a note on robinson ursescu and lyusternik graves theorem the aim of this note is twofold first we prove an analogue of the well known robinson ursescu theorem on the relative openness with a linear rate restrictive metric regularity of a multivalued mapping second we prove a generalization of graves open mapping theorem for a class of mappings which can be approximated at a reference point by a bunch of linear mappings the approximated non linear mapping is restricted to a closed convex subset of a banach space

modelling multivariate volatilities via conditionally uncorrelated components summary we propose to model multivariate volatility processes on the basis of the newly defined conditionally uncorrelated components cucs this model represents a parsimonious representation for matrix valued processes it is flexible in the sense that each cuc may be fitted separately with any appropriate univariate volatility model computationally it splits one high dimensional optimization problem into several lower dimensional subproblems consistency for the estimated cucs has been established a bootstrap method is proposed for testing the existence of cucs the methodology proposed is illustrated with both simulated and real data sets

exact duals and short certificates of infeasibility and weak infeasibility in conic linear programming in conic linear programming in contrast to linear programming the lagrange dual is not an exact dual it may not attain its optimal value or there may be a positive duality gap the corresponding farkas lemma is also not exact it does not always prove infeasibility we describe exact duals and certificates of infeasibility and weak infeasibility for conic lps which are nearly as simple as the lagrange dual but do not rely on any constraint qualification some of our exact duals generalize the sdp duals of ramana and klep and schweighofer to the context of general conic lps some of our infeasibility certificates generalize the row echelon form of a linear system of equations they consist of a small trivially infeasible subsystem obtained by elementary row operations we prove analogous results for weakly infeasible systems we obtain some fundamental geometric corollaries an exact characterization of when the linear image of a closed convex cone is closed and an exact characterization of nice cones our infeasibility certificates provide algorithms to generate all infeasible conic lps over several important classes of cones and all weakly infeasible sdps in a natural class using these algorithms we generate a public domain library of infeasible and weakly infeasible sdps the status of our instances can be verified by inspection in exact arithmetic but they turn out to be challenging for commercial and research codes

slex analysis of multivariate nonstationary time series we develop a procedure for analyzing multivariate nonstationary time series using the slex library smooth localized complex exponentials which is a collection of bases each basis consisting of waveforms that are orthogonal and time localized versions of the fourier complex exponentials under the slex framework we build a family of multivariate models that can explicitly characterize the time varying spectral and coherence properties every model has a spectral representation in terms of a unique slex basis before selecting a model we first decompose the multivariate time series into nonstationary components with uncorrelated nonredundant spectral information the best slex model is selected using the penalized log energy criterion which we derive in this article to be the kullback leibler distance between a model and the slex principal components of the multivariate time series the model selection criterion takes into account all of the pairwise cross correlation simultaneously in the multivar

approximate cone factorizations and lifts of polytopes in this paper we show how to construct inner and outer convex approximations of a polytope from an approximate cone factorization of its slack matrix this provides a robust generalization of the famous result of yannakakis that polyhedral lifts of a polytope are controlled by exact nonnegative factorizations of its slack matrix our approximations behave well under polarity and have efficient representations using second order cones we establish a direct relationship between the quality of the factorization and the quality of the approximations and our results extend to generalized slack matrices that arise from a polytope contained in a polyhedron

complementarity and nondegeneracy in semidefinite programming primal and dual nondegeneracy conditions are defined for semidefinite programming given the existence of primal and dual solutions it is shown that primal nondegeneracy implies a unique dual solution and that dual nondegeneracy implies a unique primal solution the converses hold if strict complementarity is assumed primal and dual nondegeneracy assumptions do not imply strict complementarity as they do in lp the primal and dual nondegeneracy assumptions imply a range of possible ranks for primal and dual solutions x and z this is in contrast with lp where nondegeneracy assumptions exactly determine the number of variables which are zero it is shown that primal and dual nondegeneracy and strict complementarity all hold generically numerical experiments suggest probability distributions for the ranks of x and z which are consistent with the nondegeneracy conditions

characterizing bad semidefinite programs normal forms and short proofs semidefinite programs sdps some of the most useful and versatile optimization problems of the last few decades are often pathological the optimal values of the primal and dual problems may d

universal duality in conic convex optimization given a primal dual pair of linear programs it is well known that if their optimal values are viewed as lying on the extended real line then the duality gap is zero unless both problems are infeasible in which case the optimal values are and in contrast for optimization problems over nonpolyhedral convex cones a nonzero duality gap can exist when either the primal or the dual is feasible for a pair of dual conic convex programs we provide simple conditions on the constraint matrices and cone under which the duality gap is zero for every choice of linear objective function and constraint right hand side we refer to this property as universal duality our conditions possess the following properties i they are necessary and sufficient in the sense that if and only if they do not hold the duality gap is nonzero for some linear objective function and constraint right hand side ii they are metrically and topologically generic and iii they can be verified by solving a single conic convex program we relate to universal duality the fact that the feasible sets of a primal convex program and its dual cannot both be bounded unless they are both empty finally we illustrate our theory on a class of semidefinite programs that appear in control theory applications

