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

exact order of hoffman s error bounds for elliptic quadratic inequalities derived from vector valued chebyshev approximation in this paper we introduce the exact order of hoffman s error bounds for approximate solutions of elliptic quadratic inequalities elliptic quadratic inequalities are closely related to chebyshev approximation of vector valued functions including complex valued functions the set of chebyshev approximations of a vector valued function defined on a finite set is shown to be hausdorff strongly unique of order exactly 2 s for some nonnegative integer s as a consequence the exact order of hoffman s error bounds for approximate solutions of elliptic quadratic inequalities is exactly 2 s for some nonnegative integer s the integer s called the order of deficiency which is computable quantifies how much the abadie constraint qualification is violated by the elliptic quadratic inequalities

the sharp lipschitz constants for feasible and optimal solutions of a perturbed linear program abstract the purpose of this paper is to derive the sharp lipschitz constants for the feasible solutions and optimal solutions of a linear program with respect to right hand side perturbations the lipschitz constants are given in terms of pseudoinverses of submatrices of the matrices involved and are proven to be sharp

error bounds for nondegenerate monotone linear complementarity problems error bounds and upper lipschitz continuity results are given for monotone linear complementarity problems with a nondegenerate solution the existence of a nondegenerate solution considerably simplifies the error bounds compared with problems for which all solutions are degenerate thus when a point satisfies the linear inequalities of a nondegenerate complementarity problem the residual that bounds the distance from a solution point consists of the complementarity condition alone whereas for degenerate problems this residual cannot bound the distance to a solution without adding the square root of the complementarity condition to it this and other simplified results are a consequence of the polyhedral characterization of the solution set as the intersection of the feasible region z mz q 0 z 0 with a single linear affine inequality constraint

lipschitz continuity of solutions of linear inequalities programs and complementarity problems it is shown that solutions of linear inequalities linear programs and certain linear complementarity problems e g those with p matrices or z matrices but not semidefinite matrices are lipschitz continuous with respect to changes in the right hand side data of the problem solutions of linear programs are not lipschitz continuous with respect to the coefficients of the objective function the lipschitz constant given here is a generalization of the role played by the norm of the inverse of a nonsingular matrix in bounding the perturbation of the solution of a system of equations in terms of a right hand side perturbation

a uniform approach to obtain diagonal sets in complexity classes abstract a uniform method for constructing sets which diagonalize over certain complexity classes while preserving other complexity properties is given we obtain some known results as well as some new ones as corollaries of our main theorem the new results concern the complexity classes p np co np pspace apt almost polynomial time r random polynomial time and the polynomial hierarchy

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

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

sublattices of the polynomial time degrees we show that any countable distributive lattice can be embedded in any interval of polynomial time degrees furthermore the embeddings can be chosen to preserve the least or the greatest element this holds for both polynomial time bounded many one and turing reducibilities as well as for all of the common intermediate reducibilities

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

bias and efficiency of multiple imputation compared with complete case analysis for missing covariate values when missing data occur in one or more covariates in a regression model multiple imputation mi is widely advocated as an improvement over complete case analysis cc we use theoretical arguments and simulation studies to compare these methods with mi implemented under a missing at random assumption when data are missing completely at random both methods have negligible bias and mi is more efficient than cc across a wide range of scenarios for other missing data mechanisms bias arises in one or both methods in our simulation setting cc is biased towards the null when data are missing at random however when missingness is independent of the outcome given the covariates cc has negligible bias and mi is biased away from the null with more general missing data mechanisms bias tends to be smaller for mi than for cc since mi is not always better than cc for missing covariate problems the choice of method should take into account what is known about the missing data mechanism in a particular substantive application importantly the choice of method should not be based on comparison of standard errors we propose new ways to understand empirical differences between mi and cc which may provide insights into the appropriateness of the assumptions underlying each method and we propose a new index for assessing the likely gain in precision from mi the fraction of incomplete cases among the observed values of a covariate fico

