The Rate of Convergence for Subexponential Distributions and Densities

A distribution function F on the nonnegative real line is called subexponential if lim_x(1-F ^*n (x)/(1 - F(x)) = n for all n 2, where F ^*n denotes the nfold Stieltjes convolution of F with itself. In this paper, we consider the rate of convergence in the above definition and in its density analogue. Among others we discuss the asymptotic behavior of the remainder term R _n (x) defined by R _n (x) = 1 - F*n(x) - n(1 - F(x)) and of its density analogue r_n (x) = -(R_n (x))'. Our results complement and complete those obtained by several authors. In an earlier paper, we obtained results of the form _n(x) = O(1)f(x)R(x), where f is the density of F and R(x) = ₀ ^x (1-F(y))dy. In this paper, among others we obtain asymptotic expressions of the form R _n(x)= ₂ ⁿ R₂(x) + O(1)(-f'(x))R²(x) where f' is the derivative of f.     

On the solution of nonlinear equations in interval arithmetic

We consider the problem of finding a simple zero of a continuously differentiable functionf:R ⁿ R ⁿ . There is given an intervalvectorX ₀ ^I containing one zero off, and we will construct a contracting sequence of intervalvectors enclosing this zero. This can be done by Newton's method, which gives quadratic convergence, but requires inversion of an intervalmatrix at each step of the iteration. Alefeld and Herzberger, [1], give a modification of Newton's method, without the necessity of inversion, the convergence being superlinear. We give a slight modification of the latter method, with the property that the sequence of interval widths is dominated by a quadratically convergent sequence.     

Contraction mappings in interval analysis

Under certain conditions, the contraction mapping fixed point theorem guarantees the convergence of the iterationx _i+1=f(x _i ) toward a fixed point of the functionf:R ⁿ →R ⁿ. When an interval extensionF off is used in a similar iteration scheme to obtain a sequence of interval vectors these conditions need not provide convergence to a degenerate interval vector representing the fixed point, even if the width of the initial interval vector is chosen arbitrarily small. We give a sufficient condition on the extensionF in order that the convergence is guaranteed. The centered form of Moore satisfies this condition.     

A remark on generalized iterates

Letf be an invertible function on the real lineR, and letZ denote the set of integers. For eachx Z, letf ^|n| denote then'th iterate off. Clearlyf ^|m|(f ^|n|(x))=f ^|m+n|(x) for allm,nZ and allxR. LetG be any group of orderc, the cardinality of the continuum, which contains (an isomorphic copy of)Z as a normal subgroup. If for eachxR, the iteration trajectory (orbit) ofx is non-periodic, then there exists a set of invertible functionsF={F ^||:G} on the real line with the properties (i)F ^||(F ^||(x))=F ^|+| (x) for allxR and (ii)F ^|n|(x)=f ^|n|(x) for allnZ andxR. That is,f can be embedded in a set ofG-generalized iterates. In particular,f can be embedded in a set of complex generalized iterates.Dedicated to Professor Janos Aczél on his 60th birthday     

A new continuation method for complementarity problems with uniformP-functions

The complementarity problem with a nonlinear continuous mappingf from the nonnegative orthantR ₊ ⁿ ofR ⁿ intoR ⁿ can be written as the system of equationsF(x, y) = 0 and(x, y) R ₊ ²ⁿ , whereF denotes the mapping from the nonnegative orthantR ₊ ²ⁿ ofR ²ⁿ intoR ₊ ⁿ × Rⁿ defined byF(x, y) = (x ₁y₁,,x_ny_n, f₁(x) – y₁,, f_n(x) – y_n) for every(x, y) R ₊ ²ⁿ . Under the assumption thatf is a uniformP-function, this paper establishes that the mappingF is a homeomorphism ofR ₊ ²ⁿ ontoR ₊ ⁿ × Rⁿ. This result provides a theoretical basis for a new continuation method of tracing the solution curve of the one parameter family of systems of equationsF(x, y) = tF(x ⁰, y⁰) and(x, y) R ₊ ²ⁿ from an arbitrary initial point(x ⁰, y⁰) R ₊ ²ⁿ witht = 1 until the parametert attains 0. This approach is an extension of the one used in the polynomially bounded algorithm recently given by Kojima, Mizuno and Yoshise for solving linear complementarity problems with positive semi-definite matrices.     

On the error of quadrature formulae for Cauchy principal value integrals based on piecewise interpolation

We consider the computation of the Cauchy principal value integral by quadrature formulae Q _n ^F [f] of compound type, which are obtained by replacing f by a piecewise defined function F_n[f]. The behaviour of the constants k_{i, n} in the estimates [R _n ^F [f]] |⩽K _i,n ‖f ⁽ⁱ⁾‖_∞ (where R _n ^F [f] is the quadrature error) is determined for fixed i and n→∞, which means that not only the order, but also the coefficient of the main term of k_{i, n} is determined. The behaviour of these error constants k_{i, n} is compared with the corresponding ones obtained for the method of subtraction of the singularity. As it turns out, these error constants have, in general, the same asymptotic behaviour.     