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

dynamic orthogonal components for multivariate time series we introduce dynamic orthogonal components doc for multivariate time series and propose a procedure for estimating and testing the existence of docs for a given time series we estimate the dynamic orthogonal components via a generalized decorrelation method that minimizes the linear and quadratic dependence across components and across time we then use ljung box type statistics to test the existence of dynamic orthogonal components when docs exist univariate analysis can be applied to build a model for each component those univariate models are then combined to obtain a multivariate model for the original time series we demonstrate the usefulness of dynamic orthogonal components with two real examples and compare the proposed modeling method with other dimension reduction methods available in the literature including principal component and independent component analyses we also prove consistency and asymptotic normality of the proposed estimator under some regularity conditions we provide some

on cone of nonsymmetric positive semidefinite matrices abstract in this paper we analyze and characterize the cone of nonsymmetric positive semidefinite matrices ns psd firstly we study basic properties of the geometry of the ns psd cone and show that it is a hyperbolic but not homogeneous cone secondly we prove that the ns psd cone is a maximal convex subcone of p 0 matrix cone which is not convex but the interior of the ns psd cone is not a maximal convex subcone of p matrix cone as the byproducts some new sufficient and necessary conditions for a nonsymmetric matrix to be positive semidefinite are given finally we present some properties of metric projection onto the ns psd cone

on the connection of facially exposed and nice cones abstract a closed convex cone k in a finite dimensional euclidean space is called nice if the set k f is closed for all f faces of k where k is the dual cone of k and f is the orthogonal complement of the linear span of f the niceness property plays a role in the facial reduction algorithm of borwein and wolkowicz and the question of whether the linear image of the dual of a nice cone is closed also has a simple answer we prove several characterizations of nice cones and show a strong connection with facial exposedness we prove that a nice cone must be facially exposed conversely facial exposedness with an added condition implies niceness we conjecture that nice and facially exposed cones are actually the same and give supporting evidence

an exact duality theory for semidefinite programming and its complexity implications in this paper we present a new and more complete duality for semidefinite programming sdp with the following features begin itemize item this dual is an explicit semidefinite program whose number of variables and the coefficient bitlengths are polynomial in those of the primal item if the primal is feasible then it is bounded if and only if the dual is feasible item the duality gap ie the difference between the primal and the dual objective function values is zero whenever the primal is feasible and bounded also in this case the dual attains its optimum item it yields a precise farkas lemma for semidefinite feasibility systems ie characterization of the it infeasibility of a semidefinite inequality in terms of the it feasibility of another polynomial size semidefinite inequality end itemize note that the standard duality for linear programming satisfies all of the above features but no such duality theory was previously known for sdp without slater like conditions being assumed then we apply the dual to derive certain complexity results for semidefinite programming problems the decision problem of semidefinite feasibility sdfp ie that of determining if a given semidefinite inequality system is feasible is the central problem of interest the complexity of sdfp is unknown but we show the following 1 in the turing machine model sdfp is not np complete unless np co np 2 in the real number model of blum shub and smale cite bss sdfp is in np cap co np we then give polynomial reductions from the following problems to sdfp 1 checking whether an sdp is bounded 2 checking whether a feasible and bounded sdp attains the optimum 3 checking the optimality of a feasible solution

stable zero duality gaps in convex programming complete dual characterisations with applications to semidefinite programs the zero duality gap that underpins the duality theory is one of the central ingredients in optimisation in convex programming it means that the optimal values of a given convex program and its associated dual program are equal it allows in particular the development of efficient numerical schemes however the zero duality gap property does not always hold even for finite dimensional problems and it frequently fails for problems with non polyhedral constraints such as the ones in semidefinite programming problems over the years various criteria have been developed ensuring zero duality gaps for convex programming problems in the present work we take a broader view of the zero duality gap property by allowing it to hold for each choice of linear perturbation of the objective function of the given problem globalising the property in this way permits us to obtain complete geometric dual characterisations of a stable zero duality gap in terms of epigraphs and conjugate functions for convex semidefinite programs we establish necessary and sufficient dual conditions for stable zero duality gaps as well as for a universal zero duality gap in the sense that the zero duality gap property holds for each choice of constraint right hand side and convex objective function zero duality gap results for second order cone programming problems are also given our approach makes use of elegant conjugate analysis and fenchel s duality

