Local properties of algorithms for minimizing nonsmooth composite functions

This paper considers local convergence and rate of convergence results for algorithms for minimizing the composite functionF(x)=f(x)+h(c(x)) wheref andc are smooth buth(c) may be nonsmooth. Local convergence at a second order rate is established for the generalized Gauss—Newton method whenh is convex and globally Lipschitz and the minimizer is strongly unique. Local convergence at a second order rate is established for a generalized Newton method when the minimizer satisfies nondegeneracy, strict complementarity and second order sufficiency conditions. Assuming the minimizer satisfies these conditions, necessary and sufficient conditions for a superlinear rate of convergence for curvature approximating methods are established. Necessary and sufficient conditions for a two-step superlinear rate of convergence are also established when only reduced curvature information is available. All these local convergence and rate of convergence results are directly applicable to nonlinearing programming problems.This work was done while the author was a Research fellow at the Mathematical Sciences Research Centre, Australian National University.     

基于输入-输出数据的概率分布和数字特征

利用模糊推理建立了一种基于输入-输出数据构造联合概率密度函数的方法.首先,将一组单输入-单输出数据转换成模糊推理规则,通过选择适当的模糊蕴涵算子生成模糊关系,再利用这种模糊关系求出二维随机变量的联合概率密度函数.当将模糊蕴涵分别取为Larsen蕴涵和Mamdani蕴涵时,分别得到了两种具体的概率密度函数(称之为Lars...     

A Friedrichs–Maz'ya inequality for functions of bounded variation

The aim of this short note is to give an alternative proof, which applies to functions of bounded variation in arbitrary domains, of an inequality by Maz'ya that improves Friedrichs inequality. A remarkable feature of such a proof is that it is rather elementary, if the basic background in the theory of functions of bounded variation is assumed. Nevertheless, it allows to extend all the previously known versions of this fundamental inequality to a completely general version. In fact the inequality presented here is optimal in several respects. As already observed in previous proofs, the crucial step is to provide conditions under which a function of bounded variation on a bounded open set, extended to zero outside, has bounded variation on the whole space. We push such conditions to their limits. In fact, we give a sufficient and necessary condition if the open set has a boundary with σ‐finite surface measure and a sufficient condition if the open set is fully arbitrary. Via a counterexample we show that such a general sufficient condition is sharp.     

An alternative solution of the scattering from an anisotropic cylindrical dielectric shell

A method based on the approximate wave functions for anisotropic media and the mode-matching approach is developed to solve the problem of the electromagnetic scattering from an anisotropic cylindrical dielectric shell. The cylindrical shell is assumed to be infinite in length, and it is illuminated by a plane wave or a cylindrical wave from a line source. The problem is two-dimensional and the solutions to both types of polarization (TE and TM) are presented. The validity of this solution is verified by comparing the numerical results with those in literatures and the previous calculations based on the exact wave functions for anisotropic media. Numerical results show the higher computational efficiency of the present method for bounded anisotropic media.     

Representation and approximation of multivariate functions with mixed smoothness by hyperbolic wavelets

In this paper, we study the representation theorems of multivariate functions with mixed smoothness by wavelet basis formed by tensor products of univariate wavelets, we also study the best approximation in the metric for some function classes with mixed smoothness by hyperbolic wavelets and obtain some asymptotic estimates of approximating order.     

Proper efficiency and duality for a class of multiobjective fractional variational problems containing arbitrary norms

Necessary and sufficient conditions are established for properly efficient solutions of a class of nonsmooth nonconvex variational problems with multiple fractional objective functions and nonlinear inequality constraints. Based on these proper efficiency criteria. two multiobjective dual problems are constructed and appropriate duality theorems are proved. These proper efficiency and duality results also contain as special cases similar rcsults fer constrained variational problems with multiplei fractional. and conventional objective functions, which are particular cases of the main variational problem considered in this paper     

Sufficient Conditions for Starlikeness and Strong-starlikeness of a Given Order

We find sharp upper bounds for $ \vert \,f^{\prime \prime }(z)/f^{\prime }(z) \vert $ such that the function f be starlike or strongly-starlike of a given order, where f is holomorphic in the unit disc U , with $ f(0)=f^{\prime }(0)-1=0 $ . These bounds are obtained by using the differential subordination technique. As a useful ingredient we obtain the radius of convexity for the function $ F(z)= z /(\exp (z)-1) $     

A Cubic Decomposition of Sequences of Orthogonal Polynomials on the Unit Circle

In this paper we will discuss the problem of generation of sequences of orthogonal polynomials with respect to measures supported on the unit circle from a given sequence of orthogonal polynomials using a perturbation of a cubic sieved process. The basic tools are the Szeg? forward recurrence relation as well as the fact of the coprimality of orthogonal polynomials on the unit circle and their corresponding reverse polynomials. We also give the connection between the associated orthogonality measures. Finally, some examples of this cubic decomposition are shown.     

Variants of the optimised random phase approximation: from one direct correlation function to many different radial distribution functions

It is shown that there are an infinite number of radial distribution functions (RDFs), corresponding to only one direct correlation function (DCF) of the optimised random phase approximation (ORPA). This observation in the thermodynamic perturbation theory is in sharp contrast to that of integral equation theories in which they uniquely correspond. By devising a new method we will be able to introduce various perturbation theories of simple liquids all coming from one DCF. Among all, we will only present analytically seven variants of the ORPA in the thermodynamic perturbation theory of liquids. The DCF of hard-core potential for all variants is assumed to be the same as the ORPA. However, interestingly enough the resulted expressions for the Helmholtz free energies and the RDF are obtained very differently. Furthermore, the resulted thermodynamic properties come out somehow the same, whereas the structural functions of some variants are found to behave much better than the standard ORPA.