the distance to a polyhedron abstract we give some exact formulas and some estimates for the distance to a polyhedron in a normed linear space e we show that in the case when e r n endowed with the l norm these estimates are in general better than a recent estimate by cook gerards schrijver and tardos 2

asymptotic constraint qualifications and global error bounds for convex inequalities in this paper we study various asymptotic constraint qualifications for the existence of global error bounds for approximate solutions of convex inequalities many known conditions that ensure the existence of such a global error bound are shown to be equivalent to one of the following three conditions i the bounded excess condition ii slater condition together with the asymptotic constraint qualification defined by auslender and crouzeix 1 and iii positivity of normal directional derivatives of the maximum of the constraint functions introduced by lewis and pang 12

tangent cone and contingent cone to the intersection of two closed sets abstract for a nonempty closed set c in a real normed vector space x and an inequality solution set we present several sufficient conditions for the tangent and contingent cones to their intersection to contain the intersections of the corresponding cones we not only express the contingent cone to a solution set of inequalities and equalities by the directional or frechet derivatives of the active inequality constraint functions and the frechet derivatives of the equality constraint functions but also the tangent cone by the clarke or lower dini or upper dini derivatives of the active inequality constraint functions and the directional derivatives of the equality constraint functions by using a simple property of the function d c d c c we characterize these cones by the hypertangent and hypercontingent vectors to the set c furthermore these results allow us to present new constraint qualifications for the karush kuhn tucker conditions

hidden dynamics in models of discontinuity and switching a method of solving discontinuous differential equations is analyzed the central result generalizes filippov s method to nonlinear sliding modes exponentially small terms are shown to create novel switching dynamics the results are shown to account naturally for stiction static friction models random effects are shown to eliminate nonlinear terms under limited conditions a b s t r a c t sharp switches in behaviour like impacts stick slip motion or electrical relays can be modelled by differential equations with discontinuities a discontinuity approximates fine details of a switching process that lie beyond a bulk empirical model the theory of piecewise smooth dynamics describes what happens assuming we can solve the system of equations across its discontinuity what this typically neglects is that effects which are vanishingly small outside the discontinuity can have an arbitrarily large effect at the discontinuity itself here we show that such behaviour can be incorporated within the standard theory through nonlinear terms and these introduce multiple sliding modes we show that the nonlinear terms persist in more precise models for example when the discontinuity is smoothed out the nonlinear sliding can be eliminated however if the model contains an irremovable level of unknown error which provides a criterion for systems to obey the standard filippov laws for sliding dynamics at a discontinuity

weighted estimators for proportional hazards regression with missing covariates missing covariate data are common in epidemiologic studies and disease prevention trials in this article regression parameter estimation in the cox proportional hazards model is considered when certain covariates are observed for all study subjects and other covariate data are collected only for a subset the article presents both simple weighted and kernel assisted fully augmented weighted estimators that use the partially incomplete data nonparametrically we use nonparametric methods to estimate selection probabilities in the simple weighted estimating functions we also use nonparametric kernel smoothing techniques to estimate certain conditional expectations in fully augmented weighted estimating functions the proposed methods are nonparametric in the sense that they require neither a model for the missing data mechanism nor specification of the conditional distribution of missing covariates given observed covariates these estimators allow the missing data mechanism to depend on outcome variables

nonlinear error bounds for lower semicontinuous functions on metric spaces as a development of the theory of linear error bounds for lower semicontinuous functions defined on complete metric spaces introduced in aze et al nonlinear anal 49 643 670 2002 and refined in aze and corvellec esaim control optim calc var 10 409 425 2004 we propose a similar approach to nonlinear error bounds based on the notion of strong slope the variational principle and the change of metric principle the latter allowing to obtain sharp estimates for such error bounds through a reduction to the linear case

global regularity theorems for a system of nondifferentiable convex inequalities and linear equalities the best bound is given for the quotient of the distance of an infeasible point and the norm of the residual vector applications to stability theory are then obtained

robustness of unmodified stochastic adaptive control algorithms originally adaptive control theory was developed for the ideal system models i e linear system models under the assumption that relative degree and upper bounds on the order of the systems are known in the early 1980s adaptive control algorithms designed for such ideal system models were strongly attacked by many researchers due to lack of robustness in the presence of unmodeled dynamics and external disturbances the purpose of the present paper is to relax existing constant pressure on the adaptive control algorithms originally designed for the ideal system models it is shown that such adaptive control algorithms are globally stable and robust with respect to unmodeled dynamics and external disturbances without any modifications such as i spl sigma modification spl epsiv sub 1 modification relative dead zone projection of the parameter estimates etc global stability of the unmodified algorithms is established by requiring the reference signal to be persistently exciting

