Least Squares Convex-Concave Data Smoothing

We consider n noisy measurements of a smooth (unknown) function, which suggest that the graph of the function consists of one convex and one concave section. Due to the noise the sequence of the second divided differences of the data exhibits more sign changes than those expected in the second derivative of the underlying function. We address the problem of smoothing the data so as to minimize the sum of squares of residuals subject to the condition that the sequence of successive second divided differences of the smoothed values changes sign at most once. It is a nonlinear problem, since the position of the sign change is also an unknown of the optimization process. We state a characterization theorem, which shows that the smoothed values can be derived by at most 2n – 2 quadratic programming calculations to subranges of data. Then, we develop an algorithm that solves the problem in about O(n ²) computer operations by employing several techniques, including B-splines, the use of active sets, quadratic programming and updating methods. A Fortran program has been written and some of its numerical results are presented. Applications of the smoothing technique may be found in scientific, economic and engineering calculations, when a potential shape for the underlying function is an S-curve. Generally, the smoothing calculation may arise from processes that show initially increasing and then decreasing rates of change.     

一种测量凹透镜焦距的方法

介绍了一种新的测量凹透镜焦距的方法：复合成象法。该方法通过凹透镜的两次成象过程中透镜位移的测量，再利用贝塞尔公式将凹透镜的焦距值间接测量出来，此方法的测量精度高，相对误差约为１％左右。     

折射型微透镜列阵的光刻热熔法研究

研究了制作折射型微透镜列阵的一种新方法光刻胶热熔成形法，获得了２０×２０的折射型微透镜列阵，单元微透镜相对口径为Ｆ／２，单元透镜直径为９０μｍ，中心间隔１００μｍ，透镜的波像差小于１．３波长。本文详细阐述了光刻热熔法的基本原理及微透镜设计方法，并讨论了工艺参数对微透镜列阵质量的影响。     

A Convex Envelope Formula for Multilinear Functions

Convex envelopes of multilinear functions on a unit hypercube arepolyhedral. This well-known fact makes the convex envelopeapproximation very useful in the linearization of non-linear 0–1programming problems and in global bilinear optimization. This paperpresents necessary and sufficient conditions for a convex envelope to be apolyhedral function and illustrates how these conditions may be used inconstructing of convex envelopes. The main result of the paper is a simpleanalytical formula, which defines some faces of the convex envelope of amultilinear function. This formula proves to be a generalization of the wellknown convex envelope formula for multilinear monomial functions.     

Distribution of Global Measures of Deviation Between the Empirical Distribution Function and Its Concave Majorant

We investigate the distribution of some global measures of deviation between the empirical distribution function and its least concave majorant. In the case that the underlying distribution has a strictly decreasing density, we prove asymptotic normality for several L _k -type distances. In the case of a uniform distribution, we also establish their limit distribution together with that of the supremum distance. It turns out that in the uniform case, the measures of deviation are of greater order and their limit distributions are different.     

Bound on integrals: Elimination of the dual and reduction of the number of equality constraints

Let L_j (j = 1, …, n + 1) be real linear functions on the convex set $\text{F}$ of probability distributions. We consider the problem of maximization of L_n+1(F) under the constraint F ? $\text{F}$ and the equality constraints L₁(F) = z₁ (i = 1, …, n). Incorporating some of the equality constraints into the basic set $\text{F}$, the problem is equivalent to a problem with less equality constraints. We also show how the dual problems can be eliminated from the statement of the main theorems and we give a new illuminating proof of the existence of particular solutions.The linearity of the functions L_j(j = 1, …, n + 1) can be dropped in several results.     

线性比式和规划问题的输出空间分支定界算法

本文旨在针对线性比式和规划这一NP-Hard非线性规划问题提出新的全局优化算法.首先,通过引入p个辅助变量把原问题等价的转化为一个非线性规划问题,这个非线性规划问题的目标函数是乘积和的形式并给原问题增加了p个新的非线性约束,再通过构造凸凹包络的技巧对等价问题的目标函数和约束条件进行相应的线性放缩,构成等价问题的一个下界线性松弛规划问题,从而提出了一个求解原问题的分支定界算法,并证明了算法的收敛性.最后,通过数值结果比较表明所提出的算法是可行有效的.     

Elliptic equations with indefinite concave nonlinearities near the origin

In this note existence of infinitely many solutions is proved for an elliptic equation with indefinite concave nonlinearities.     

φ凹(-φ凸)算子不动点定理及其应用

讨论了φ凹(-φ凸)算子,得到了φ凹增(-φ凸减)算子不动点存在唯一性结果,并且给出了收敛到该不动点的迭代序列.该结果去掉了以往文献中的φ→0(t→0~+)这一条件,从而改进和推广了相关结果.作为应用,给出了一类的Sturm-Liouville边值问题的正解的存在唯一性结果.     

无穷凹角区域各向异性问题的自然边界元与有限元耦合法

本文研究无穷凹角区域上一类各向异性问题的自然边界元与有限元耦合法.利用自然边界归化原理,获得圆弧或椭圆弧人工边界上的自然积分方程,给出了耦合的变分形式及其数值方法,以及逼近解的收敛性和误差估计,最后给出了数值例子,以示方法的可行性和有效性.