Special Pair of Dual Parametric Nonlinear Optimization Problems with Common Set of Optimal Points

On the base of a given strictly convex function defined on the Euclidean space E n ( n S 2) we can-without the assumption that it is differentiable - introduce some manifolds in topologic sense. Such manifolds are sets of all optimal points of a certain parametric non-linear optimization problem. This paper presents above all certain generalization of some results of [F. No ? i ) ka and L. Grygarová (1991). Some topological questions connected with strictly convex functions. Optimization , 22 , 177-191. Akademie Verlag, Berlin] and [L. Grygarová (1988). Über Lösungsmengen spezieller konvexer parametrischer Optimierungsaufgaben . Optimization 19 , 215-228. Akademie Verlag Berlin], under less strict assumptions. The main results are presented in Sections 3 and 4, in Section 3 the geometrical characterization of the set of optimal points of a certain parametric minimization problem is presented; in Section 4 we study a maximization non-linear parametric problem assigned to it. It seems that it is a certain pair of parametric optimization problems with the same set of their optimal points, so that this pair of problems can be denoted as a pair of dual parametric non-linear optimization problems. This paper presents, most of all in Section 2, a number of interesting geometric facts about strictly convex functions. From the point of view of non-smooth analysis the present article is a certain complement to Chapter 4.3 of the book [B. Bank, J. Guddat, D. Klatte, B. Kummer and K. Tammer (1982). Nonlinear Parametric Optimization . Akademie Verlag, Berlin] where a convex parametric minimization problem is considered under more general and stronger conditions (but without any assumptions concerning strict convexity and without geometrical aspects).     

Optimal times for constrained nonlinear control problems without local controllability

We study optimal times to reach a given closed target for controlled systems with a state constraint. Our goal is to characterize these optimal time functions in such a way that it is possible to compute them numerically and we do not need to compute trajectories of the controlled system. In this paper we provide new results using viability theory. This allows us to study optimal time functions free from the controllability assumptions classically made in the partial differential equations approach.     

Second order optimality conditions for generalized semi-infinite programming problems

We consider generalized semi-infinite programming problems. Second order necessary and sufficient conditionsfor local optimality are given. The conditions are derived under assumptions such that the feasible set can be described by means of a finite number of optimal value functions. Since we do not require a strict complementary condition for the local reduction these functions are only of class C¹ A sufficient condition for optimality is proven under much weaker assumptions.     

A note on negative dynamic programming for risk-sensitive control

Negative dynamic programming for risk-sensitive control is studied. Under some compactness and semicontinuity assumptions the following results are proved: the convergence of the value iteration algorithm to the optimal expected total reward, the Borel measurability or upper semicontinuity of the optimal value functions, and the existence of an optimal stationary policy.     

On Generic Existence and Uniqueness in Nonconvex Optimal Control Problems

We derive conditions for generic existence and uniqueness of optimal control and trajectories for some class of finite-dimensional optimal control problems in the absence of traditional convexity assumptions. It is shown that for these problems existence and uniqueness of optimal control for a given initial point x is equivalent to the differentiability of optimal value functions at x. These results are obtained in the general framework of representation formulas for (sub-)gradients of inf-envelope functions which appear in nonsmooth analysis.     

On strong product integration

Product integration is defined for a very general class of bounded-operator-valued functions on a Banach space X. Previous assumptions of continuity or Riemann-integrability of such functions are not needed. Properties of the product integral in the new setting are derived, including material on improper product integration. The theory is applied to the study of an equation of evolution involving unbounded operators, and a stronger form of a theorem due to T. Kato and K. Yosida is obtained.     

随机截尾试验变量抽样接收方案的贝叶斯决策

本运用Bayes决策理论研究指数分布和随机截尾试验的抽样接收方案的一般模型,我们证明了最优Bayes法则具有单调性,并对二个特殊的决策损失函数给出了最优Bayes法则和Bayes风险的具体表达式。     

The optimal value and optimal solutions of the proximal average of convex functions

The proximal average of a finite collection of convex functions is a parameterized convex function that provides a continuous transformation between the convex functions in the collection. This paper analyzes the dependence of the optimal value and the minimizers of the proximal average on the weighting parameter. Concavity of the optimal value is established and implies further regularity properties of the optimal value. Boundedness, outer semicontinuity, single-valuedness, continuity, and Lipschitz continuity of the minimizer mapping are concluded under various assumptions. Sharp minimizers are given further attention. Several examples are given to illustrate our results.     

Nonlinear Lagrange Duality Theorems and Penalty Function Methods In Continuous Optimization

We propose a general dual program for a constrained optimization problem via generalized nonlinear Lagrangian functions. Our dual program includes a class of general dual programs with explicit structures as special cases. Duality theorems with the zero duality gap are proved under very general assumptions and several important corollaries which include some known results are given. Using dual functions as penalty functions, we also establish that a sequence of approximate optimal solutions of the penalty function converges to the optimal solution of the original optimization problem.     

一类二层凸规划的对偶二层规划

本文讨论上层目标函数以下层子系统目标函数的最优值作为反馈的一类二层凸规划的对偶规划问题 ,在构成函数满足凸连续可微等条件的假设下 ,建立了二层凸规划的 Lagrange对偶二层规划 ,并证明了基本对偶定理 .     

