optimal control problems with set valued control and state constraints in this paper a general optimal control problem with pure state and mixed control state constraints is considered these constraints are of the form of set inclusions second order necessary optimality conditions for weak local minimum are derived for this problem in terms of the original data in particular the nonemptiness of the set of critical directions and the evaluation of its support function are expressed in terms of the given functions and set valued maps in order that the lagrange multiplier corresponding to the mixed control state inclusion constraint be represented via an integrable function a strong normality condition involving the notion of the critical tangent cone is introduced

efficient minimum distance estimation with multiple rates of convergence abstract this paper extends the asymptotic theory of gmm inference to allow sample counterparts of the estimating equations to converge at multiple rates different from the usual square root of the sample size in this setting we provide consistent estimation of the structural parameters in addition we define a convenient rotation in the parameter space or reparametrization to disentangle the different rates of convergence more precisely we identify special linear combinations of the structural parameters associated with a specific rate of convergence finally we demonstrate the validity of usual inference procedures like the overidentification test and wald test with standard formulas it is important to stress that both estimation and testing work without requiring the knowledge of the various rates however the assessment of these rates is crucial for asymptotic power considerations possible applications include econometric problems with two dimensions of asymptotics due to trimming tail estimation infill asymptotic social interactions kernel smoothing or any kind of regularization

weak maximum principle for optimal control problems with mixed constraints optimality conditions for control problems with mixed state control constraints have been the focus of attention for a long time in particular the subject of necessary conditions in the form of maximum principles have been addressed by a number of authors see for example 1 4 to name but a few weak maximum principles which apply to weak local solutions covering problems with possibly nonsmooth data have been considered in 5 and in a more general setting in 3 for nonsmooth problems strong maximum principles which in turn apply to strong local solutions have also received some attention recently see 6 7 various recent results including those of the present paper can be captured as special cases of the following optimal control problem with mixed constraints also known as

mixed constrained control problems necessary optimality conditions are derived in the form of a weak maximum principle for optimal control problems with mixed state control equality and inequality constraints in contrast to previous work these conditions hold when the jacobian of the active constraints with respect to the unconstrained control variable has full rank a feature of these conditions is that they are stated in terms of a joint clarke subdifferential furthermore the use of the joint subdifferential gives sufficiency for nonsmooth normal linear convex problems the main point of interest is not only the full rank condition assumption but also the nature of the analysis employed in this paper a key element is the removal of the constraints and application of ekeland s variational principle

optimal control problems with mixed constraints we develop necessary conditions of broad applicability for optimal control problems in which the state and control are subject to mixed constraints we unify subsume and significantly extend most of the results on this subject notably in the three special cases that comprise the bulk of the literature calculus of variations differential algebraic systems and mixed constraints specified by equalities and inequalities our approach also provides a new and unified calibrated formulation of the appropriate constraint qualifications and shows how to extend them to nonsmooth data other features include a very weak hypothesis concerning the type of local minimum nonrestrictive hypotheses on the data and stronger conclusions notably as regards the maximum or weierstrass condition the necessary conditions are stratified in the sense that they are asserted on precisely the domain upon which the hypotheses and the optimality are assumed to hold this leads to local intermediate and global versions of the necessary conditions according to how the hypotheses are formulated

first and second order necessary conditions for control problems with constraints second order necessary conditions are developed for an abstract nonsmooth control problem with mixed state control equality and inequality constraints as well as a constraint of the form g x u e if where f is a closed convex set of a banach space with nonempty interior the inequality constraints g s x u 0 depend on a parameter s belonging to a compact metric space s the equality constraints are split into two sets of equations k x u 0 and h x u 0 where the first equation is an abstract control equation and h is assumed to have a full rank property in u the objective function is maxtet f t x u where t is a compact metric space f is upper semicontinuous in t and lipschitz in x u the results are in terms of a function a that disappears when the parameter spaces t and s are discrete we apply these results to control problems governed by ordinary differential equations and having pure state inequality constraints and control state equality and inequality constraints thus we obtain a generalization and extension of the existing results on this problem

