Stability of multistage stochastic programming

In this paper, we study the stability of multistage stochastic programming with recourse in a way that is different from that used in studying stability of two-stage stochastic programs. Here, we transform the multistage programs into mathematical programs in the space ⁿ ×L _p with a simple objective function and multistage stochastic constraints. By investigating the continuity of the multistage multifunction defined by the multistage stochastic constraints and applying epi-convergence theory we obtain stability results for linear and linear-quadratic multistage stochastic programs.Project supported by the National Natural Science Foundation of China.     

Randomized algorithms for the separation of point sets and for solving quadratic programs

A randomized algorithm for finding a hyperplane separating two finite point sets in the Euclidean space ^d and a randomized algorithm for solving linearly constrained general convex quadratic problems are proposed. The expected running time of the separating algorithm isO(dd! (m + n)), wherem andn are cardinalities of sets to be separated. The expected running time of the algorithm for solving quadratic problems isO(dd! s) wheres is the number of inequality constraints. These algorithms are based on the ideas of Seidel's linear programming algorithm [6]. They are closely related to algorithms of [8], [2], and [9] and belong to an abstract class of algorithms investigated in [1]. The algorithm for solving quadratic problems has some features of the one proposed in [7].This research was done when the author was supported by the Alexander von Humboldt Foundation, Germany.On leave from the Institute of Mathematics and Mechanics, Ural Department of the Russian Academy of Sciences, 620219 Ekaterinburg, S. Kovalevskaya str. 16, Russia.     

Banach spaces that admit support sets

It is shown that the existence of a closed convex set all of whose points are properly supported in a Banach space is equivalent to the existence of a certain type of uncountable ordered one-sided biorthogonal system. Under the continuum hypothesis, we deduce that this notion is weaker than the existence of an uncountable biorthogonal system.

     

SOME BLENDING INTERPOLATION METHODS IN CAGD

Some of the blending interpolation methods (which) stated by a convex combination technique are presented. This includes C1 ,C2 blending triangular interpolanls and C1 -blending quadrilateral inter polanls for surfaces. The mentioned methods can be generalized to the construction of C" blending interpolants.     

Quantitative stability of variational systems: III.ε-approximate solutions

We prove that the-optimal solutions of convex optimization problems are Lipschitz continuous with respect to data perturbations when these are measured in terms of the epi-distance. A similar property is obtained for the distance between the level sets of extended real valued functions. We also show that these properties imply that the-subgradient mapping is Lipschitz continuous.Research supported in part by the National Science Foundation and the Air Force Office of Scientific Research.     

Generalizations of Slater's constraint qualification for infinite convex programs

In this paper we study constraint qualifications and duality results for infinite convex programs (P) = inf{f(x): g(x) – S, x C}, whereg = (g ₁,g ₂) andS = S ₁ ×S ₂,S _i are convex cones,i = 1, 2,C is a convex subset of a vector spaceX, andf andg _i are, respectively, convex andS _i -convex,i = 1, 2. In particular, we consider the special case whenS ₂ is in afinite dimensional space,g ₂ is affine andS ₂ is polyhedral. We show that a recently introduced simple constraint qualification, and the so-called quasi relative interior constraint qualification both extend to (P), from the special case thatg = g ₂ is affine andS = S ₂ is polyhedral in a finite dimensional space (the so-called partially finite program). This provides generalized Slater type conditions for (P) which are much weaker than the standard Slater condition. We exhibit the relationship between these two constraint qualifications and show how to replace the affine assumption ong ₂ and the finite dimensionality assumption onS ₂, by a local compactness assumption. We then introduce the notion of strong quasi relative interior to get parallel results for more general infinite dimensional programs without the local compactness assumption. Our basic tool reduces to guaranteeing the closure of the sum of two closed convex cones.     

Bayes estimation with spherically symmetric,convex loss

Summary It is desired to estimate a parameter with the loss function of the formL(θ, a)=W(‖θ−a‖), where is convex, differentiable, and non-decreasing. With this structure a characterization of Bayes estimators is given. Also it is noted that if the sample space, , for the observation,X, is a complete separable metric space then a Bayes estimator exists.     

Banach空间半线性发展方程的周期解

把上、下解方法引入到有序Ｂａｎａｃｈ空间中的半线性发展方程周期解问题，利用正算子半群特征与单调迭代程序，获得了最大周期解与最小周期解的存在性．所得的结果概括和推广了常微分方程与偏微分方程中的有关结论     

Convexifactors,generalized convexity and vector optimization

In this article we study a recently introduced notion of non-smooth analysis, namely convexifactors. We study some properties of the convexifactors and introduce two new chain rules. A new notion of non-smooth pseudoconvex function is introduced and its properties are studied in terms of convexifactors. We also present some optimality conditions for vector minimization in terms of convexifactors.