Preparation of papers

LetF be a family of real-valued maps onR ⁿ, and letY be a subset ofR ⁿ. Denote byS(Y|F) the set of ally* Y such that, for somef F,f(y)f(y*) for ally inY. Let us say thatF is a scalarization family if, for any subsetY,S(Y|F) is equal to the set of properly efficient points inY. General conditions forF to be a scalarization family were given in Ref. 1. However, scalarization families must contain nondifferentiable functions. In this note, it is shown that, if the condition of Ref. 1 which forces nondifferentiability is dropped, thenS(Y|F) is dense in the set of properly efficient points.     

Minimizing and Stationary Sequences of Convex Constrained Minimization Problems

In the asymptotic analysis of the minimization problem for a nonsmooth convex function on a closed convex set X in ⁿ, one can consider the corresponding problem of minimizing a smooth convex function F on ⁿ, where F denotes the Moreau–Yosida regularization of f. We study the interrelationship between the minimizing/stationary sequence for f and that for F. An algorithm is given to generate iteratively a possibly unbounded sequence, which is shown to be a minimizing sequence of f under certain regularity and uniform continuity assumptions.     

Grouped coordinate minimization using Newton's method for inexact minimization in one vector coordinate

Letf(x,y) be a function of the vector variablesx R ⁿ andy R ^m. The grouped (variable) coordinate minimization (GCM) method for minimizingf consists of alternating exact minimizations in either of the two vector variables, while holding the other fixed at the most recent value. This scheme is known to be locally,q-linearly convergent, and is most useful in certain types of statistical and pattern recognition problems where the necessary coordinate minimizers are available explicitly. In some important cases, the exact minimizer in one of the vector variables is not explicitly available, so that an iterative technique such as Newton's method must be employed. The main result proved here shows that a single iteration of Newton's method solves the coordinate minimization problem sufficiently well to preserve the overall rate of convergence of the GCM sequence.The authors are indebted to Professor R. A. Tapia for his help in improving this paper.     

On the central limit theorem in R k

Summary Let F ^*n denote the n th convolution of a distribution function F on R ^k and suppose that F has zero moments of the first order and finite second order moment matrix. It is well-known that F ^*n (n·) converges to a Gaussian d.f. as n + t8. These d.f.s determine measures F ^*n (nA) and (A) for Borelsets A, We present a method that admits the estimation of the remainder-term F ^*n (n A)- (A) when A belongs to a certain class of Borelsets. This class contains all convex sets. If F has finite absolute third order moments then the remainder-term is of the order n ^–1/2. Also the remainder term's dependence on the dimension k is given. These results strengthen and generalize earlier results in the same direction.This paper was first communicated at the Scandinavian mathematical congress in Oslo, August 1968.     

On the Problem of Describing Sequences of Best Trigonometric Rational Approximations

For a strictly decreasing sequence a_{n n=0} of nonnegative real numbers converging to zero, we construct a continuous 2-periodic function f such that R^T _n(f) = a_n, n=0,1,2,..., where R^T _n(f) are best approximations of the function f in uniform norm by trigonometric rational functions of degree at most n.     

Kinematics of Mechanisms to the Second Order – Application to the Closed Mechanisms

The degree of freedom of a closed mechanism is the dimension of a subset M of R ⁿ , M being the inverse image of the unity by the closure function f : (q ₁, ..., q _n ) f(q ₁, ..., q _n ), where q ₁, ..., q _n are the articular coordinates. We first study the regular points for the mapping f from R ⁿ into the Lie group of displacements and, second, study the singularities of the mapping f. The classical theory of mechanisms considers, often implicitly, that f is a subimmersion. Here, the calculations are made in a larger case, up to second order, and the results are then slightly different. The case of such classical mechanisms as Bennett, Bricard, and Goldberg mechanisms, justify the considerations of this more general framework and the example of a Bricard mechanism is chosen as an application of the method.     

Fast Construction of Optimal Circulant Preconditioners for Matrices from the Fast Dense Matrix Method

In this paper, we consider solving non-convolution type integral equations by the preconditioned conjugate gradient method. The fast dense matrix method is a fast multiplication scheme that provides a dense discretization matrix A approximating a given integral equation. The dense matrix A can be constructed in O(n) operations and requires only O(n) storage where n is the size of the matrix. Moreover, the matrix-vector multiplication A xcan be done in O(n log n) operations. Thus if the conjugate gradient method is used to solve the discretized system, the cost per iteration is O(n log n) operations. However, for some integral equations, such as the Fredholm integral equations of the first kind, the system will be ill-conditioned and therefore the convergence rate of the method will be slow. In these cases, preconditioning is required to speed up the convergence rate of the method. A good choice of preconditioner is the optimal circulant preconditioner which is the minimizer of C – A _F in Frobenius norm over all circulant matrices C. It can be obtained by taking arithmetic averages of all the entries of A and therefore the cost of constructing the preconditioner is of O(n ²) operations for general dense matrices. In this paper, we develop an O(n log n) method of constructing the preconditioner for dense matrices A obtained from the fast dense matrix method. Application of these ideas to boundary integral equations from potential theory will be given. These equations are ill-conditioned whereas their optimal circulant preconditioned equations will be well-conditioned. The accuracy of the approximation A, the fast construction of the preconditioner and the fast convergence of the preconditioned systems will be illustrated by numerical examples.     