bayesian analysis of covariance matrices and dynamic models for longitudinal data summary parsimonious modelling of the within subject covariance structure while heeding its positive definiteness is of great importance in the analysis of longitudinal data using the cholesky decomposition and the ensuing unconstrained and statistically meaningful reparameterisation we provide a convenient and intuitive framework for developing conditionally conjugate prior distributions for covariance matrices and show their connections with generalised inverse wishart priors our priors offer many advantages with regard to elicitation positive definiteness computations using gibbs sampling shrinking covariances toward a particular structure with considerable flexibility and modelling covariances using covariates bayesian estimation methods are developed and the results are compared using two simulation studies these simulations suggest simpler and more suitable priors for the covariance structure of longitudinal data

necessary conditions for constrained problems under mangasarian fromowitz conditions the focus of this paper is on first order necessary conditions for optimal control problems with mixed state control equality and inequality constraints we consider the case when the cost and dynamics are nonsmooth and the constraints satisfy mangasarian fromowitz type assumptions that weaken the commonly used hypothesis that the jacobian of the active constraints with respect to the free variable is of full rank the results are formulated as unmaximized hamiltonian inclusion type conditions involving not the customary product of partial subdifferentials but the joint subdifferential with respect to the state and control variables

optimal control problems with partially polyhedric constraints this paper discusses a class of state constrained optimal control problems for which it is possible to formulate second order necessary or sufficient conditions for local optimality or quadratic growth that do not involve all curvature terms for the constraints this kind of result is classical in the case of polyhedric control constraints our theory of optimization problems with partially polyhedric constraints allows to extend these results to the case when the control constraints are polyhedric in the presence of state constraints satisfying some specific hypotheses the analysis is based on the assumption that some strict semilinearized qualification condition is satisfied we apply the theory to some optimal control problems of elliptic equations with state and control constraints

the conjugate point condition for smooth control sets abstract in this paper we present for optimal control problems with smooth control constraints second order necessary conditions stated in terms of conjugate points and a riccati type equation we show explicitly how to construct a set of feasible directions and how to derive the corresponding accessory problem

estimation of average treatment effect with incompletely observed longitudinal data application to a smoking cessation study we study the problem of estimation and inference on the average treatment effect in a smoking cessation trial where an outcome and some auxiliary information were measured longitudinally and both were subject to missing values dynamic generalized linear mixed effects models linking the outcome the auxiliary information and the covariates are proposed the maximum likelihood approach is applied to the estimation and inference on the model parameters the average treatment effect is estimated by the g computation approach and the sensitivity of the treatment effect estimate to the nonignorable missing data mechanisms is investigated through the local sensitivity analysis approach the proposed approach can handle missing data that form arbitrary missing patterns over time we applied the proposed method to the analysis of the smoking cessation trial

the upper and lower second order directional derivatives of a sup type function the purpose of this paper is to give a formula for expressing the second order directional derivatives of the sup type functions x sup f x t t t in terms of the first and second derivatives off x t wheret is a compact set in a metric space and we assume thatf f x and 2f x2 are continuous on n t we will give a geometrical meaning of the formula we will moreover give a sufficient condition fors x to be directionally twice differentiable

new second order optimality conditions for variational problems with c 2 hamiltonians the generalized problem of bolza with a c2 hamiltonian is considered necessary and sufficient conditions are obtained respectively in terms of the accessory problem the existence of conjoined basis and the existence of a solution to a riccati equation with boundary conditions no strengthened legendre clebsch condition is required when applied to a general optimal control problem these results include and generalize known results

legendre type optimality conditions for a variational problem with inequality state constraints they differ from the legendre clebsch condition they give information about the hesse matrix of the integrand at not only inactive points but also active points on the other hand since the inequality state constraints can be regarded as an infinite number of inequality constraints they sometimes form an envelope according to a general theory 9 one has to take the envelope into consideration when one consider second order necessary optimality conditions for an abstract optimization problem with a generalized inequality constraint however we show that we do not need to take it into account when we consider legendre type conditions finally we show that any inequality state constraint forms envelopes except two trivial cases we prove it by presenting an envelope in a visible form

