Extended convergence results for the method of multipliers for nonstrictly binding inequality constraints

In this paper, we extend the classical convergence and rate of convergence results for the method of multipliers for equality constrained problems to general inequality constrained problems, without assuming the strict complementarity hypothesis at the local optimal solution. Instead, we consider an alternative second-order sufficient condition for a strict local minimum, which coincides with the standard one in the case of strict complementary slackness. As a consequence, new stopping rules are derived in order to guarantee a local linear rate of convergence for the method, even if the current Lagrangian is only asymptotically minimized in this more general setting. These extended results allow us to broaden the scope of applicability of the method of multipliers, in order to cover all those problems admitting loosely binding constraints at some optimal solution. This fact is not meaningless, since in practice this kind of problem seems to be more the rule rather than the exception.In proving the different results, we follow the classical primaldual approach to the method of multipliers, considering the approximate minimizers for the original augmented Lagrangian as the exact solutions for some adequate approximate augmented Lagrangian. In particular, we prove a general uniform continuity property concerning both their primal and their dual optimal solution set maps, a property that could be useful beyond the scope of this paper. This approach leads to very simple proofs of the preliminary results and to a straight-forward proof of the main results.The author gratefully acknowledges the referees for their helpful comments and remarks. This research was supported by FONDECYT (Fondo Nacional de Desarrollo Científico y Technológico de Chile).     

Degenerate Principal Series Representations of U(p, q) and Spin0(p, q)

Let p>q and let G be the group U(p, q) or Spin₀(p, q). Let P=LN be the maximal parabolic subgroup of G with Levi subgroup where

有限变形弹性动力学的非传统Gurtin型变分原理

根据古典阴阳互补和现代对偶互补的基本思想,通过作者早已提出的一条简单而统一的途径,系统地建立了有限变形弹性动力学的各类非传统Gurtin型变分原理.给出一个以卷积表示的重要关系式,可以认为,该式是有限变形动力学的广义虚功原理的表式.从该式出发,不仅能得到有限变形动力学的虚功原理,而且通过给出的一系列广义Legendre变换,还能系统地成对导出5类变量、3类变量、2类变量和1类变量非传统Gurtin型变分原理的互补泛函.通过这条途径还能清楚地阐明这些原理之间的内在联系.     

关于线性互补问题解的存在性

讨论线性互补问题解的存在性。证明关于解的唯一性定理。用反例表明:对于线性互补问题解的存在性,"M是半正定矩阵"既不是充分条件,也不是必要条件。     

克尔微腔光频梳理论分析模型及多波长光源应用探讨

基于高品质因子微谐振腔的参量四波混频实现光学频率梳是一种新的频率梳实现方法,拓展了传统固体及非线性光纤飞秒激光器等光频梳的应用范围,在精密频率标定、天文光谱校准、任意波形产生、光学存储和孤子传输、片上通信用光源等方面具有较高的优势。本文简要总结、评述了几种主要的光频梳动力学分析模型及数值方法,以及这些不同方法的内在联系。基于描述光频梳动态行为的非线性Lugiato-Lefever方程分析了可能存在的动力学过程,并据此对不同特点光频梳进行了分类。通过设计反馈结构理论上研究了正常色散微腔和反常色散微腔的光梳特点,探讨了作为片上光互连用多波长光源应满足的条件及可能的实现途径。     

Ring resonators with N × M couplers

Multiple-zero multiple-pole optical filter transfer functions may be implemented more efficiently in an integrated optics architecture if higher order N × M optical couplers are utilized. For example, a coherent ring resonator made from two 3×3 couplers offers some advantages over the three mirror Fabry-Perot etalon, which is its analog. To this end we develop the formalism for obtaining the transfer functions and scattering matrices of ring resonators made from two N × M couplers. We then present a methodology for analyzing serial and parallel systems of N × M optical coupler ring resonators.     

Chapter 15 DEA/AR analysis of the 1988–1989 performance of the Nanjing textiles corporation

This article employs new data envelopment analysis/assurance region (DEA/AR) methods to evaluate the efficiency of the 35 textile factories of the Nanjing Textiles Corporation (NTC), Nanjing, China. The returns to scale (RTS) of these factories were studied without assuming that the optimal DEA solutions were unique. All DMUs are identified with pointsE (Extreme Efficient),E (Efficient but not an extreme point) andF (Frontier but not efficient). We then further identify the nonfrontier DMUs with pointsNE, NE andNF according to whether they are projected onto a point inE, E, orF en route to evaluating their performances. All of the inefficient factories were in classNF and had unique optimal primal-dual solution pairs. Consequently, the solution pairs satisfy the strong complementary slackness condition (SCSC). Application of cone-ratio (CR) ARs reduced significantly the number of factories in classE, and showed that some AR-efficient factories were more flexible in adopting the mixture of central planning and market economies that China currently is trying to use. Also, linked-cone (LC) ARs were applied to measure maximum and minimum profit ratios. The SCSC multiplier space approach was utilized to analyze the sensitivity of the efficiency results to potential errors in the data with and without ARs. The results in this article suggest that collective units had a better performance than state-owned units in the two consecutive years analyzed.This paper was written while the author was at the School of Economics and Management, Southeast University, Nanjing 210018, P.R. China.     

四分之一超导谐振腔的仿真对比

超导量子电路已成为实现量子计算机的主流技术路线之一,其中四分之一波长超导谐振腔主要用于读取量子比特的状态信息,是实现超导量子电路的关键器件.本文设计了四分之一波长超导谐振腔,利用两种电磁仿真算法(有限元法以及矩量法),对超导谐振腔的传输特性进行建模仿真验证.制备出了设计的超导谐振腔样品,在20±5 mK的低温环境下对其传输特性进行测量.通过仿真结果与设计值和实测值进行对比研究,发现基于矩量法的sonnet软件在仿真准确性、仿真速率以及资源消耗等方面都优于基于有限元法的HFSS软件.同时研究了谐振腔之间的串扰对仿真精度的影响,当谐振腔数目不多时,其相互之间串扰的影响几乎可以忽略.     

Nonlinear optics in spheres: from second harmonic scattering to quasi‐phase matched generation in whispering gallery modes

Over the last fifteen years, a series of theoretical and experimental investigations have demonstrated the usefulness of circular geometries to tailor second‐order nonlinear optical effects. However, until recently, such effects have remained rather weak, calling for their enhancement. In parallel, developments in the field of high quality factor spherical or ring resonators have shown that many different types of light‐matter interactions can be dramatically amplified when light is coupled in the whispering gallery modes of such resonators. In high‐quality spherical micro‐resonators, close to one million interactions can occur between a nonlinear molecule and a circulating light pulse. Recent research on nonlinear optics in spherical geometry is reviewed, from micrometer‐size spheres to whispering gallery mode resonators.