A note on an approximate mean value theorem for Fréchet subgradients

In this note, we derive a new characterization of Asplund spaces and give a clarification of the proof of the approximate mean value theorem in Mordukhovich (2006) [5] and Mordukhovich and Shao (1996) [1].     

Dykstras algorithm with bregman projections: A convergence proof

Dykstra’s algorithm and the method of cyclic Bregman projections are often employed to solve best approximation and convex feasibility problems, which are fundamental in mathematics and the physical sciences. Censor and Reich very recently suggested a synthesis of these methods, Dykstra’s algorithm with Bregman projections, to tackle a non-orthogonal best approximation problem, They obtained convergence when each constraint is a halfspace. It is shown here that this new algorithm works for general closed convex constraints; this complements Censor and Reich’s result and relates to a framework by Tseng. The proof rests on Boyle and Dykstra’s original work and on strong properties of Bregman distances corresponding to Legendre functions. Special cases and observations simplifying the implementation of the algorithm are aiso discussed     

基于蒙特卡罗模拟的无创生化检测最佳探测距离的研究

人体内各种血液生化成分含量的变化是评价健康状况的重要信息。人体皮肤中,血糖、胆固醇等生化成分主要存在于真皮层的血液中,本文通过研究最佳的探测距离,提高真皮层的近红外光谱信号强度,排除表皮层、皮下组织产生的干扰。首先,通过对人体皮肤的组织结构进行分析,计算得到了皮肤各层的组织光学特性参数,以葡萄糖在合频波段吸收峰2 270 nm处的组织光学参数为例建立了皮肤的蒙特卡罗模型;然后,利用蒙特卡罗方法模拟了光在皮肤中的传输规律,得到了平均路径长度、平均探测深度以及各层吸收光子能量比例随入射角度、探测距离的变化情况。结果表明,光子以小于45°角入射时,可以忽略入射角度对光子传输路径的影响。探测距离为1 mm时,真皮层吸收光子能量所占比例最大,同时又能保证探测器接收到较多的能量。本文确定1 mm作为最佳的探测距离,可以有效降低皮肤中其他组织的干扰,获取真皮层中血液的光谱信息,有利于生化成分的近红外无创检测,为后续实验奠定理论基础。     

Estimates for multilinear singular integral operators with nonsmooth kernels

Let (, d, μ) be a space of homogeneous type in the sense of Coifman and Weiss. In this paper, we consider the behavior on × ⋯ × for the m-linear singular integral operators with nonsmooth kernels which were first introduced by Duong, Grafakos and Yan.     

Approximation of infinitely differentiable multivariate functions is intractable

We prove that L_∞-approximation of C^∞-functions defined on [0,1]^d is intractable and suffers from the curse of dimensionality. This is done by showing that the minimal number of linear functionals needed to obtain an algorithm with worst case error at most ε(0,1) is exponential in d. This holds despite the fact that the rate of convergence is infinite.     

Nonconvex piecewise linear knapsack problems

This paper considers the minimization version of a class of nonconvex knapsack problems with piecewise linear cost structure. The items to be included in the knapsack have a divisible quantity and a cost function. An item can be included partially in the given quantity range and the cost is a nonconvex piecewise linear function of quantity. Given a demand, the optimization problem is to choose an optimal quantity for each item such that the demand is satisfied and the total cost is minimized. This problem and its close variants are encountered in manufacturing planning, supply chain design, volume discount procurement auctions, and many other contemporary applications. Two separate mixed integer linear programming formulations of this problem are proposed and are compared with existing formulations. Motivated by different scenarios in which the problem is useful, the following algorithms are developed: (1) a fast polynomial time, near-optimal heuristic using convex envelopes; (2) exact pseudo-polynomial time dynamic programming algorithms; (3) a 2-approximation algorithm; and (4) a fully polynomial time approximation scheme. A comprehensive test suite is developed to generate representative problem instances with different characteristics. Extensive computational experiments show that the proposed formulations and algorithms are faster than the existing techniques.     

Two simple constant ratio approximation algorithms for minimizing the total weighted completion time on a single machine with a fixed non-availability interval

In this note, we consider the scheduling problem of minimizing the sum of the weighted completion times on a single machine with one non-availability interval on the machine under the non-resumable scenario. Together with a recent 2-approximation algorithm designed by Kacem [I. Kacem, Approximation algorithm for the weighted flow-time minimization on a single machine with a fixed non-availability interval, Computers & Industrial Engineering 54 (2008) 401–410], this paper is the first successful attempt to develop a constant ratio approximation algorithm for this problem. We present two approaches to designing such an algorithm. Our best algorithm guarantees a worst-case performance ratio of 2+ε$2+\epsilon$.     

神经网络的加权本质逼近阶

证明了具有单一隐层的神经网络在L_ω~q的逼近,获得了网络逼近的上界估计和下界估计.这一结果揭示了神经网络在加权逼近的意义下,网络的收敛阶与隐层单元个数之间的关系,为神经网络的应用提供了重要的理论基础.     

模糊形式背景下的类近似算子

分别在形式背景和模糊形式背景下定义类下近似算子和类模糊下近似算子,并研究它们的性质.证明这两种算子分剐等价于形式背景和模糊形式背景下的*算子和模糊*算子.进一步给出类下近似算子与类模糊下近似算子的公理刺画.最后,对偶地讨论类上近似算子和类模糊上近似算子的定义和性质.