optimality conditions via exact penalty functions in this paper we study kkt optimality conditions for constrained nonlinear programming problems and strong and mordukhovich stationarities for mathematical programs with complementarity constraints using l p penalty functions with 0 leq p leq1 we introduce some optimality indication sets by using contingent derivatives of penalty function terms some characterizations of optimality indication sets are obtained by virtue of the original problem data we show that the kkt optimality condition holds at a feasible point if this point is a local minimizer of some l p penalty function with p belonging to the optimality indication set our result on constrained nonlinear programming includes some existing results from the literature as special cases

stability of error bounds for semi infinite convex constraint systems in this paper we are concerned with the stability of the error bounds for semi infinite convex constraint systems roughly speaking the error bound of a system of inequalities is said to be stable if all its small perturbations admit a local or global error bound we first establish subdifferential characterizations of the stability of error bounds for semi infinite systems of convex inequalities by applying these characterizations we extend some results established by aze and corvellec siam j optim 12 2002 pp 913 927 on the sensitivity analysis of hoffman constants to semi infinite linear constraint systems

cal l _ 2 gain of periodic linear switched systems fast switching behavior we investigate the l 2 gain of periodic linear switched systems under fast switching for systems that possess a suitable notion of a time average system we characterize the relationship between the l 2 gain of the switched system and the l 2 gain of its induced time average system when the switching rate is sufficiently fast we show that the switched system l 2 gain is in general different from the average system l 2 gain if the input or output coefficient matrix switches if only the state coefficient matrix switches the input output energy gain for a fixed l 2 input signal is bounded by the l 2 gain of the average system as the switching rate grows large additionally for a fixed l 2 input the maximum pointwise in time difference between the switched and average system outputs approaches zero as the switching rate grows

error bounds in the averaging of hybrid systems the authors analyze the error introduced by the averaging of hybrid systems these systems involve linear systems which can take a number of different realizations based on the state of an underlying finite state process the averaging technique based on a formula from lie algebras known as the backer campbell hausdorff bch formula provides a single system matrix as an approximation to the hybrid system the two errors discussed are 1 the error induced by the truncation of the bch series expansion and 2 the error between the actual hybrid system and its average a simple sufficient stability test is proposed to check the asymptotic behavior of this error in addition conditions are derived that allow the use of state feedback instead of averaging to arrive at a time invariant system matrix

error bounds for piecewise convex quadratic programs and applications in this paper we establish a local error estimate for feasible solutions of a piecewise convex quadratic program and a global error estimate for feasible solutions of a convex piecewise quadratic program these error estimates provide a unified approach for deriving many old and new error estimates for linear programs linear complementarity problems convex quadratic programs and affine variational inequality problems the approach reveals the fact that each error estimate is a consequence of some reformulation of the original problem as a piecewise convex quadratic program or a convex piecewise quadratic program in a sense even robinson s result on the upper lipschitz continuity of a polyhedral mapping can be considered as a special case of error estimates for approximate solutions of a piecewise convex quadratic program as an application we derive new global error estimates for iterates of the proximal point algorithm for solving a convex piecewise quadratic program

analysis of adaptive identifiers in the presence of unmodeled dynamics averaging and tuned parameters the authors analyze the behavior of a standard identifier when the plant contains additional dynamics called unmodeled dynamics which invalidate the known order assumption the first result of the analysis is an input richness condition which does not depend on the order of the unmodeled dynamics to guarantee persistency of excitation of the regressor then it is shown that the persistently exciting pe condition leads to a bibo bounded input bounded output stability property for the identifier the method of averaging is used to formally define the notion of tuned parameters as the equilibrium of the identifier averaged system it is shown that the tuned parameters always exist and that the actual parameters converge to some neighborhood of the tuned parameters from the definition of the tuned parameters an explicit expression to calculate and interpret them as the fixed parameter values that minimize the mean squared output error is derived

global error bounds for piecewise convex polynomials in this paper by examining the recession properties of convex polynomials we provide a necessary and sufficient condition for a piecewise convex polynomial to have a holder type global error bound with an explicit holder exponent our result extends the corresponding results of li siam j control optim 33 5 1510 1529 1995 from piecewise convex quadratic functions to piecewise convex polynomials

error bounds for degenerate cone inclusion problems error bounds for cone inclusion problems in banach spaces are established under conditions weaker than robinson s constraint qualification the results allow the cone to be more general than the origin and therefore also generalize a classical error bound result concerning equality constrained sets in optimization applications in finite dimensional differentiable inequalities and in tangent cones are discussed