variational analysis of a composite function a formula for the lower second order epi derivative an exact formula is established for the lower second order epi derivative of a function of the form g f x where f is a smooth map from one banach space into another and g is a convex function generally not everywhere finite unconstrained minimization of such functions typically arise as an equivalent in one or another sense reduction form for many important classes of constrained optimization problems the formula is further applied to study epi differentiability of the max function x max q x q q

dynamic latent trait models for multidimensional longitudinal data this article presents a new approach for analysis of multidimensional longitudinal data motivated by studies using an item response battery to measure traits of an individual repeatedly over time a general modeling framework is proposed that allows mixtures of count categorical and continuous response variables each response is related to age specific latent traits through a generalized linear model that accommodates item specific measurement errors a transition model allows the latent traits at a given age to depend on observed predictors and on previous latent traits for that individual following a bayesian approach to inference a markov chain monte carlo algorithm is proposed for posterior computation the methods are applied to data from a neurotoxicity study of the pesticide methoxychlor and evidence of a dose dependent increase in motor activity is presented

high order approximations and generalized necessary conditions for optimality in this paper we derive generalized necessary conditions for optimality for an optimization problem with equality and inequality constraints in a banach space the equality constraints are given in operator form as q x in x f x 0 where f x rightarrow y is an operator between banach spaces the inequality constraints are given by smooth functionals or by closed convex sets models of this type are common in the optimal control problem the paper addresses the case when the frechet derivative f x is not onto and hence the classical lyusternik theorem does not apply to describe the tangent space to q in this case the classical euler lagrange type necessary conditions are trivially satisfied generating abnormal cases a high order generalization of the lyusternik theorem derived earlier u ledzewicz and h schattler nonlinear anal 34 1998 pp 793 815 is used to calculate high order tangent cones to the equality constraint at points x in q where f x is not onto combining these with high order approximating cones related to the other constraints of the problem feasible cones respectively cones of decrease a high order generalization of the dubovitskii milyutin theorem is given and then applied to derive generalized necessary conditions for optimality these conditions reduce to classical conditions for normal cases but they give new and nontrivial conditions for abnormal cases

causal inference using potential outcomes design modeling decisions causal effects are defined as comparisons of potential outcomes under different treatments on a common set of units observed values of the potential outcomes are revealed by the assignment mechanism a probabilistic model for the treatment each unit receives as a function of covariates and potential outcomes fisher made tremendous contributions to causal inference through his work on the design of randomized experiments but the potential outcomes perspective applies to other complex experiments and nonrandomized studies as well as noted by kempthorne in his 1976 discussion of savage s fisher lecture fisher never bridged his work on experimental design and his work on parametric modeling a bridge that appears nearly automatic with an appropriate view of the potential outcomes framework where the potential outcomes and covariates are given a bayesian distribution to complete the model specification also this framework crisply separates scientific inference for causal effects and decisions based on s

variational approach to second order optimality conditions for control problems with pure state constraints for optimal control problems with set valued control constraints and pure state constraints we propose new second order necessary optimality conditions in addition to the usual second order derivative of the hamiltonian these conditions contain extra terms involving second order tangents to the set of feasible trajectory control pairs at the extremal process under consideration the second order necessary optimality conditions of the present work are obtained by using a variational approach in particular we present a new second order variational equation this approach allows us to make direct proofs as opposed to the classical way of obtaining second order necessary conditions by using an abstract infinite dimensional optimization problem no convexity assumptions on the constraints are imposed and optimal controls are required to be merely measurable

necessary optimality conditions for local minimizers of stochastic optimal control problems with state constraints the main purpose of this work is to establish some first and second order necessary optimality conditions for local minimizers of stochastic optimal control problems with state constraints the control may affect both the drift and the diffusion terms of the systems and the control regions are allowed to be nonconvex a stochastic inward pointing condition is proposed to ensure the normality of the corresponding necessary conditions

