Some Characterizations of V MO(Rn)

In this paper we give three characterizations of VMO(Rn) space, which are of John-Nirenberg type, Uchiyama-type and Miyachi-type, respectively.     

When null energy condition meets ADM mass

We give a conjecture on the lower bound of the ADM mass M by using the null energy condition. The conjecture includes a Penrose-like inequality $3M\geqslant \kappa { \mathcal A }/(4\pi )+\sqrt{{ \mathcal A }/4\pi }$ and the Penrose inequality $2M\geqslant \sqrt{{ \mathcal A }/4\pi }$ with ${ \mathcal A }$ the event horizon area and κ the surface gravity. Both the conjecture in the static spherically symmetric case and the Penrose inequality for a dynamical spacetime with spherical symmetry are proved by imposing the null energy condition. We then generalize the conjecture to a general dynamical spacetime. Our results raise a new challenge for the famous unsettled question in general relativity: in what general case can the null energy condition replace other energy conditions to ensure the Penrose inequality?     

非均布随机参数结构非线性响应的概率密度演化简

首先考察了概率密度演化理论中的点演化和群演化与概率空间剖分的关系. 继而,讨论了点集筛选的基本准则. 在此基础上推广了点集偏差的概念,对非均匀、非正态的一般多维分布,提出了广义F 偏差（GF 偏差）的概念,避免了偏差计算的NP 难解问题. 探索了GF 偏差与EF 偏差的关系. 以GF 偏差最小化为准则,建议了概率空间最优剖分与点集重整的新策略. 结果表明,上述方法能够处理包含多达数10 个随机变量的结构动力响应概率密度演化分析问题. 最后,指出了需要进一步研究的问题.     

A new iterative algorithm for the variational inequality problem over the fixed point set of a firmly nonexpansive mapping

Many problems to appear in signal processing have been formulated as the variational inequality problem over the fixed point set of a nonexpansive mapping. In particular, convex optimization problems over the fixed point set are discussed, and operators which are considered to the problems satisfy the monotonicity. Hence, the uniqueness of the solution of the problem is not always guaranteed. In this article, we present the variational inequality problem for a monotone, hemicontinuous operator over the fixed point set of a firmly nonexpansive mapping. The main aim of the article is to solve the proposed problem by using an iterative algorithm. To this goal, we present a new iterative algorithm for the proposed problem and its convergence analysis. Numerical examples for the proposed algorithm for convex optimization problems over the fixed point set are provided in the final section.     

On the solution existence of implicit quasivariational inequalities with discontinuous multifunctions

In this article, we established some solution existence theorems for implicit quasivariational inequalities. We first established some results in finite-dimensional spaces and then a solution existence result in infinite-dimensional spaces was derived. Our theorems are proved for discontinuous mappings and sets which may be unbounded. The results presented in this article are improvements of results in Cubiotti, and Yao (Cubiotti, P. and Yao, J.C., 1997, Discontinuous implicit quasi-variational inequalities with applications to fuzzy mappings. Mathematical Methods of Operations Research, 46, 213–328; Cubiotti, P. and Yao, J.C., 2007, Discontinuous implicit generalized quasi-variational inequalities in Banach spaces. Journal of Global Optimization (To appear))     

A robust optimization model for dynamic market with uncertain production cost

This article presents a robust optimization formulation for dealing with production cost uncertainty in an oligopolistic market scenario. It is not uncommon that players in the market face an equilibrium selling price but uncertain production costs. We show that, based on a nominal problem, the robust optimization formulation can be derived as a variational inequality with control and state variables. This convenient approach may be applied for computing optimal solutions efficiently, which help manufacturers dramatically and rapidly reform production and distribution schedules such that they can compete in the market successfully.     

Accelerated reflection projection algorithm and its application to the LMI problem

We discuss accelerated version of the alternating projection method which can be applied to solve the linear matrix inequality (LMI) problem. The alternating projection method is a well-known algorithm for the convex feasibility problem, and has many generalizations and extensions. Bauschke and Kruk proposed a reflection projection algorithm for computing a point in the intersection of an obtuse cone and a closed convex set. We carry on this research in two directions. First, we present an accelerated version of the reflection projection algorithm, and prove its weak convergence in a Hilbert space; second, we prove the finite termination of an algorithm which is based on the proposed algorithm and provide an explicit upper bound for the required number of iterations under certain assumptions. Numerical experiments for the LMI problem are provided to demonstrate the effectiveness and merits of the proposed algorithms.     

A new method to build confidence regions for solutions of stochastic variational inequalities

Stochastic variational inequalities model a large class of equilibrium problems subject to data uncertainty. The true solution to such a problem is usually estimated by a solution to its sample average approximation (SAA) problem. This article proposed a new method to build asymptotically exact confidence regions for the true solution that are computable from the SAA solution.     

Recurrence set-relations in stochastic multiobjective dynamic decision processes

Finite horizon stochastic dynamic decision processes with R^p valued additive returns are considered. The optimization criterion is a partial-order preference relation induced from a convex cone in R^p . The state space is a countable set, and the action space is a compact metric spaces. The optimal value function, which is of a set-valued mapping, is defined. Under certain assumptions on the continuity of the reward vector and the transition probability, a system of a recurrence set-relations concerning the optimal value functions is given.