global error bounds for systems of convex polynomials over polyhedral constraints this paper is devoted to studying the lipschitzian holderian type global error bound for systems of finitely many convex polynomial inequalities over a polyhedral constraint first for systems of this type we show that under a suitable asymptotic qualification condition the lipschitzian type global error bound property is equivalent to the abadie qualification condition in particular the lipschitzian type global error bound is satisfied under the slater condition second without regularity conditions the holderian global error bound with an explicit exponent is investigated

improving efficiency of parameter estimation in case cohort studies with multivariate failure time data the case cohort study design is an effective way to reduce cost of assembling and measuring expensive covariates in large cohort studies recently several weighted estimators were proposed for the case cohort design when multiple diseases are of interest however these existing weighted estimators do not make effective use of the covariate information available in the whole cohort furthermore the auxiliary information for the expensive covariates which may be available in the studies cannot be incorporated directly in this article we propose a class of updated estimators we show that by making effective use of the whole cohort information the proposed updated estimators are guaranteed to be more efficient than the existing weighted estimators asymptotically furthermore they are flexible to incorporate the auxiliary information whenever available the advantages of the proposed updated estimators are demonstrated in simulation studies and a real data analysis

error bounds for monotone linear complementarity problems we give a bound on the distance between an arbitrary point and the solution set of a monotone linear complementarity problem in terms of a condition constant that depends on the problem data only and a residual function of the violations of the complementary problem conditions by the point considered when the point satisfies the linear inequalities of the complementarity problem the residual consists of the complementarity condition plus its square root this latter term is essential and without it the error bound cannot hold we also show that another natural residual that has been employed to bound errors for strictly monotone linear complementarity problems fails to bound errors for the monotone case considered here

stability theory for adaptive systems method of averaging and persistency of excitation a method of averaging is developed for the stability analysis of linear differential equations with small time varying coefficients which do not necessarily possess an average value the technique is then applied to determine the stability of a linear equation which arises in the study of adaptive systems where the adaptive parameters are slowly varying the stability conditions are stated in the frequency domain which shows the relation between persistent excitation and unmodeled dynamics

bounded linear regularity of convex sets in banach spaces and its applications this paper deals with bounded linear regularity linear regularity and the strong conical hull intersection property chip of a collection of finitely many closed convex intersecting sets in banach spaces it is shown that as in finite dimensional space setting see 6 the standard constraint qualification implies bounded linear regularity which in turn yields the strong conical hull intersection property and that the collection of closed convex sets c1 cn is bounded linearly regular if and only if the tangent cones of c1 cn has the chip and the normal cones of c1 cn has the property g uniformly on a neighborhood in the intersection c as applications we study the global error bounds for systems of linear and convex inequalities

some equivalent formulations of the generalized ekeland s variational principle and their applications abstract in this paper we first give a generalized takahashi s existence theorem from the existence theorem we establish some equivalence relations between generalized caristi s fixed point theorems and generalized ekeland s variational principles some applications to the existence theorems of weak sharp minima and global error bounds for lower semicontinuous functions are also given

perturbation analysis of a condition number for convex inequality systems and global error bounds for analytic systems in this paper several types of perturbations on a convex inequality system are considered and conditions are obtained for the system to be well conditioned under these types of perturbations where the well conditionedness of a convex inequality system is defined in terms of the uniform boundedness of condition numbers under a set of perturbations it is shown that certain types of perturbations can be used to characterize the well conditionedness of a convex inequality system in which either the system has a bounded solution set and satisfies the slater condition or an associated convex inequality system which defines the recession cone of the solution set for the system satisfies the slater condition finally sufficient conditions are given for the existence of a global error bound for an analytic system it is shown that such a global error bound always holds for any inequality system defined by finitely many convex analytic functions when the zero vector is in the relative interior of the domain of an associated convex conjugate function

strong chip normality and linear regularity of convex sets we extend the property n introduced by jameson for closed convex cones to the normal property for a finite collection of convex sets in a hilbert space variations of the normal property such as the weak normal property and the uniform normal property are also introduced a dual form of the normal property is derived when applied to closed convex cones the dual normal property is the property g introduced by jameson normality of convex sets provides a new perspective on the relationship between the strong conical hull intersection property strong chip and various regularity properties in particular we prove that the weak normal property is a dual characterization of the strong chip and the uniform normal property is a characterization of the linear regularity moreover the linear regularity is equivalent to the fact that the normality constant for feasible direction cones of the convex sets at x is bounded away from 0 uniformly over all points in the intersection of these convex sets