error bound and convergence analysis of matrix splitting algorithms for the affine variational inequality problem consider the affine variational inequality problem it is shown that the distance to the solution set from a feasible point near the solution set can be bounded by the norm of a natural residual at that point this bound is then used to prove linear convergence of a matrix splitting algorithm for solving the symmetric case of the problem this latter result improves upon a recent result of luo and tseng that further assumes the problem to be monotone

error bounds for the linear complementarity problem with a p matrix abstract we introduce a fundamental quantity associated with a p matrix and show how this quantity is useful in deriving error bounds for the linear complementarity problem of the p type we also obtain upper and lower bounds for the quantity introduced

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

error bound and reduced gradient projection algorithms for convex minimization over a polyhedral set consider the problem of minimizing over a polyhedral set the composition of an affine mapping with a strongly convex differentiable function the polyhedral set is expressed as the intersection of an affine set with a simpler polyhedral set and a new local error bound for this problem based on projecting the reduced gradient associated with the affine set onto the simpler polyhedral set is studied a class of reduced gradient projection algorithms for solving the case where the simpler polyhedral set is a box is proposed and this bound is used to show that algorithms in this class attain a linear rate of convergence included in this class are the gradient projection algorithm of goldstein and levitin and poljak and an algorithm of bertsekas a new algorithm in this class reminiscent of active set algorithms is also proposed some of the results presented here extend to problems where the objective function is extended real valued and to variational inequality problems

sensitivity theorems in integer linear programming we consider integer linear programming problems with a fixed coefficient matrix and varying objective function and right hand side vector among our results we show that for any optimal solution to a linear program max wx ax b the distance to the nearest optimal solution to the corresponding integer program is at most the dimension of the problem multiplied by the largest subdeterminant of the integral matrixa using this we strengthen several integer programming proximity results of blair and jeroslow graver and wolsey we also show that the chvatal rank of a polyhedron x ax b can be bounded above by a function of the matrixa independent of the vectorb a result which as blair observed is equivalent to blair and jeroslow s theorem that each integer programming value function is a gomory function

error estimates and lipschitz constants for best approximation in continuous function spaces abstract we use a structural characterization of the metric projection pg f from the continuous function space to its one dimensional subspace g to derive a lower bound of the hausdorff strong unicity constant or weak sharp minimum constant for pg and then show this lower bound can be attained then the exact value of lipschitz constant for pg is computed the process is a quantitative analysis based on the g teaux derivative of pg a representation of local lipschitz constants the equivalence of local and global lipschitz constants for lower semicontinuous mappings and construction of functions

minimum principle sufficiency we characterize the property of obtaining a solution to a convex program by minimizing over the feasible region a linearization of the objective function at any of its solution points minimum principle sufficiency for the case of a monotone linear complementarity problem this mps property is completely equivalent to the existence of a nondegenerate solution to the problem for the case of a convex quadratic program the mps property is equivalent to the span of the hessian of the objective function being contained in the normal cone to the feasible region at any solution point plus the cone generated by the gradient of the objective function at any solution point this in turn is equivalent to the quadratic program having a weak sharp minimum an important application of the mps property is that minimizing on the feasible region a linearization of the objective function at a point in a neighborhood of a solution point gives an exact solution of the convex program this leads to finite termination of convergent algorithms that periodically minimize such a linearization

on the linear convergence of descent methods for convex essentially smooth minimization consider the problem of minimizing over a polyhedral set the composition of an affine mapping with a strictly convex essentially smooth function a general result on the linear convergence of descent methods for solving this problem is presented by applying this result the linear convergence of both the gradient projection algorithm of goldstein and levitin and polyak and a matrix splitting algorithm using regular splitting is established the results do not require that the cost function be strongly convex or that the optimal solution set be bounded the key to the analysis lies in a new error bound for estimating the distance from a feasible point to the optimal solution set