facially exposed cones are not always nice we address the conjecture proposed by gabor pataki that every facially exposed cone is nice we show that the conjecture is true in the three dimensional case however there exists a four dimensional counterexample of a cone that is facially exposed but is not nice

on the second order feasibility cone primal dual representation and efficient projection we study the second order feasibility cone mathcal f y in mathbb r n my leq g ty for given data m g we construct a new representation for this cone and its dual based on the spectral decomposition of the matrix m tm gg t this representation is used to efficiently solve the problem of projecting an arbitrary point x in mathbb r n onto mathcal f min y y x my leq g ty which aside from theoretical interest also arises as a necessary subroutine in the rescaled perceptron algorithm we develop a method for solving the projection problem to an accuracy varepsilon whose computational complexity is bounded by o mn 2 n ln ln 1 varepsilon n ln ln 1 min operatorname width mathcal f operatorname width mathcal f operations here operatorname width mathcal f and operatorname width mathcal f denote the width of mathcal f and mathcal f respectively we also perform computational tests that indicate that the method is extremely efficient in practice

bad semidefinite programs they all look the same conic linear programs among them semidefinite programs often behave pathologically the optimal values of the primal and dual programs may differ and may not be attained we present a novel analysis of these pathological behaviors we call a conic linear system cal a x leq k b badly behaved if the value of sup langle c x rangle cal a x leq k b is finite but the dual program has no solution with the same value for some c we describe simple and intuitive geometric characterizations of badly behaved conic linear systems our main motivation is the striking similarity of badly behaved semidefinite systems in the literature we characterize such systems by certain excluded matrices which are easy to spot in all published examples we prove several other structural results about badly behaved semidefinite systems for example we show that they are in cal np cap mbox rm co cal np in the real number model of computing

a conic duality frank wolfe type theorem via exact penalization in quadratic optimization the famous frank wolfe theorem ensures attainability of the optimal value for quadratic objective functions over a possibly unbounded polyhedron if the feasible values are bounded this theorem does not hold in general for conic programs where linear constraints are replaced by more general convex constraints like positive semidefiniteness or copositivity conditions despite the fact that the objective can be even linear this paper studies exact penalizations of classical quadratic programs i e optimization of quadratic functions over a polyhedron and applies the results to establish a frank wolfe type theorem for the primal dual pair of a class of conic programs that frequently arises in applications one result is that uniqueness of the solution of the primal ensures dual attainability i e existence of the solution of the dual

lyapunov type least squares problems over symmetric cones abstract the lyapunov type least squares problem over symmetric cone is to find the least squares solution of the lyapunov equation with a constraint of symmetric cone in the euclidean jordan algebra and it contains the lyapunov type least squares problem over cone of semidefinite matrices as a special case in this paper we first give a detailed analysis for the image of lyapunov operator in the euclidean jordan algebra relying on these properties together with some characterizations of symmetric cone we then establish some necessary and or sufficient conditions for solution existence of the lyapunov type least squares problem finally we study uniqueness of the least squares solution

strong duality for semidefinite programming it is well known that the duality theory for linear programming lp is powerful and elegant and lies behind algorithms such as simplex and interior point methods however the standard lagrangian for nonlinear programs requires constraint qualifications to avoid duality gaps semidefinite linear programming sdp is a generalization of lp where the nonnegativity constraints are replaced by a semidefiniteness constraint on the matrix variables there are many applications e g in systems and control theory and combinatorial optimization however the lagrangian dual for sdp can have a duality gap we discuss the relationships among various duals and give a unified treatment for strong duality in semidefinite programming these duals guarantee strong duality i e a zero duality gap and dual attainment this paper is motivated by the recent paper by ramana where one of these duals is introduced

generalized autoregressive conditional heteroskedasticity a natural generalization of the arch autoregressive conditional heteroskedastic process introduced in engle 1982 to allow for past conditional variances in the current conditional variance equation is proposed stationarity conditions and autocorrelation structure for this new class of parametric models are derived maximum likelihood estimation and testing are also considered finally an empirical example relating to the uncertainty of the inflation rate is presented

on the duality operator of a convex cone let c be a convex set in r for each y c the cone of c at y denoted by cone y c is the cone a x y a 0 and x e c if k is a cone in r we shall denote by k its dual cone and by k the lattice of faces of k then the duality operator of k is the mapping dk k k given by dn f span f a cq k properties of the duality operator d k of a closed pointed hill cone k have been studied before in this paper we study da for a general cone k especially m relation to d o c v k where y e k our main result says that for any closed cone k in r the duality operator d k is injective surjective if and only if the duality operator d o u n is injective surjective for each vector y e k k n k in the last part of the paper we obtain some partial results on the problem of constructing a compact convex set c which contains the zero vector such that cone 0 c is equal to a given cone