non linear dynamics in adaptive control chaotic and periodic stabilization abstract in this paper the non linear features of adaptive control algorithms are highlighted using a simple linear discrete time example based on classical design it is demonstrated that in the presence of undermodelling errors non linear phenomena in the feedback gain such as limit cycles and even chaos arise despite these complicated dynamics robust stabilization of the plant can still occur for the class of undermodelling errors considered the set of plants stabilizable by this adaptive controller is completely characterized

applications of a kushner and clark lemma to general classes of stochastic algorithms two general classes of stochastic algorithms are considered including algorithms considered by ljung as well as algorithms of the form theta n 1 theta n gamma n 1 v n 1 theta n z where z is a stationary ergodic process it is shown how one can apply a lemma of kushner and clark to obtain properties of these algorithms this is done by using in particular martingale arguments in the generalized ljung case in these various situations the convergence is obtained by the method of the associated ordinary differential equation under the classical boundedness assumptions in the case of linear algorithms the boundedness assumptions are dropped

global error bounds for convex inequality systems in banach spaces we study conditions under which a global error bound in terms of a natural residual exists for a convex inequality system specifically we obtain an error bound result which unifies many existing results assuming a slater condition we also derive two characterizations for a convex inequality system to possess a global error bound one is in terms of metric regularity and the other is in terms of an associated convex inequality system as a consequence we show that in font bbbabstract msbm10 at 8pt mbox bbbabstract r n a global error bound holds for such a system under the assumption of the zero vector in the relative interior of the domain of an associated conjugate function along with metric regularity at every point of the feasible set defined by the system finally we discuss some applications of these results to convex programs

stability of error bounds for convex constraint systems in banach spaces this paper studies stability of error bounds for convex constraint systems in banach spaces we show that certain known sufficient conditions for local and global error bounds actually ensure error bounds for the family of functions being in a sense small perturbations of the given one a single inequality as well as semi infinite constraint systems are considered

on first and second order conditions linebreak for error bounds we establish first order and second order sufficient conditions ensuring that a proper lower semicontinuous function f on a banach space x has an error bound we also consider similar problems with constraint namely that f is replaced by its restriction to a subset of x these results are employed to identify exactly when a quadratic function on x has an error bound

global error bounds with fractional exponents using the partial order induced by a proper weakly lower semicontinuous function on a reflexive banach space x we give a sufficient condition for f to have error bounds with fractional exponents application is given to identify the set of such exponents for quadratic functions

a case cohort design for epidemiologic cohort studies and disease prevention trials summary suppose that a cohort of individuals is to be followed in order to relate failure rates to preceding covariate histories a design is proposed which involves covariate data only for cases experiencing failure and for members of a randomly selected subcohort odds ratio and relative risk estimation procedures are presented for such a case cohort design a small simulation study compares case cohort relative risk estimation procedures to full cohort and synthetic case control analyses relevance to epidemiologic cohort studies and disease prevention trials is discussed

paper an input output view of robustness in adaptive control the stability and robustness properties of adaptive control systems are examined using input output stability theory i e passivity and small gain theory a generic adaptive error system is developed based on the concept of a tuned system an ideal converged nonadaptive closed loop system using this error system with passivity theory gives conditions for global stability where only boundedness in norm is required on the external inputs e g disturbance reference and initial conditions small gain theory is used to develop local stability results where the magnitudes of the external inputs are restricted in the global case a particular system operator not the plant is required to be strictly passive a condition which is unlikely to hold in actual use due to unmodeled dynamics the local results however are not so restricted and allow for unmodeled dynamics in this latter case an estimate of the stability margin is given under a persistent excitation condition

averaging analysis of local stability of a real constant modulus algorithm adaptive filter the constant modulus algorithm cma of adaptive filtering was developed for usage when the modulus of the desired signal is known but its specific value at every sample instant is not known a real arithmetic version of cma was recently proposed and studied using simulations by j r treichler and m g larimore ibid vol assp 33 p 420 31 apr 1985 local stability of their arithmetic version of cma is proven in two applications channel equalization for a transmitted sequence of plus and minus ones and the separation of a sinusoidal signal from its sum with a number of sinusoidal interferers at separate frequencies the proofs utilize dynamic system stability theorems from averaging theory which is a technique currently being utilized in the stability analysis of a variety of adaptive systems

computable error bounds for convex inequality systems in reflexive banach spaces in 1952 a j hoffman proved a fundamental result of an error bound on the distance from any point to the solution set of a linear system in hbox bbb r n in siam j control 13 1975 pp 271 273 robinson extended hoffman s theorem to any system of convex inequalities in a normed linear space which satisfies the slater constraint qualification and has a bounded solution set this paper studies any system of convex inequalities in a reflexive banach space which has an unbounded solution set it is shown that hoffman s error bound holds for such a system when a related convex system which defines the recession cone of the solution set for the system satisfies the slater constraint qualification