computation times of np sets of different densities abstract in this paper we study the computational complexity of sets of different densities in np we show that the deterministic computation time for sets in np can depend on their density if and only if there is a collapse or partial collapse of the corresponding higher nondeterministic and deterministic time bounded complexity classes we show also that for np sets of different densities there exist complete sets of the corresponding density under polynomial time turing reductions finally we show that these results can be interpreted as results about the complexity of theorem proving and proof presentation in axiomatized mathematical systems this interpretation relates fundamental questions about the complexity of our intellectual tools to basic structural problems about p np conp and pspace discussed in this paper

hyper polynomial hierarchies and the polynomial jump abstract assuming that the polynomial hierarchy ph does not collapse we show the existence of ascending sequences of ptime turing degrees of length 1 ck in pspace such that successors are polynomial jumps of their predecessors moreover these ptime degrees are all uniformly hard for ph this is analogous to the hyperarithmetic hierarchy which is defined similarly but with the computable turing degrees the lack of uniform least upper bounds for ascending sequences of ptime degrees causes the limit levels of our hyper polynomial hierarchy to be inherently non canonical this problem is investigated in depth and various possible structures for hyper polynomial hierarchies are explicated as are properties of the polynomial jump operator on the languages which are in pspace but not in ph

sparse sets in np p exptime versus nexptime this paper investigates the structural properties of sets in np p and shows that the computational difficulty of lower density sets in np depends explicitly on the relations between higher deterministic and nondeterministic time bounded complexity classes the paper exploits the recently discovered upward separation method which shows for example that there exist sparse sets in np p if and only if exptime nexptime in addition the paper uses relativization techniques to determine logical possibilities limitations of these proof techniques and exhibits one of the first natural structural differences between relativized np and conp

weak sharp minima in mathematical programming the notion of a sharp or strongly unique minimum is extended to include the possibility of a nonunique solution set these minima will be called weak sharp minima conditions necessary for the solution set of a minimization problem to be a set of weak sharp minima are developed in both the unconstrained and constrained cases these conditions are also shown to be sufficient under the appropriate convexity hypotheses the existence of weak sharp minima is characterized in the cases of linear and quadratic convex programming and for the linear complementarity problem in particular a result of mangasarian and meyer is reproduced that shows that the solution set of a linear program is always a set of weak sharp minima whenever it is nonempty consequences for the convergence theory of algorithms are also examined especially conditions yielding finite termination

on the relative complexity of hard problems for complexity classes without complete problems abstract we show that any recursive sequence of recursive sets which is ascending with respect to the standard polynomial time reducibility notions has no minimal upper bound as a consequence any complexity class with certain natural closure properties possesses either complete problems or no easiest hard problems a further corollary is that assuming p np the partial ordering of the polynomial time degrees of np sets is not complete and that there are no degree invariant approximations to np complete problems

sharp lipschitz constants for basic optimal solutions and basic feasible solutions of linear programs the main purpose of this paper to give lipschitz constants for basic optimal solutions or vertices of solution sets and basic feasible solutions or vertices of feasible sets of linear programs with respect to right hand side perturbations the lipschitz constants are given in terms of norms of pseudoinverses of submatrices of the matrices involved and are sharp under very general assumptions there are two mathematical principles involved in deriving the lipschitz constants 1 the local upper lipschitz constant of a hausdorff lower semicontinuous mapping is equal to the lipschitz constant of the mapping and 2 the lipschitz constant of a finite set valued mapping can be inherited by its continuous submappings moreover it is proved that any lipschitz constant for basic feasible solutions can be used as an lipschitz constant for basic optimal solutions feasible solutions and optimal solutions

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

global error bounds for monotone affine variational inequality problems abstract a global gradient projection type error bound is given for monotone affine variational inequality problems this new bound is based on a previously obtained global error bound for monotone linear complementarity problems and requires no assumptions on the monotone affine variational inequality other than solvability the error bound can be used in conjunction with recent results of tseng and luo to establish global linear convergence of any regular splitting method for solvable symmetric monotone affine variational inequality problems

