区间值函数的可微性及其在区间值规划中的应用

利用实值函数的全微分思想,讨论了区间值函数的可微性,建立了区间值函数的$D$-可微性的概念及其一些基本性质. 通过讨论无约束区间规划的最优性条件,给出了一类约束函数为实值函数的约束区间值规划问题取得最优解的必要条件. 同时给出了具有实值函数约束的凸区间值规划问题取得最优解的充分条件.     

Sufficient Optimality Conditions for Optimal Control Problems with State Constraints

It is well-known in optimal control theory that the maximum principle, in general, furnishes only necessary optimality conditions for an admissible process to be an optimal one. It is also well-known that if a process satisfies the maximum principle in a problem with convex data, the maximum principle turns to be likewise a sufficient condition. Here an invexity type condition for state constrained optimal control problems is defined and shown to be a sufficient optimality condition. Further, it is demonstrated that all optimal control problems where all extremal processes are optimal necessarily obey this invexity condition. Thus optimal control problems which satisfy such a condition constitute the most general class of problems where the maximum principle becomes automatically a set of sufficient optimality conditions.     

A saddle-point characterization of Pareto optima

This paper provides an answer to the following basic problem of convex multi-objective optimization: Find a saddle-point condition that is both necessary and sufficient that a given point be Pareto optimal. No regularity condition is assumed for the constraints or the objectives.Research partly supported by the Natural Sciences and Engineering Research Council of Canada.Corresponding author.Contribution of this author is a part of her M.Sc. Thesis in Applied Mathematics.     

Optimality Conditions for Bilevel Vector Optimization Problems with a Variable Ordering Structure

The article is concerned with the optimistic formulation of a multiobjective bilevel optimization problem with locally Lipschitz continuous inclusion constraints. Using a variable ordering structure defined by a Bishop–Phelps cone, we investigate necessary optimality conditions for locally weakly nondominated solutions. Reducing the problem into a one-level nonlinear and nonsmooth program, we use the extremal principle by Mordukhovich to get fuzzy optimality conditions. More explicit conditions with the initial data are obtained using both the Ekeland’s variational principle and the support function. Fortunately, the Lipschitz property of a set-valued mapping is conserved for its support function. An appropriate regularity condition is given to help us discern the Lagrange-Kuhn-Tucker multipliers.     

Stochastic maximum principle for mean-field forward-backward stochastic control system with terminal state constraints

In this paper,we consider an optimal control problem with state constraints,where the control system is described by a mean-field forward-backward stochastic differential equation(MFFBSDE,for short)and the admissible control is mean-field type.Making full use of the backward stochastic differential equation theory,we transform the original control system into an equivalent backward form,i.e.,the equations in the control system are all backward.In addition,Ekeland’s variational principle helps us deal with the state constraints so that we get a stochastic maximum principle which characterizes the necessary condition of the optimal control.We also study a stochastic linear quadratic control problem with state constraints.     

Lipschitz Regularity in Some Geometric Problems

The Lipschitz regularity is perhaps the most natural, and surely the most geometrical among all the types of regularities. For example, the Lipschitz character of an ordinary differential equation (vector field) is the natural classical sufficient condition for the (unique) integrability of this equation. The goal here is to show that, in some sense, the Lipschitz regularity is also necessary, if one assumes (geometric) individual conditions on the trajectories. In other words, we show that tangential rigidity leads to a transversal regularity.     

Optimal endpoints

A complete set of necessary and sufficient conditions for selecting optimal endpoints for extremals obtained from the variational Bolza problem in control notation has been developed. The method used to obtain these conditions is based on a seldom used concept of performing a dichotomy on the general optimization problem. With this concept, the problem of Bolza is decomposed into two problems, the first of which involves the selection of optimal paths with the endpoints considered fixed. The second problem involves the selection of optimal endpoints with the paths between the endpoints taken to be stationary curves. The convenience of the dichotomy in deriving the necessary and sufficient conditions for endpoints lies in its simplicity and elementary character; well-known necessary and sufficient conditions from the theory of ordinary maxima and minima are used.An endpoint necessary condition is first obtained which is simply the well-known transversality condition. An additional condition is then developed which, together with the transversality condition, leads to a set of necessary and sufficient conditions for a given extremal to be locally optimal with respect to endpoint variations. While the second condition presented is akin to the classical focal-point condition, the result is new in form and is directly applicable to the optimal control problem. In addition, it is relatively simple to apply and is easy to implement numerically when an analytical solution is not possible. It should be useful in situations where the transversality conditions yield more than one choice for an optimal endpoint.An analytic solution for a simple geodetics problem is presented to illustrate the theory. A discussion of numerical implementation of the sufficiency conditions and its application to an orbit transfer example is also included.This work was supported in part by the National Aeronautics and Space Administration, Grant No. NGR-03-002-001.     

平衡损失下约束线性回归模型中回归系数的可容许估计

本文在平衡损失函数下得到等式约束模型中回归系数在齐次(非齐次)估计类中存在可容许估计的充要条件,给出带有不完全椭球约束模型中回归系数的线性估计在一切估计类中为可容许估计的充要条件.     

State-constrained optimal control of the three-dimensional stationary Navier-Stokes equations

In this paper, an optimal control problem for the stationary Navier-Stokes equations in the presence of state constraints is investigated. Existence of optimal solutions is proved and first order necessary conditions are derived. The regularity of the adjoint state and the state constraint multiplier is also studied. Lipschitz stability of the optimal control, state and adjoint variables with respect to perturbations is proved and a second order sufficient optimality condition for the case of pointwise state constraints is stated.     