regularities and their relations to error bounds in this paper we mainly study various notions of regularity for a finite collection c1 cm of closed convex subsets of a banach space x and their relations with other fundamental concepts we show that a proper lower semicontinuous function f on x has a lipschitz error bound resp error bound if and only if the pair epi f x 0 of sets in the product space x is linearly regular resp regular similar results for multifunctions are also established next we prove that c1 cm is linearly regular if and only if it has the strong chip and the collection nc1 z ncm z of normal cones at z has property g for each z c i 1mci provided that c1 is a closed convex cone and that c2 y is a closed vector subspace of x we show that c1 y is linearly regular if and only if there exists 0 such that each positive relative to the order induced by c1 linear functional on y of norm one can be extended to a positive linear functional on x with norm bounded by similar characterization is given in terms of normal cones

equivalent formulations of ekeland s variational principle abstract we prove that for some 0 inf x f inf x f e there exists y x such that 0

a condition number for differentiable convex inequalities for a system of differentiable convex inequalities a new bound is given for the absolute error in an infeasible point in terms of the absolute residual by using this bound a condition number is defined for the system of inequalities which gives a bound for the relative error in an infeasible point in terms of the relative residual

sharp estimates for hoffman s constant for systems of linear inequalities and equalities we extend the formulae given by aze and corvellec belousov and andronov and ng and zheng for the hoffman constant associated to the system of linear inequalities ax leq b and relate them to that established by li

error bounds for convex differentiable inequality systems in banach spaces the paper is devoted to studying the hoffman global error bound for convex quadratic affine inequality equality systems in the context of banach spaces we prove that the global error bound holds if the hoffman local error bound is satisfied for each subsystem at some point of the solution set of the system under consideration this result is applied to establishing the equivalence between the hoffman error bound and the abadie qualification condition as well as a general version of wang pang s result 30 on error bound of holderian type the results in the present paper generalize and unify recent works by luo luo in 17 li in 16 and wang pang in 30

fast switching analysis of linear switched systems using exponential splitting stability of periodic switched linear systems with fast switching patterns is studied by combining the method of averaging with exponential splitting this approach yields less conservative bounds on stabilizing fast switching rates than can be obtained with prior approaches such bounds can be useful for analysis and design of switching control laws the method is also generalized to arbitrary including nonperiodic nonswitched time varying systems in particular the effects of time varying state perturbations on the stability of linear periodic switched systems are analyzed

extension of hoffman s error bound to polynomial systems given any linear system defined by linear inequalities equalities in re n hoffman s error bound says that the distance from any point x in re n to the solution set of the linear system is bounded by a constant independent of x times a certain residual function evaluated at x this paper considers arbitrary possibly nonconvex polynomial systems defined by polynomial inequalities equalities it is shown that the solution sets of such systems are in general holder continuous as the right hand side changes in contrast to the lipschitzian continuity for the linear case also for any convex quadratic inequality system possessing an interior solution the authors show that its solution set behaves in a lipschitzian manner as the right hand side varies this result sharpens the earlier error bound results by mangasarian and robinson for general convex differentiable inequality systems which in addition to the interiority assumption require either an asymptotic constraint qualification con

analysis of stochastic gradient algorithms for linear regression problems parameter estimation problems that can be formulated as linear regressions are quite common in many applications recursive on line sequential estimation of such parameters can be performed using the recursive least squares rls algorithm or a stochastic gradient version of this algorithm in this paper the convergence properties of the gradient algorithm are analyzed under the assumption that the gain tends to zero the technique is the same as the so called ordinary differential equation approach but the treatment here is self contained and includes a proof of the boundedness of the estimates a main result is that the convergence conditions for the gradient algorithm are the same as those for the recursive least squares algorithm

how exciting can a signal really be abstract the rate of parameter convergence in a number of adaptive estimation schemes is related to the smallest eigenvalue of the average information matrix determined by the regression vector using a very simple examples we illustrate that the input signals that maximize this minimum eigenvalue may be quite different from the input signals that optimize more classical input design criteria e g d optimal criterion

linear semi infinite programming theory an updated survey abstract this paper presents an state of the art survey on linear semi infinite programming theory and its extensions in particular convex semi infinite programming this review updates a previous survey semi infinite programming non convex optim appl 25 1998 of the same authors on the same topic which was published in 1998

on generalized second order derivatives and taylor expansions in nonsmooth optimization a representation of cominetti and correa s generalized second order directional derivative siam j control optim 28 1990 pp 789 809 is given and then applied to obtain a taylor theorem type result a conjecture in siam j control optim 28 1990 pp 789 809 concerning functions of the form max 1 leq i leq n g i x is proved under a strengthened assumption but not true otherwise