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

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

expected estimating equations for missing data measurement error and misclassification with application to longitudinal nonignorable missing data missing data measurement error and misclassification are three important problems in many research fields such as epidemiological studies it is well known that missing data and measurement error in covariates may lead to biased estimation misclassification may be considered as a special type of measurement error for categorical data nevertheless we treat misclassification as a different problem from measurement error because statistical models for them are different indeed in the literature methods for these three problems were generally proposed separately given that statistical modeling for them are very different the problem is more challenging in a longitudinal study with nonignorable missing data in this article we consider estimation in generalized linear models under these three incomplete data models we propose a general approach based on expected estimating equations eees to solve these three incomplete data problems in a unified fashion this eee approach can be easily implemented and its asymptotic covariance can be obtained by sandwich estimation intensive simulation studies are performed under various incomplete data settings the proposed method is applied to a longitudinal study of oral bone density in relation to body bone density

a comparison of multiple imputation and fully augmented weighted estimators for cox regression with missing covariates several approaches exist for handling missing covariates in the cox proportional hazards model the multiple imputation mi is relatively easy to implement with various software available and results in consistent estimates if the imputation model is correct on the other hand the fully augmented weighted estimators fawes recover a substantial proportion of the efficiency and have the doubly robust property in this paper we compare the fawes and the mi through a comprehensive simulation study for the mi we consider the multiple imputation by chained equation and focus on two imputation methods bayesian linear regression imputation and predictive mean matching simulation results show that the imputation methods can be rather sensitive to model misspecification and may have large bias when the censoring time depends on the missing covariates in contrast the fawes allow the censoring time to depend on the missing covariates and are remarkably robust as long as getting either the conditional expectations or the selection probability correct due to the doubly robust property the comparison suggests that the fawes show the potential for being a competitive and attractive tool for tackling the analysis of survival data with missing covariates

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

estimation of regression coefficients when some regressors are not always observed abstract in applied problems it is common to specify a model for the conditional mean of a response given a set of regressors a subset of the regressors may be missing for some study subjects either by design or happenstance in this article we propose a new class of semiparametric estimators based on inverse probability weighted estimating equations that are consistent for parameter vector 0 of the conditional mean model when the data are missing at random in the sense of rubin and the missingness probabilities are either known or can be parametrically modeled we show that the asymptotic variance of the optimal estimator in our class attains the semiparametric variance bound for the model by first showing that our estimation problem is a special case of the general problem of parameter estimation in an arbitrary semiparametric model in which the data are missing at random and the probability of observing complete data is bounded away from 0 and then deriving a representation for the efficient score

multinomial logistic regression with missing outcome data an application to cancer subtypes many diseases such as cancer and heart diseases are heterogeneous and it is of great interest to study the disease risk specific to the subtypes in relation to genetic and environmental risk factors however due to logistic and cost reasons the subtype information for the disease is missing for some subjects in this article we investigate methods for multinomial logistic regression with missing outcome data including a bootstrap hot deck multiple imputation bhmi simple inverse probability weighted sipw augmented inverse probability weighted aipw and expected estimating equation eee estimators these methods are important approaches for missing data regression the bhmi modifies the standard hot deck multiple imputation method such that it can provide valid confidence interval estimation under the situation when the covariates are discrete the sipw aipw and eee estimators are numerically identical when the covariates are continuous nonparametric smoothers can be applied to estimate the selection probabilities and the estimating scores these methods perform similarly extensive simulations show that all of these methods yield unbiased estimators while the complete case cc analysis can be biased if the missingness depends on the observed data our simulations also demonstrate that these methods can gain substantial efficiency compared with the cc analysis the methods are applied to a colorectal cancer study in which cancer subtype data are missing among some study individuals

finite termination of the proximal point algorithm this paper concerns the notion of a sharp minimum on a set and its relationship to the proximal point algorithm we give several equivalent definitions of the property and use the notion to prove finite termination of the proximal point algorithm