A smoothing Levenberg–Marquardt method for the extended linear complementarity problem

We consider the extended linear complementarity problem (XLCP), of which the linear and horizontal linear complementarity problems are two special cases. We reformulate the XLCP to a smooth equation by using some smoothing functions and propose a Levenberg–Marquardt method to solve the system of smooth equation. The global convergence and local superlinear convergence rate are established under certain conditions. Numerical tests show the effectiveness of the proposed algorithm.     

NOTES ON REFINABLE FUNCTIONS

1.IntroductionItiswell--knownthatrefinablefunctionsplayanimportantroleinthestudyingofwavelet.Usually,onehopesthatrefinablefunctionshavesomeparticularpropertiessuchassmoothnessandintegrability.Inthisnote,thezerosofanintegrablerefinablefunctionareobtained.Inparticalarbyexamplesoneshowsthatthelinearspaceassociatingthetranslatesoverthelatticepointsofarefinablefunctioncouldincludepolynomialspaceofdegreehigherthanitssmoothorder.LetsbeapositiveintegerandletR'(resp.C')bethes-dimensionalreal(qgmplex)…     

On Sensitivity in Linear Multiobjective Programming

In this paper, we prove that, if the data of a linear multiobjectiveprogramming problem are smooth functions of a parameter, then in theparameter space there is an open dense subset where the efficient solutionset of the problem can be locally represented as a union of some faces whosevertices and directions are smooth functions of the parameter.     

On the law of the iterated logarithm for Brownian motion on compact manifolds

By taking a functional analytic point of view,we consider a family of distributions(continuous linear functionals on smooth functions),denoted by{μt,t0},associated to the law of the iterated logarithm for Brownian motion on a compact manifold.We give a complete characterization of the collection of limiting distributions of{μt,t0}.     

Some remarks on bilinear Littlewood-Paley theory

In this work, some bilinear analogues of linear Littlewood-Paley theory are explored. Paraproducts with functions decomposed at different scales are shown to be bounded on certain products of Lebesgue spaces. Results concerning square functions associated with smooth and nonsmooth cutoffs are also obtained.     

Approximation to linear functionals on H1 by point evaluations

The degree of approximation to a bounded linear functional on H¹ having a smooth kernel by finite sums of point evaluations is shown to depend on the smoothness of that kernel. Use is made of a Jackson-Bernstein type theorem concerning the approximation of continuous unimodular functions by finite Blaschke products.     

Polynomial spline estimation for partial functional linear regression models

Because of its orthogonality, interpretability and best representation, functional principal component analysis approach has been extensively used to estimate the slope function in the functional linear model. However, as a very popular smooth technique in nonparametric/semiparametric regression, polynomial spline method has received little attention in the functional data case. In this paper, we propose the polynomial spline method to estimate a partial functional linear model. Some asymptotic results are established, including asymptotic normality for the parameter vector and the global rate of convergence for the slope function. Finally, we evaluate the performance of our estimation method by some simulation studies.     

Optimal recovery and average n-k widths for some function classes in L2(R)*

In this paper the optimal recovery of a linear differential operator on some classes of smooth functions and the average n-k widths of these classes in L₂ R are considered. Supported by National Natural Science Foundation of China     

The Whitney problem of existence of a linear extension operator

We prove that a linear bounded extension operator exists for the trace of C ^1·ω (R ⁿ )to an arbitrary closed subset of R ⁿ .The similar result is obtained for some other spaces of multivariate smooth functions. We also show that unlike the one-dimensional case treated by Whitney, for some trace spaces of multivariate smooth functions a linear bounded extension operator does not exist. The proofs are based on a relation between the problem under consideration and a similar problem for Lipschitz spaces defined on hyperbolic Riemannian manifolds.     

Estimating a linear functional relationship under linear constraints for errors

Maximum likelihood estimates for parameters in a linear functional relationship are derived when the errors are also linearly related for each observation. This is approximately the case, for example, when both variables are smooth functions of time and their values are recorded as an experimental unit reaches a certain state. This kind of model specification was needed to describe how the timing of growth cessation of trees depends on night length and temperature sum. If the slope in the constraint equation for errors varies, then an iterative estimation procedure is needed. The estimation method is extended for a two-phase linear model.     

Convergence property of the Iri-Imai algorithm for some smooth convex programming problems

In this paper, the Iri-Imai algorithm for solving linear and convex quadratic programming is extended to solve some other smooth convex programming problems. The globally linear convergence rate of this extended algorithm is proved, under the condition that the objective and constraint functions satisfy a certain type of convexity, called the harmonic convexity in this paper. A characterization of this convexity condition is given. The same convexity condition was used by Mehrotra and Sun to prove the convergence of a path-following algorithm.The Iri-Imai algorithm is a natural generalization of the original Newton algorithm to constrained convex programming. Other known convergent interior-point algorithms for smooth convex programming are mainly based on the path-following approach.     