dynamic conditionally linear mixed models for longitudinal data we develop a new class of models dynamic conditionally linear mixed models for longitudinal data by decomposing the within subject covariance matrix using a special cholesky decomposition here dynamic means using past responses as covariates and conditional linearity means that parameters entering the model linearly may be random but nonlinear parameters are nonrandom this setup offers several advantages and is surprisingly similar to models obtained from the first order linearization method applied to nonlinear mixed models first it allows for flexible and computationally tractable models that include a wide array of covariance structures these structures may depend on covariates and hence may differ across subjects this class of models includes e g all standard linear mixed models antedependence models and vonesh carter models second it guarantees the fitted marginal covariance matrix of the data is positive definite we develop methods for bayesian inference and motivate the usefulness of these models using a series of longitudinal depression studies for which the features of these new models are well suited

first and second order optimality conditions for a strong local minimum in control problems with pure state constraints in this paper first and second order optimality conditions for a strong local minimum are presented for optimal control problems with pure state set inclusion constraints the first order condition is of pontryagin type while the second order condition is of the form of an accessory problem associated with the strong local minimality this latter condition contains an extra term reflecting the presence of the pure state constraints

nonsmooth optimum problems with constraints this paper develops second order necessary conditions for nonsmooth infinite dimensional optimization problems with banach space valued equality and real valued inequality constraints another constraint in the form of a closed convex set is also present the objective function is the maximum over a parameter of functions f t z that are lipschitz in z and upper semicontinuous in t the inequality constraints g s z depend on a parameter s the technique we use is a generalization of that of dubovitskii and milyutin the second order conditions obtained here are in terms of a certain function sigma that disappears when the parameters and a certain set that derives from the given convex set are absent the presence of the function sigma and that set is due to the envelope like effect discovered by kawasaki

second order necessary optimality conditions for minimizing a sup type function the second derivative of an envelope cannot be expressed only by second derivatives of the constituent functions by taking account of this fact we derive new second order necessary optimality conditions for minimization of a sup type function the conditions involve an extra term besides the second derivative of the lagrange function furthermore we will comment on the relationship between the extra term and a kind of second order directional derivative of the sup type function

generalized hessian for c 1 1 functions in infinite dimensional normed spaces the subject of this paper is the systematic study of second order notions concerning differentiable functions with lipschitz derivative the results and notions are motivated by recent papers of cominetti correa and hiriart urruty the first goal of this paper is the comparison of several known second order directional derivatives the second goal is the introduction of a generalized hessian which is a set of certain symmetric bilinear forms the relation of this generalized hessian to other existing second order derivatives is also described

an extension of the schwarzkopf multiplier rule in optimal control the search for multiplier rules in dynamic optimization has been an important theme in the subject for over a century it was central in the classical calculus of variations and the pontryagin maximum principle of optimal control theory is part of this quest a more recent thread has involved problems with so called mixed constraints involving the control and state variables jointly a subject which now boasts a considerable literature recently clarke and de pinho proved a general multiplier rule for such problems that extends and subsumes rather directly most of the available results namely those which postulate some kind of rank condition or more generally a constraint qualification or generalized mangasarian fromowitz condition an exception to this approach is due to schwarzkopf whose well known theorem replaces the rank hypothesis for relaxed problems by one of covering the purpose of this article is to show how to obtain this type of theorem from the general multiplier rule of clarke and de pinho in so doing we subsume extend and correct the currently available versions of schwarzkopf s result

pointwise second order necessary optimality conditions and second order sensitivity relations in optimal control abstract this paper is devoted to pointwise second order necessary optimality conditions for the mayer problem arising in optimal control theory we first show that with every optimal trajectory it is possible to associate a solution p of the adjoint system as in the pontryagin maximum principle and a matrix solution w of an adjoint matrix differential equation that satisfy a second order transversality condition and a second order maximality condition these conditions seem to be a natural second order extension of the maximum principle we then prove a jacobson like necessary optimality condition for general control systems and measurable optimal controls that may be only partially singular and may take values on the boundary of control constraints finally we investigate the second order sensitivity relations along optimal trajectories involving both p and w