On some conditions on mappings of spaces onto cylindrical domains

Criteria are given for a local homeomorphismf: Rⁿ Rⁿ to be a homeomorphism of Rⁿ onto a strip. In the planar case, this gives a condition forf to be a transformation of R² onto a band.Translated from Matematicheskie Zametki, Vol. 11, No. 4, pp. 459–462, April, 1972.     

A quadratically convergent method for minimizing a sum of euclidean norms

We consider the problem of minimizing a sum of Euclidean norms. \(F(x) = \sum\nolimits_{i = 1}^m {||r_i } (x)||\) here the residuals {r _i(x)} are affine functions fromR ⁿ toR ¹ (n≥1≥2,m>-2). This arises in a number of applications, including single-and multi-facility location problems. The functionF is, in general, not differentiable atx if at least oner _i (x) is zero. Computational methods described in the literature converge quite slowly if the solution is at such a point. We present a new method which, at each iteration, computes a direction of search by solving the Newton system of equations, projected, if necessary, into a linear manifold along whichF is locally differentiable. A special line search is used to obtain the next iterate. The algorithm is closely related to a method recently described by Calamai and Conn. The new method has quadratic convergence to a solutionx under given conditions. The reason for this property depends on the nature of the solution. If none of the residuals is zero at^* x, thenF is differentiable at^* x and the quadratic convergence follows from standard properties of Newton's method. If one of the residuals, sayr _i ^* x), is zero, then, as the iteration proceeds, the Hessian ofF becomes extremely ill-conditioned. It is proved that this illconditioning, instead of creating difficulties, actually causes quadratic convergence to the manifold (x?r _i (x)=0}. If this is a single point, the solution is thus identified. Otherwise it is necessary to continue the iteration restricted to this manifold, where the usual quadratic convergence for Newton's method applies. If several residuals are zero at^* x, several stages of quadratic convergence take place as the correct index set is constructed. Thus the ill-conditioning property accelerates the identification of the residuals which are zero at the solution. Numerical experiments are presented, illustrating these results.     

A note on a third‐order multi‐point boundary value problem at resonance

Based on the coincidence degree theory of Mawhin, we prove some existence results for the following third‐order multi‐point boundary value problem at resonance where f: [0, 1] × R³ → R is a continuous function, 0 < ξ₁ < ??? < ξ_m < 1, α_i ∈ R, i = 1, …, m, m ≥ 1 and 0 < η₁ < η₂ < ??? < η_n < 1, β_j ∈ R, j = 1, 2, …, n, n ≥ 2. In this paper, the dimension of the linear space Ker L (linear operator L is defined by Lx = x′^′′) is equal to 2. Since all the existence results for third‐order differential equations obtained in previous papers are for the case dim Ker L = 1, our work is new.     

Kantorovič operators of second order

The Kantorovi? operators of second order are introduced byQ _n f= =(B _n+2 F)″ whereF is the double indefinite integraloff andB _n+2 the (n+2)-th Bernstein operator. The operatorsQ _n will reveal a close affinity to the so-called modified Bernstein operatorsC _n introduced bySchnabl [10] on a quite different way. The article contains investigations concerning the asymptotic behavior ofQ _n ^kn f (asn → ∞), where (k _n) is a sequence of natural numbers.     

An implicit function theorem

Suppose thatF:DR ⁿ×R^mRⁿ, withF(x ⁰,y ⁰)=0. The classical implicit function theorem requires thatF is differentiable with respect tox and moreover that ₁ F(x ⁰,y ⁰) is nonsingular. We strengthen this theorem by removing the nonsingularity and differentiability requirements and by replacing them with a one-to-one condition onF as a function ofx.     

Bi-Lipschitz concordance implies bi-Lipschitz isotopy

We modify the proof of an earlier result of ours to deforming topological, bi-Lipschitz, and quasiconformal embeddings of an open subsetU ofR ⁿ which now are of small uniform distance from the inclusion map. As an application we show that two bi-Lipschitz homeomorphismsf ₀,f ₁:R ⁿRⁿ are bi-Lipschitz isotopic if and only ifd(f ₀,f ₁)<.Research supported in part by a grant from the Institut Mittag-Leffler.     

Cutting Plane Algorithms for Nonlinear Semi-Definite Programming Problems with Applications

We will propose an outer-approximation (cutting plane) method for minimizing a function f X subject to semi-definite constraints on the variables XR ⁿ. A number of efficient algorithms have been proposed when the objective function is linear. However, there are very few practical algorithms when the objective function is nonlinear. An algorithm to be proposed here is a kind of outer-approximation(cutting plane) method, which has been successfully applied to several low rank global optimization problems including generalized convex multiplicative programming problems and generalized linear fractional programming problems, etc. We will show that this algorithm works well when f is convex and n is relatively small. Also, we will provide the proof of its convergence under various technical assumptions.