integer programs we specialize our lifting results to derive facet defining inequalities for mixed integer bilinear knapsack sets finally we demonstrate the strength of nonlinear lifting by showing that these inequalities cannot be obtained using a single round of traditional integer programming cut generation techniques applied on a tight reformulation of the problem

optimization problems with perturbations a guided tour this paper presents an overview of some recent and significant progress in the theory of optimization problems with perturbations we put the emphasis on methods based on upper and lower estimates of the objective function of the perturbed problems these methods allow one to compute expansions of the optimal value function and approximate optimal solutions in situations where the set of lagrange multipliers is not a singleton may be unbounded or is even empty we give rather complete results for nonlinear programming problems and describe some extensions of the method to more general problems we illustrate the results by computing the equilibrium position of a chain that is almost vertical or horizontal

non commutative banach function spaces in this paper we survey some aspects of the theory of non commutative banach function spaces that is spaces of measurable operators associated with a semi finite von neumann algebra these spaces are also known as non commutative symmetric spaces the theory of such spaces emerged as a common generalization of the theory of classical commutative rearrangement invariant banach function spaces in the sense of w a j luxemburg and a c zaanen and of the theory of symmetrically normed ideals of bounded linear operators in hilbert space in the sense of i c gohberg and m g krein these two cases may be considered as the two extremes of the theory in the first case the underlying von neumann algebra is the commutative algebra l on some measure space with integration as trace in the second case the underlying von neumann algebra is b the algebra of all bounded linear operators on a hilbert space with standard trace important special cases of these non commutative spaces are the non commutative l p spaces which correspond in the commutative case with the usual l p spaces on a measure space and in the setting of symmetrically normed operator ideals they correspond to the schatten p classes mathfrak s p