Biorthogonal Rational Functions and the Generalized Eigenvalue Problem

We present some general results concerning so-called biorthogonal polynomials of R_II type introduced by M. Ismail and D. Masson. These polynomials give rise to a pair of rational functions which are biorthogonal with respect to a linear functional. It is shown that these rational functions naturally appear as eigenvectors of the generalized eigenvalue problem for two arbitrary tri-diagonal matrices. We study spectral transformations of these functions leading to a rational modification of the linear functional. An analogue of the Christoffel–Darboux formula is obtained.     

Compositions of convex functions and fully linear models

Derivative-free optimization (DFO) is the mathematical study of the optimization algorithms that do not use derivatives. One branch of DFO focuses on model-based DFO methods, where an approximation of the objective function is used to guide the optimization algorithm. Proving convergence of such methods often applies an assumption that the approximations form fully linear models—an assumption that requires the true objective function to be smooth. However, some recent methods have loosened this assumption and instead worked with functions that are compositions of smooth functions with simple convex functions (the max-function or the \(\ell _1\) norm). In this paper, we examine the error bounds resulting from the composition of a convex lower semi-continuous function with a smooth vector-valued function when it is possible to provide fully linear models for each component of the vector-valued function. We derive error bounds for the resulting function values and subgradient vectors.     

On Factorizations of Smooth Nonnegative Matrix-Values Functions and on Smooth Functions with Values in Polyhedra

We discuss the possibility to represent smooth nonnegative matrix-valued functions as finite linear combinations of fixed matrices with positive real-valued coefficients whose square roots are Lipschitz continuous. This issue is reduced to a similar problem for smooth functions with values in a polyhedron. The work was partially supported by NSF Grant DMS-0653121.     

强相依数据的函数系数部分线性模型的估计

本文讨论在数据是强相依的情况下函数系数部分线性模型的估计．首先,采用局部线性方法,给出该模型函数项函数的估计;然后,使用两阶段方法给出系数函数的估计．并且讨论了函数项函数估计的渐近正态性,以及系数函数估计的弱相合性和渐近正态性.模拟研究显示,这些估计是较为理想的．     

The not-quite non-atomic game: Homogeneous games on two measures

The space of continuous, piecewise smooth homogeneous functions of degree one in two variables can be generated by linear functions and by functions which are the minimum of two linear expressions. This permits a representation of the value for homogeneous games on two measures in terms of the values of additive games and of “shoe-like” games. We treat several examples in detail.     

Modeling linear logic with implicit functions

Just as intuitionistic proofs can be modeled by functions, linear logic proofs, being symmetric in the inputs and outputs, can be modeled by relations (for example, cliques in coherence spaces). However generic relations do not establish any functional dependence between the arguments, and therefore it is questionable whether they can be thought as reasonable generalizations of functions. On the other hand, in some situations (typically in differential calculus) one can speak in some precise sense about an implicit functional dependence defined by a relation. It turns out that it is possible to model linear logic with implicit functions rather than general relations, an adequate language for such a semantics being (elementary) differential calculus. This results in a non-degenerate model enjoying quite strong completeness properties.     

Application of a delta-method for random operators to testing equality of two covariance operators

In this paper some useful mathematical tools for the analysis of functional data are applied to the problem of testing the equality of two covariance operators. The test to be used is derived from a univariate likelihood ratio test in conjunction with Roy’s union-intersection principle. The asymptotic distribution and asymptotic power against rather general local alternatives are calculated. Perturbation theory of operators, in particular, a delta-method for smooth functions of operators, is exploited to obtain these theoretical results that are corroborated by some simulations.     

Local Convergence Properties of Douglas–Rachford and Alternating Direction Method of Multipliers

The Douglas–Rachford and alternating direction method of multipliers are two proximal splitting algorithms designed to minimize the sum of two proper lower semi-continuous convex functions whose proximity operators are easy to compute. The goal of this work is to understand the local linear convergence behaviour of Douglas–Rachford (resp. alternating direction method of multipliers) when the involved functions (resp. their Legendre–Fenchel conjugates) are moreover partly smooth. More precisely, when the two functions (resp. their conjugates) are partly smooth relative to their respective smooth submanifolds, we show that Douglas–Rachford (resp. alternating direction method of multipliers) (i) identifies these manifolds in finite time; (ii) enters a local linear convergence regime. When both functions are locally polyhedral, we show that the optimal convergence radius is given in terms of the cosine of the Friedrichs angle between the tangent spaces of the identified submanifolds. Under polyhedrality of both functions, we also provide conditions sufficient for finite convergence. The obtained results are illustrated by several concrete examples and supported by numerical experiments.     

Relations between directional derivatives for smooth functions in 2D and 3D spaces

In this paper, relations between directional derivatives are considered for smooth functions both in 2D and 3D spaces. These relations are established in the form of linear combinations of directional derivatives with their coefficients having simple form and structural regularity. By them, expressions based on directional derivatives for some typical differential operators are derived. This builds up a solid mathematical foundation for further study on numerical computation by the finite point method based on directional difference.