on using the cox proportional hazards model with missing covariates summary we propose two methods for handling missing covariates in using the cox proportional hazards model the maximum partial likelihood estimator based only on study subjects having complete covariates does not utilise all available information also it is biased when the probability of missingness depends on the failure or censoring time our suggestion is to impute the conditional expectation of the statistic involving missing covariates given the available information the proposed method provides a consistent regression parameter estimator when the probability of missingness depends on the failure or censoring time as well as on the observed covariates also the proposed estimator is more efficient than the estimator suggested previously by lin ying 1993 when data are missing completely at random

error bounds for vector valued functions necessary and sufficient conditions abstract in this paper we attempt to extend the definition and existing local error bound criteria to vector valued functions or more generally to functions taking values in a normed linear space some new derivative like objects slopes and subdifferentials are introduced and a general classification scheme of error bound criteria is presented

case cohort studies with interval censored failure time data the case cohort design has been widely used as a means of cost reduction in assembling or measuring expensive covariates in large cohort studies the existing literature on the case cohort design is mainly focused on right censored data in practice however the failure time is often subject to interval censoring it is known only to fall within some random time interval in this paper we consider the case cohort study design for interval censored failure time and develop a sieve semiparametric likelihood approach for analyzing data from this design under the proportional hazards model we construct the likelihood function using inverse probability weighting and build the sieves with bernstein polynomials the consistency and asymptotic normality of the resulting regression parameter estimator are established and a weighted bootstrap procedure is considered for variance estimation simulations show that the proposed method works well for practical situations and an application to real data is provided

multiple imputation analysis of case cohort studies the usual methods for analyzing case cohort studies rely on sometimes not fully efficient weighted estimators multiple imputation might be a good alternative because it uses all the data available and approximates the maximum partial likelihood estimator this method is based on the generation of several plausible complete data sets taking into account uncertainty about missing values when the imputation model is correctly defined the multiple imputation estimator is asymptotically unbiased and its variance is correctly estimated we show that a correct imputation model must be estimated from the fully observed data cases and controls using the case status among the explanatory variable to validate the approach we analyzed case cohort studies first with completely simulated data and then with case cohort data sampled from two real cohorts the analyses of simulated data showed that when the imputation model was correct the multiple imputation estimator was unbiased and efficient the observed gain in precision ranged from 8 to 37 per cent for phase 1 variables and from 5 to 19 per cent for the phase 2 variable when the imputation model was misspecified the multiple imputation estimator was still more efficient than the weighted estimators but it was also slightly biased the analyses of case cohort data sampled from complete cohorts showed that even when no strong predictor of the phase 2 variable was available the multiple imputation was unbiased as precised as the weighted estimator for the phase 2 variable and slightly more precise than the weighted estimators for the phase 1 variables however the multiple imputation estimator was found to be biased when because of interaction terms some coefficients of the imputation model had to be estimated from small samples multiple imputation is an efficient technique for analyzing case cohort data practically we suggest building the analysis model using only the case cohort data and weighted estimators multiple imputation can eventually be used to reanalyze the data using the selected model in order to improve the precision of the results

multiple imputation for interval estimation from simple random samples with ignorable nonresponse abstract several multiple imputation techniques are described for simple random samples with ignorable nonresponse on a scalar outcome variable the methods are compared using both analytic and monte carlo results concerning coverages of the resulting intervals for the population mean using m 2 imputations per missing value gives accurate coverages in common cases and is clearly superior to single imputation m 1 in all cases the performances of the methods for various m can be predicted well by linear interpolation in 1 m 1 between the results for m 2 and m as a rough guide to assure coverages of interval estimates within 2 of the nominal level when using the preferred methods the number of imputations per missing value should increase from 2 to 3 as the nonresponse rate increases from 10 to 60

a generalized second order derivative in nonsmooth optimization in this work a new notion of generalized second order directional derivative and generalized hessian for nonsmooth real valued functions is studied the general properties of these mathematical objects are investigated together with some calculus rules that may facilitate their practical computation two applications of these derivatives in optimization theory are considered first to obtaining necessary and sufficient second order optimality conditions for problems with or without constraints and second to extending the newton method for the minimization of a mathcal c 1 1 function

error bounds of regularized gap functions for nonsmooth variational inequality problems we study the clarke rockafellar directional derivatives of the regularized gap functions and of some modified ones for the variational inequality problem vip defined by a locally lipschitz but not necessarily differentiable function on a closed convex set in an euclidean space as applications we show that under the strong monotonicity assumption the regularized gap functions have fractional exponent error bounds and consequently that the sequences provided by an algorithm of armijo type converge to the solution of the vip

