Über Extremalpunkte des numerischen Wertebereichs eines linearen Operators

For a bounded linear operator T in a Hilbert space denote by (T) the set of the extreme points of its numerical range W(T), and by K(T) the subset of the points of (T) which lie on the boundary of a disc being a spectral set for T in the sense of J. von Neumann. Each corner of W(T) belongs to K(T). This implies: K(T) is dense in (T) if T is compact and the convex hull of its spectrum contains W(T). If Re T0 and o (T) then a necessary and sufficient condition is given that O is a eigenvalue of T. This is applied to show that W(T) (T) is a subset of the point spectrum of T for a class of operators containing the hyponormal operators.     

Die ɛ-Störung linearer und quadratischer Optimierungsaufgaben und ihre Anwendung auf das Hildreth-Verfahren

Zusammenfassung Durch eine -Störung in der Diagonalen der quadratischen Form kann man eine lineare oder quadratisch semidefinite Optimierungsaufgabe zu einer streng definiten quadratischen Aufgabe machen, so daß Lösungsverfahren, die die Formmatrix als nichtsingulär voraussetzen müssen, anwendbar werden. Bekanntlich konvergiert die Lösungx der -gestörten Aufgabe für 0 gegen den Lösungsvektorx ^m von minimalem Betrag der ursprünglichen Aufgabe. Wir zeigen darüber hinaus, daß im linearen Fall immer und im eigentlich quadratischen in gewissen Fällen schon für 0<<* die beiden Lösungenx undx ^m übereinstimmen. Im linearen Fall ist die obere Grenze * durch die Lösung eines linearen Ungleichungssystems gegeben.Im zweiten Abschnitt wenden wir dasHildreth-Verfahren mittels der -Störung auf lineare und quadratisch semidefinite Aufgaben an, diskutieren Konvergenz- und Genauigkeitsfragen und kommen zu dem Schluß, daß man in der Praxis sowohl bei Rechnung von Hand als auch bei maschineller Rechnung zu befriedigenden Ergebnissen kommt.

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Summary Linear and quadratic semidefinite programming problems may be transformed into strongly definite quadratic problems by means of an -perturbation of the quadratic form so that procedures which presuppose the matrix of the form to be nonsingular, may be applied. As is well known, the solutionx of the -perturbated problem converges to the solutionx ^m of minimal length of the original problem as 0. We show that always in the linear case and in the quadratic case under certain circumstances, both solutionsx andx ^m are equal if 0 <<*. In the linear case, the upper limit * is given by the solution of a system of linear inequalities.In the second part of this paper we apply the method ofHildreth to linear and quadratic semidefinite programming problems by the -perturbation. We discuss questions of convergence and exactness, and conclude that in practice calculation by hand as well as by computer leads to satisfying results.

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Der Verfasser ist Herrn Prof. Dr.W. Vogel, Bonn, für einen Hinweis zu Dank verpflichtet.

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Vorgel. v.:H. P. Künzi     

ɛ-duality theorem of nondifferentiable nonconvex multiobjective programming

Necessary Kuhn-Tucker conditions up to precision without constraint qualification for -Pareto optimality of multiobjective programming are derived. This article suggests the establishment of a Wolfe-type -duality theorem for nondifferentiable, nonconvex, multiobjective minimization problems. The -vector Lagrangian and the generalized -saddle point for Pareto optimality are studied.     

Asymptotic bounds for the eigenvalues and eigenvectors of a perturbed linear non-self-conjugate operator

We obtain asymptotic bounds for the perturbed eigenvalues and eigenvectors of a perturbed linear bounded operator A(), in a Hilbert space under the assumption that A() is holomorphic at the point =₀ and the eigenvalue ₀= gl(₀) of the operator A(₀) is isolated and of finite multiplicity. We study certain cases of high degeneracy in the limiting problem, i.e., the case when there are generalized associated vectors.Translated from Matematicheskie Zametki, Vol. 12, No. 4, pp. 403–412, October, 1972.The author wishes to express his sincere gratitude to his scientific director T. Sabirov for valuable observations and advice.     

Convergence in probability for perturbed stochastic integral equations

Summary In this work, one considers two stochastic integral equations indexed by some parameter and one studies the contiguity of their solutions when the parameter converges to some ₀. Two types of behaviour are described; they lead to the notion of regular and singular perturbations. The method which is used also enables a study of the rate of convergence. Applications to time discretization of equations are given.     

Stefan problem with a kinetic and the classical conditions at the free boundary

The Stefan problem is considered with the kinetic condition u⁺=u^–=k(y, )-v at the phase interface, where k(y, ) is the half-sum of the principal curvatures of the free boundary and v is the speed of its shifting in the direction of a normal. The solvability of a modified Stefan problem in spaces of smooth functions and the convergence of its solutions as 0 to a solution of the classical Stefan problem are proved.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 2, pp. 155–166, February, 1992.     

A result from diophantine approximation which has applications in economics

We say that a real number allows poor approximations if we can find 0<=()<1 and a sequence of integers n₁2<... such that for all rationals p/q with qn. we have |–.p/q| > Kn _j ^–l– where K is a constant depending only on .In this note we prove that the set of numbers which allow poor approximations are precisely the very well-approximable numbers.The existence of numbers with poor approximations has been used by Cheng [1] to show the existence of a dense set of economies whose cone converges to the Walras equilibrium as slowly as 0(n^–1/2–) after n replications.     

On the completeness of a system of functions on a segment

The question is considered of the completeness of the systems of functions {Xⁿ[1÷_n]}, where _n(x) are small, in the spaces C and L_p on the segment [0, a].Translated from Matematicheskie Zametki, Vol. 4, No. 5, pp. 557–568, November, 1968.     

Probabilities of moderate deviations

Let X₁, X₂, ... be a sequence of independent, identically distributed random variables, and let. The rate of convergence of probabilities of the form andis studied for any > ₀ and some r and ₀. Moreover, necessary and sufficient conditions are given that the relations be satisfied uniformly with respect to x in the region 0 x clog n, where and c are some positive constants, and. Local limit theorems are also presented.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituts im. V. A. Steklova AN SSSR, Vol. 85, pp. 6–16, 1979.     

Enlargement of Monotone Operators with Applications to Variational Inequalities

Given a point-to-set operator T, we introduce the operator T defined as T(x)= {u: u – v, x – y – for all y Rn, v T(y)}. When T is maximal monotone T inherits most properties of the -subdifferential, e.g. it is bounded on bounded sets, T(x) contains the image through T of a sufficiently small ball around x, etc. We prove these and other relevant properties of T, and apply it to generate an inexact proximal point method with generalized distances for variational inequalities, whose subproblems consist of solving problems of the form 0 H(x), while the subproblems of the exact method are of the form 0 H(x). If k is the coefficient used in the kth iteration and the k's are summable, then the sequence generated by the inexact algorithm is still convergent to a solution of the original problem. If the original operator is well behaved enough, then the solution set of each subproblem contains a ball around the exact solution, and so each subproblem can be finitely solved.