Stochastic Multiproduct Inventory Models with Limited Storage

This paper studies multiproduct inventory models with stochastic demands and a warehousing constraint. Finite horizon as well as stationary and nonstationary discounted-cost infinite-horizon problems are addressed. Existence of optimal feedback policies is established under fairly general assumptions. Furthermore, the structure of the optimal policies is analyzed when the ordering cost is linear and the inventory/backlog cost is convex. The optimal policies generalize the base-stock policies in the single-product case. Finally, in the stationary infinite-horizon case, a myopic policy is proved to be optimal if the product demands are independent and the cost functions are separable.     

B-spline estimation for varying coefficient regression with spatial data

This paper considers a nonparametric varying coefficient regression with spatial data. A global smoothing procedure is developed by using B-spline function approximations for estimating the coefficient functions. Under mild regularity assumptions,the global convergence rates of the B-spline estimators of the unknown coefficient functions are established. Asymptotic results show that our B-spline estimators achieve the optimal convergence rate. The asymptotic distributions of the B-spline estimators of the u...     

The Mayer and Minimum Time Problems with Stratified State Constraints

This paper studies optimal control problems with state constraints by imposing structural assumptions on the constraint domain coupled with a tangential restriction with the dynamics. These assumptions replace pointing or controllability assumptions that are common in the literature, and provide a framework under which feasible boundary trajectories can be analyzed directly. The value functions associated with the state constrained Mayer and minimal time problems are characterized as solutions to a pair of Hamilton-Jacobi inequalities with appropriate boundary conditions. The novel feature of these inequalities lies in the choice of the Hamiltonian.     

Successive approximation technique for a class of large-scale NLP problems and its application to dynamic programming

In this paper, two successive approximation techniques are presented for a class of large-scale nonlinear programming problems with decomposable constraints and a class of high-dimensional discrete optimal control problems, respectively. It is shown that: (a) the accumulation point of the sequence produced by the first method is a Kuhn-Tucker point if the constraint functions are decomposable and if the uniqueness condition holds; (b) the sequence converges to an optimum solution if the objective function is strictly pseudoconvex and if the constraint functions are decomposable and quasiconcave; and (c) similar conclusions for the second method hold also for a class of discrete optimal control problems under some assumptions.     

Dirichlet Boundary Control of Hyperbolic Equations in the Presence of State Constraints

We study optimal control problems for hyperbolic equations (focusing on the multidimensional wave equation) with control functions in the Dirichlet boundary conditions under hard/pointwise control and state constraints. Imposing appropriate convexity assumptions on the cost integral functional, we establish the existence of optimal control and derive new necessary optimality conditions in the integral form of the Pontryagin Maximum Principle for hyperbolic state-constrained systems.     

On Hsu's theorem in multivariate regression

The paper deals with optimal quadratic unbiased estimation of the unknown dispersion matrix in multivariate regression models without assuming normality of the errors. We show that Hsu's theorem for univariate regression models continues to multivariate models with no additional assumptions. Furthermore optimal quadratic plus linear estimating functions for regression coefficients are considered, and we investigate whether the ordinary linear estimates are the best. This leads to a new theorem which is similar to that of Hsu.     

Singular optimal controls of stochastic recursive systems and Hamilton–Jacobi–Bellman inequality

In this paper, we study the optimal singular controls for stochastic recursive systems, in which the control has two components: the regular control, and the singular control. Under certain assumptions, we establish the dynamic programming principle for this kind of optimal singular controls problem, and prove that the value function is a unique viscosity solution of the corresponding Hamilton–Jacobi–Bellman inequality, in a given class of bounded and continuous functions. At last, an example is given for illustration.     

Adaptive and Optimal Point-Wise Estimations for Densities in GARCH-Type Model by Wavelets

This paper considers adaptive point-wise estimations of density functions in GARCH-type model under the local Holder condition by wavelet methods.A point-wise lower bound estimation of that model is first investigated;then we provide a linear wavelet estimate to obtain the optimal convergence rate,which means that the convergence rate coincides with the lower bound.The non-linear wavelet estimator is introduced for adaptivity,although it is nearly-optimal.However,the non-linear wavelet one depends on an upper bound of the smoothness index of unknown functions,we finally discuss a data driven version without any assumptions on the estimated functions.     

Optimal impulse control problems for degenerate diffusions with jumps

Optimal stopping and impulse control problems for degenerate diffusion with jumps are studied in this paper. Lipschitzian coefficients for the diffusion process, data with polynomial growth, and evolution in the whole space are the main assumptions on the models. Several characterizations of the optimal cost functions are given. Existence of optimal policies is obtained.This research has been supported in part by Army Research Office Contract DAAG29-83-K-0014 and by National Science Foundation Grant DMS-8601998.     

Optimal scaling of random-walk metropolis algorithms on general target distributions

One main limitation of the existing optimal scaling results for Metropolis–Hastings algorithms is that the assumptions on the target distribution are unrealistic. In this paper, we consider optimal scaling of random-walk Metropolis algorithms on general target distributions in high dimensions arising from practical MCMC models from Bayesian statistics. For optimal scaling by maximizing expected squared jumping distance (ESJD), we show the asymptotically optimal acceptance rate 0.234 can be obtained under general realistic sufficient conditions on the target distribution. The new sufficient conditions are easy to be verified and may hold for some general classes of MCMC models arising from Bayesian statistics applications, which substantially generalize the product i.i.d. condition required in most existing literature of optimal scaling. Furthermore, we show one-dimensional diffusion limits can be obtained under slightly stronger conditions, which still allow dependent coordinates of the target distribution. We also connect the new diffusion limit results to complexity bounds of Metropolis algorithms in high dimensions.