minimizing and stationary sequences of constrained optimization problems we study some fundamental asymptotic properties associated with the well posedness of constrained optimization problems emphasis is placed on the relation between minimizing and stationary sequences and their characterizations in terms of a set of asymptotic karush kuhn tucker kkt optimality conditions unlike the subdifferential approach used by auslender cominetti and crouzeix we use a residual function approach that is closely tied to the theory of error bounds this approach handles constraints explicitly and allows the effective treatment of infeasible stationary sequences the asymptotic kkt conditions provide an asymptotic optimality certificate for inequality constrained programs specializations of the results to convex quadratically constrained convex quadratic spline minimization problems and convex programs with holderian minima are discussed an application of the results to the convergence of a family of newton type iterative descent methods involving singular quasi newton matrices for solving a constrained minimization problem is also presented

robust adaptive control conditions for global stability an input output approach is presented for analyzing the global stability and robustness properties of adaptive controllers to unmodeled dynamics the concept of a tuned system is introduced i e the control system that could be obtained if the plant were known comparing the adaptive system to the tuned system results in the development of a generic adaptive error system passivity theory is used to derive conditions which guarantee global stability of the error system associated with the adaptive controller and ensure boundedness of the adaptive gains specific bounds are presented for certain significant signals in the control systems limitations of these global results are discussed particularly the requirement that a certain operator be strictly positive real spr a condition that is unlikely to hold due to unmodeled dynamics

ekeland s variational principle for set valued functions we establish several set valued function versions of ekeland s variational principle and hence provide some sufficient conditions ensuring the existence of error bounds for inequality systems defined by finitely many lower semicontinuous functions

global weak sharp minima on banach spaces we consider a proper lower semicontinuous function f on a banach space x with lambda inf f x x in x infty let alpha geq lambda and s alpha x in x f x leq alpha we define the lower derivative of f at the set s alpha by underline d f s alpha liminf x rightarrow s alpha frac f x alpha dist x s alpha where x rightarrow s alpha can be interpreted in various ways we show that when f is convex and alpha lambda it is equal to the largest weak sharp minima constant in terms of these derivatives and subdifferentials we present several characterizations for convex f to have global weak sharp minima some of these results are also shown to be valid for nonconvex f as applications we give error bound results for abstract linear inequality systems

error bounds for convex polynomials the purpose of this paper is to investigate error bounds for convex polynomials we prove that for a convex polynomial f in n variables which is not everywhere positive and which is not constant on any affine subspace either f is a sum of a convex polynomial in fewer variables and a linear form with negative coefficients or the negativity set of f is compact as an application we deduce various types of error bounds for unconstrained and polyhedral constrained convex polynomials

error bounds for eigenvalue and semidefinite matrix inequality systems in this paper we give sufficient conditions for existence of error bounds for systems expressed in terms of eigenvalue functions such as in eigenvalue optimization or positive semidefiniteness such as in semidefinite programming

error bounds for nondifferentiable convex inequalities under a strong slater constraint qualification a global error bound is given on the distance between an arbitrary point in then dimensional real spacern and its projection on a nonempty convex set determined bym convex possibly nondifferentiable inequalities the bound is in terms of a natural residual that measures the violations of the inequalities multiplied by a new simple condition constant that embodies a single strong slater constraint qualification cq which implies the ordinary slater cq a very simple bound on the distance to the projection relative to the distance to a point satisfying the ordinary slater cq is given first and then used to derive the principal global error bound

merit functions in vector optimization we study the weak domination property and weakly efficient solutions in vector optimization problems in particular scalarization of these problems is obtained by virtue of some suitable merit functions some natural conditions to ensure the existence of error bounds for merit functions are also given

weak sharp minima for piecewise linear multiobjective optimization in normed spaces abstract in a general normed space we consider a piecewise linear multiobjective optimization problem we prove that a cone convex piecewise linear multiobjective optimization problem always has a global weak sharp minimum property by a counter example we show that the weak sharp minimum property does not necessarily hold if the cone convexity assumption is dropped moreover under the assumption that the ordering cone is polyhedral we prove that a not necessarily cone convex piecewise linear multiobjective optimization problem always has a bounded weak sharp minimum property

second order sufficient conditions for error bounds in banach spaces recently huang and ng presented second order sufficient conditions for error bounds of continuous and ga teaux differentiable functions in banach spaces wu and ye dropped the assumption of huang and ng on ga teaux differentiability but required the space to be a hilbert space we carry on this research in two directions first we extend wu and ye s result to some non hilbert spaces second same as huang and ng we work on banach spaces but provide different second order sufficient conditions that may allow the function to be non ga teaux differentiable