a fresh cp look at mixed binary qps new formulations and relaxations triggered by burer s seminal characterization from 2009 many copositive reformulations of mixed binary qps have been discussed by now most of them can be used as proper relaxations if the intractable co mpletely positive cones are replaced by tractable approximations while the widely used approximation hierarchies have the disadvantage to use positive semidefinite psd matrices of orders which rapidly increase with the level of approximation alternatives focus on the problem of keeping psd matrix orders small with the aim to avoid memory problems in the interior point algorithms this work continues this approach proposing new reformulations and relaxations moreover we provide a thorough comparison of the respective duals and establish a monotonicity relation among their duality gaps we also identify sufficient conditions for strong duality zero duality gap in some of these formulations and generalize some of our observations to general conic problems

on the closedness of the linear image of a closed convex cone when is the linear image of a closed convex cone closed we present very simple and intuitive necessary conditions that 1 unify and generalize seemingly disparate classical sufficientconditions such as polyhedrality of the cone and slater type conditions 2 are necessary and sufficient when the dual cone belongs to a class that we call nice cones nice cones subsume all cones amenable to treatment by efficient optimization algorithms for instance polyhedral semidefinite and p cones and 3 provide similarly attractive conditions for an equivalent problem the closedness of the sum of two closed convex cones

time dependent diffeomorphisms as quantum canonical transformations and the time dependent harmonic oscillator quantum canonical transformations corresponding to time dependent diffeomorphisms of the configuration space are studied a special class of these transformations which correspond to time dependent dilatations is used to identify a previously unknown class of exactly solvable time dependent harmonic oscillators the caldirola kanai oscillator belongs to this class for a general time dependent harmonic oscillator it is shown that choosing the dilatation parameter to satisfy the classical equation of motion one obtains the solution of the schrodinger equation a simple generalization of this result leads to the reduction of the schrodinger equation to a second order ordinary differential equation whose special case is the auxiliary equation of the lewis riesenfeld invariant theory the time evolution operator is expressed in terms of a positive real solution of this equation in a closed form and the time dependent position and momentum operators are calculated

regularizing the abstract convex program abstract characterizations of optimality for the abstract convex program inf p x g x s x p where s is an arbitrary convex cone in a finite dimensional space is a convex set and p and g are respectively convex and s convex on were given in 10 these characterizations hold without any constraint qualification they use the minimal cone s f of p and the cone of directions of constancy d g s f in the faithfully convex case these cones can be used to regularize p i e transform p into an equivalent program p r for which slater s condition holds we present an algorithm that finds both s f and d g s f the main step of the algorithm consists in solving a particular complementarity problem we also present a characterization of optimality for p in terms of the cone of directions of constancy of a convex functional d g rather than d g s f

coordinate shadows of semidefinite and euclidean distance matrices we consider the projected semidefinite and euclidean distance cones onto a subset of the matrix entries these two sets are precisely the input data defining feasible semidefinite and euclidean distance completion problems we classify when these sets are closed and use the boundary structure of these two sets to elucidate the krislock wolkowicz facial reduction algorithm in particular we show that under a chordality assumption the minimal cones of these problems admit combinatorial characterizations as a by product we record a striking relationship between the complexity of the general facial reduction algorithm singularity degree and facial exposedness of conic images under a linear mapping

inferential theory for factor models of large dimensions this paper develops an inferential theory for factor models of large dimensions the principal components estimator is considered because it is easy to compute and is asymptotically equivalent to the maximum likelihood estimator if normality is assumed we derive the rate of convergence and the limiting distributions of the estimated factors factor loadings and common components the theory is developed within the framework of large cross sections n and a large time dimension t to which classical factor analysis does not apply we show that the estimated common components are asymptotically normal with a convergence rate equal to the minimum of the square roots of n and t the estimated factors and their loadings are generally normal although not always so the convergence rate of the estimated factors and factor loadings can be faster than that of the estimated common components these results are obtained under general conditions that allow for correlations and heteroskedasticities in both dimensions stronger results are obtained when the idiosyncratic errors are serially uncorrelated and homoskedastic a necessary and sufficient condition for consistency is derived for large n but fixed t

set intersection theorems and existence of optimal solutions the question of nonemptiness of the intersection of a nested sequence of closed sets is fundamental in a number of important optimization topics including the existence of optimal solutions the validity of the minimax inequality in zero sum games and the absence of a duality gap in constrained optimization we consider asymptotic directions of a sequence of closed sets and introduce associated notions of retractive horizon and critical directions based on which we provide new conditions that guarantee the nonemptiness of the corresponding intersection we show how these conditions can be used to obtain simple and unified proofs of some known results on existence of optimal solutions and to derive some new results including a new extension of the frank wolfe theorem for nonconvex quadratic programming