An optimal control problem for a parabolic equation in the class of smooth controls

We consider an optimal control problem for a parabolic equation with a differential constraint on the boundary. We study this problem in the class of smooth controls satisfying certain pointwise constraints. Such problems describe mass transfer processes in column-type apparatuses, taking into account the longitudinal mixing. Control functions in these problems represent flows of raw materials or finished products. For the problem under consideration we obtain a necessary optimality condition, propose a method for improving admissible controls, and carry out the numerical experiment.     

Properties of the Lagrange multipliers in the Pontryagin maximum principle for optimal control problems with state constraints

We study the Pontryagin maximum principle for an optimal control problem with state constraints. We analyze the continuity of a vector function µ (which is one of the Lagrange multipliers corresponding to an extremal by virtue of the maximum principle) at the points where the extremal trajectory meets the boundary of the set given by the state constraints. We obtain sufficient conditions for the continuity of µ in terms of the smoothness of the extremal trajectory.     

Non-compact-valued stochastic control under state constraints

In the present paper, we study a necessary condition under which the solutions of a stochastic differential equation governed by unbounded control processes, remain in an arbitrarily small neighborhood of a given set of constraints. We prove that, in comparison to the classical constrained control problem with bounded control processes, a further assumption on the growth of control processes is needed in order to obtain a necessary and sufficient condition in terms of viscosity solution of the associated Hamilton-Jacobi-Bellman equation. A rather general example illustrates our main result.     

线性模型中不可估参数的线性估计的可容许性

对一般线性模型在平方损失函数下，得到了一维不可估参数函数的线性估计为可容许估计的充要条件，以及模型中参数向量（非线性可估）的线性估计为可容许估计的两个充要条件，并得到了多维参数函数（可估或不可估）的线性估计为可容许估计的一个充分条件以及特殊情况下的一个充要条件．     

Lagrange principle in quadratic optimal control problems with infinite horizon

We consider a quadratic optimal control problem on an infinite time interval with integral quadratic equality and inequality constraints. For this (generally, nonconvex) problem, we justify the Lagrange constraint removal principle and the duality relation. The obtained result is based on the general theory of extremal problems, namely, on necessary second-order extremum conditions.     

A note on Hamilton sequences for extremal Beltrami coefficients

F. P. Gardiner gave a sufficient condition for a sequence to be a Hamilton sequence for an extremal Beltrami coefficient. In this note, we shall consider the converse problem, proving that the condition is not necessary.

     

A Comparison of Restricted and Unrestricted Estimators in Estimating Linear Functions of Ordered Scale Parameters of Two Gamma Distributions

The problem of estimating linear functions of ordered scale parameters of two Gamma distributions is considered. A necessary and sufficient condition on the ratio of two coefficients is given for the maximum likelihood estimator (MLE) to dominate the crude unbiased estimator (UE) in terms of mean square error. A modified MLE which satisfies the restriction is also suggested, and a necessary and sufficient condition is also given for it to dominate the admissible estimator based solely on one sample. The estimation of linear functions of variances in two sample problem and also of variance components in a one-way random effect model is mentioned.     

On necessary optimality conditions for systems Governed by a two point boundary value problem. I

The paper concerns a necessary optimality condition in form of a Pontryagin Minimum Principle for a system governed by a linear two point boundary value problem with homogeneous Dibichlet conditions, whereby the control vector occurs in all coefficients of the differential equation. Without any convexity assumption the optimality condition is derived using a needle-like variation of the optimal control. In case of convex local control constraints the optimality condition implies the linearized minimum principle, which we have proved in [2]. An example shows that for this linearized optimality condition the convexity of the set of all admissible controls is essential.     

Geometric Criterion for Controllability under Arbitrary Constraints on the Control

In this paper, necessary and sufficient conditions for null-controllability of a linear system under geometric constraints on the control are given, without the assumption that the origin is an equilibrium point of the system. The criterion for controllability uses the concept of a return condition on an interval which is introduced in the paper. This condition generalizes the existence of an equilibrium point.     

Necessary Conditions for Impulsive Nonlinear Optimal Control Problems without a priori Normality Assumptions

First-order and second-order necessary conditions of optimality for an impulsive control problem that remain informative for abnormal control processes are presented and derived. One of the main features of these conditions is that no a priori normality assumptions are required. This feature follows from the fact that these conditions rely on an extremal principle which is proved for an abstract minimization problem with equality constraints, inequality constraints, and constraints given by an inclusion in a convex cone. Two simple examples illustrate the power of the main result.The first author was partially supported by the Russian Foundation for Basic Research Grant 02-01-00334. The second author was partially supported by the Russian Foundation for Basic Research Grant 00-01-00869. The third author was partially supported by Fundacao para a Ciencia e Tecnologia and by INVOTAN Grant.     

Lipschitz continuity and local semiconcavity for exit time problems with state constraints

In the classical time optimal control problem, it is well known that the so-called Petrov condition is necessary and sufficient for the minimum time function to be locally Lipschitz continuous. In this paper, the same regularity result is obtained in the presence of nonsmooth state constraints. Moreover, for a special class of control systems we obtain a local semiconcavity result for the constrained minimum time function.