Extension of Subgradient Techniques for Nonsmooth Optimization in Banach Spaces

In this paper we continue the study of the subgradient method for nonsmooth convex constrained minimization problems in a uniformly convex and uniformly smooth Banach space. We consider the case when the stepsizes satisfy _{k=1 k} =, lim _{k k} =0.     

Stability of unconditional convergence almost everywhere

We will investigate the properties of series of functions which are unconditionally convergent almost everywhere on [0, 1]. We will establish the following theorem: If the series _k=1 f _k(x) converges unconditionally almost everywhere, then there exists a sequence {_k} ₁ ,_k , such that if _{k k} , k=1, 2,..., the series _{k=1 k}/k(x) converges unconditionally almost every-where.Translated from Mate matte heskie Zametki, Vol. 14, No. 5, pp. 645–654, November, 1973.The author wishes to thank Professor P. L. Ul'yanov for his help.     

The problem of conformal transformations of a circle into nonoverlapping regions

Let a, a0, a, be a fixed point in the z-plane, (a, 0, ), the class of all systemsf _k()_l ³ of functions z=f ^k(), k=1, 2, 3, of which the first two map conformally and in a s ingle-sheeted manner the circle ¦¦<1, and the third maps in a similar manner the region ¦¦>1, into pair-wise nonintersecting regions B_k, k=1, 2, 3, containing the points a, 0, and , respectively, so thatf ₁(0)=a,f ₂(0)=0 andf ₃()=. The region of values (a, 0, ) of the system M(¦f ₁'(0)¦, ¦f ₂'(0)¦, 1/¦f ₃'()¦) in the class (a, 0, ) is determined.Translated from Matematicheskie Zametki, Vol. 6, No. 4, pp. 417–424, October, 1969.     

Limiting behavior of weighted central paths in linear programming

We study the limiting behavior of the weighted central paths{(x(), s())} > 0 in linear programming at both = 0 and = . We establish the existence of a partition (B ,N ) of the index set { 1, ,n } such thatx _i() ands _j () as fori B , andj N , andx _N (),s _B () converge to weighted analytic centers of certain polytopes. For allk 1, we show that thekth order derivativesx ^(k) () ands ^(k) () converge when 0 and . Consequently, the derivatives of each order are bounded in the interval (0, ). We calculate the limiting derivatives explicitly, and establish the surprising result that all higher order derivatives (k 2) converge to zero when .     

Characterization,structure and analysis on abelian ℒ∞ groups

An abelian topological group is an group if and only if it is a locally -compactk-space and every compact subset in it is contained in a compactly generated locally compact subgroup. Every abelian groupG is topologically isomorphic to G ₀ where ₀ andG ₀ is an abelian group where every compact subset is contained in a compact subgroup. Intrinsic definitions of measures, convolution of measures, measure algebra,L ₁-algebra, Fourier transforms of abelian groups are given and their properties are studied.     

Smoothness of solutions of a nonlinear ode

Smoothness of aC -functionf is measured by (Carleman) sequence {M _k} ₀ ; we sayfC _M [0, 1] if|f ^(k) (t)|CR ^k M _k,k=0, 1, ... withC, R>0. A typical statement proven in this paper isTHEOREM: Let u, b be two C -functions on [0, 1]such that (a) u=u ²+b, (b) |b ^(k) (t)|CR ^k (k!) , >1,k_–.Then |u^(k)(t)|C₁R^k((k–1)!),k_–.The first author acknowledges the hospitality of Mathematical Research Institute of the Ohio State University during his one month visit there in the spring of 1999     

A strong approximation for logarithmic averages of partial sums of random variables

LetS _n be the partial sums of -mixing stationary random variables and letf(x) be a real function. In this note we give sufficient conditions under which the logarithmic average off(S _n / _n ) converges almost surely to _– f(x)d(x). We also obtain strong approximation forH(n)= _k=1 ⁿ k ^–1 f(S _k /k)=logn _– f(x)d(x) which will imply the asymptotic normality ofH(n)/log^1/2 n. But for partial sums of i.i.d. random variables our results will be proved under weaker moment condition than assumed for -mixing random variables.     

On the projected subgradient method for nonsmooth convex optimization in a Hilbert space

We consider the method for constrained convex optimization in a Hilbert space, consisting of a step in the direction opposite to an _k -subgradient of the objective at a current iterate, followed by an orthogonal projection onto the feasible set. The normalized stepsizes _k are exogenously given, satisfying _{k=0 k} = , _{k=0 k} ² < , and _k is chosen so that _{k k} for some > 0. We prove that the sequence generated in this way is weakly convergent to a minimizer if the problem has solutions, and is unbounded otherwise. Among the features of our convergence analysis, we mention that it covers the nonsmooth case, in the sense that we make no assumption of differentiability off, and much less of Lipschitz continuity of its gradient. Also, we prove weak convergence of the whole sequence, rather than just boundedness of the sequence and optimality of its weak accumulation points, thus improving over all previously known convergence results. We present also convergence rate results. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.Research of this author was partially supported by CNPq grant nos. 301280/86 and 300734/95-6.     