error bounds for lower semicontinuous functions in normed spaces without the convexity or analyticity assumption we study error bounds for an inequality system defined by a general lower semicontinuous function and establish sufficient necessary conditions on the existence of error bounds in infinite dimensional normed spaces some characterizations for a convex inequality system to possess an error bound in a reflexive banach space are also given as applications in dealing with the hoffman error bound result in normed spaces we give a computable lipschitz bound constant which is better than previous lipschitz bound constants in some examples we also consider error bounds for quadratic functions on rn

error bound moduli for conic convex systems on banach spaces we give two explicit formulas which express the error bound moduli for conic convex systems one in terms of directional derivative and the other in terms of the coderivative as applications we study error bounds for systems of infinitely many convex inequalities

metric subregularity of piecewise linear multifunctions and applications to piecewise linear multiobjective optimization in this paper we consider the metric subregularity property for a piecewise linear multifunction with respect to a piecewise linear constraint as well as the weak sharp minimum property for a piecewise linear constrained multiobjective optimization problem of these properties we pay special attention to the global ones we first provide a result on a certain relationship between two nonnegative piecewise linear numerical functions for which the kernel of one of them is contained in the kernel of the other using this result we establish the bounded global metric subregularity results for a piecewise linear multifunction with respect to a piecewise linear set as applications we study the weak sharp minimum property for a piecewise linear constrained multiobjective optimization problem

a unified analysis of hoffman s bound via fenchel duality in 1952 hoffman showed that the distance from any point to the solution set of a linear system is bounded above by a constant times the norm of the residual subsequently this bound has been studied extensively and has found many practical uses in this paper we consider an extension of hoffman s bound to a partially infinite dimensional setting in which the norm defining the distance is replaced by a positively homogeneous convex function and the linear system is replaced by a convex inclusion of the form ax a in k where a is a continuous linear operator from some normed linear space into mathbf r m a is an element of the range space of a and k is a nonempty closed convex cone in mathbf r m when specialized to the finite dimensional case we unify and extend many existing results on hoffman s bound our analysis is based on the use of fenchel duality to express the distance as the supremum of a certain concave function over a bounded subset of the polar of k much of our analysis als

on minimizing and critical sequences in nonsmooth optimization let f be a bounded below lower semicontinuous function from a banach space into r cup infty we study the relationships between minimizing and critical sequences of f where the criticality condition is given in terms of some subdifferential partial here the objective function f is not supposed to be convex or smooth our work extends that of auslender and crouzeix and that of chou ng and pang

nonlinear averaging theorems and the determination of parameter convergence rates in adaptive control abstract the paper presents nonlinear averaging theorems for two time scale systems where the dynamics of the fast system are allowed to vary with the slow system the results are applied to the narendra valavani adaptive control algorithm and estimates of the parameter convergence rates are obtained which do not rely on a linearization of the system around the equilibrium and therefore are valid in a larger region in the parameter space

metric regularity of epigraphical multivalued mappings and applications to vector optimization in this work we combine in a meaningful way two techniques of variational analysis and nonsmooth optimization on one hand we use the error bound approach to study the metric regularity of some special types of multifunctions and on the other hand we exploit the incompatibility between the metric regularity and the pareto minimality this method allows us to present some varepsilon fermat rules for set valued optimization problem in the setting of general banach spaces our results are comparable to several recent results in literature

perturbation of error bounds our aim in the current article is to extend the developments in kruger et al siam j optim 20 6 3280 3296 2010 doi 10 1137 100782206 and more precisely to characterize in the banach space setting the stability of the local and global error bound property of inequalities determined by lower semicontinuous functions under data perturbations we propose new concepts of arbitrary convex and linear perturbations of the given function defining the system under consideration which turn out to be a useful tool in our analysis the characterizations of error bounds for families of perturbations can be interpreted as estimates of the radius of error bounds the definitions and characterizations are illustrated by examples

dominant richness and improvement of performance of robust adaptive control abstract several recent robust adaptive control schemes guarantee stability and residual tracking errors which are small in the mean for any bounded initial conditions independent of any persistence of excitation condition since smallness in the mean does not always guarantee small bounds for the tracking error at steady state the tracking performance of these robust schemes may not be acceptable in this paper we show that the convergence and the tracking performance of these globally stable adaptive control schemes can be considerably improved if the reference input signal is chosen to be dominantly rich we show that dominantly rich signals maintain a high level of persistence of excitation relative to the level of the modeling error which guarantees exponential convergence and small bounds for the tracking and parameter error at steady state