The interpolation ofℓ r sequences by hp functions

The sequence spaceH ^P (z)={{f (z_h)}:f H ^p} is defined for a fixed sequence Z={z_k} of different points of the open unit disk and the Hardy class HP of analytic functions in the disk. For an arbitrary p[1, ) is constructed a point sequence Z= {z_k} such that ¹h ^p(z), but ^r h^p (Z) for r > 1. It follows from a well-known result of L. Carleson that the inclusions ^r h (Z) for all r[1,] are equivalent.Translated from Matematicheskie Zametki, Vol. 21, No. 4, pp. 503–508, April, 1977.     

On a property of functional series

The question of the convergence of functional series everywhere in the segment [0, 1] is considered. Let F=f be the set of such functions in [0, 1] for each of which there is a transposition of the series _k=1 f_k(x), which converges to it everywhere in [0, 1]. An example of a series is constructed such that the set F consists just of an identical zero, but _k=1 |f _k (x ₀)ü=,(x₀ [0,1]) for any point of the segment [0, 1].Translated from Matematicheskie Zametki, Vol. 11, No. 5, pp. 481–490, May, 1972.     

Random Walks Crossing High Level Curved Boundaries

Let {S _n} be a random walk, generated by i.i.d. increments X _i which drifts weakly to in the sense that as n . Suppose k0, k1, and E|X ₁|^1\k = if k>1. Then we show that the probability that S. crosses the curve nan ^K before it crosses the curve n –an ^k tends to 1 as a . This intuitively plausible result is not true for k = 1, however, and for 1/2 <k<1, the converse results are not true in general, either. More general boundaries g(n) than g(n) = n ^k are also considered, and we also prove similar results for first passages out of regions like { (n, y): n1, |y| (a + n) ^k } as a .     

Matrix orthogonal Laurent polynomials and two-point Padé approximants

This paper is concerned with double sequencesC={C _n} _n ^=–/ of Hermitian matrices with complex entriesC _n M _s×s ) and formal Laurent seriesL ₀(z)=– _k=1 C _–k z ^k andL (z)= _k=0 C _k z ^–k . Making use of a Favard-type theorem for certain sequences of matrix Laurent polynomials which was obtained previously in [1] we can establish the relation between the matrix counterpart of the so-calledT-fractions and matrix orthogonal Laurent polynomials. The connection with two-point Padé approximants to the pair (L ₀,L ) is also exhibited proving that such approximants are Hermitian too. Finally, error formulas are also given.     

A Criterion for Convergence of Weak Greedy Algorithms

We find a class V of sequences such that the condition V is necessary and sufficient for convergence of weak greedy algorithm with weakness sequence for each f and all Hilbert spaces H and dictionaries D. We denote by V the class of sequences x={x _{k k=1} , x _k 0, k=1,2,..., with the following property: there exists a sequence 0=q ₀<q ₁< such that _s=1 2 ^s /q _s )< and _s=1 2^–s _k=1 ^q s ^x _k ²<, where q _s :=q _s –q _s–1.     

On the convergence of orthogonal series to +∞

For any sequence of numbers _n0, _n=1 a _n ² =, a uniformly bounded orthonormal system of continuous functions _n(x) which is complete in L₂ (0, 1), and a sequence of numbers b_n(0< b_na_n) are constructed such that _n=1 ^Emphasis> b_nn(x)= everywhere on (0, 1).Translated from Matematicheskie Zametki, Vol. 11, No. 5, pp. 499–508, May, 1972.     

Remarks on ergodicity of stationary irreducible transient Markov chains

Summary Consider a stationary process {X _n(), – < n < . If the measure of the process is finite (the measure of the whole sample space finite), it is well known that ergodicity of the process {X _n(), - < n < and of each of the subprocesses {X _n(), 0 n < , {X _n(), – < n 0 are equivalent (see [3]). We shall show that this is generally not true for stationary processes with a sigma-finite measure, specifically for stationary irreducible transient Markov chains. An example of a stationary irreducible transient Markov chain {X _n(), - < n <} with {itX_n(), 0 n < < ergodic but {X _n(), < n 0 nonergodic is given. That this can be the case has already been implicitly indicated in the literature [4]. Another example of a stationary irreducible transient Markov chain with both {X _n(), 0 n < and {itX _n(),-< < n 0} ergodic but {X _n(), - < n < nonergodic is presented. In fact, it is shown that all stationary irreducible transient Markov chains {X _n(), - < n < < are nonergodic.This research was supported in part by the Office of Naval Research.John Simon Guggenheim Memorial Fellow.     

On the asymptotic and invariantσ-algebras of random walks on locally compact groups

Summary Consider a random walk of law on a locally compact second countable groupG. Let the starting measure be equivalent to the Haar measure and denote byQ the corresponding Markov measure on the space of pathsG . We study the relation between the spacesL (G , ^a ,Q) andL (G , ⁱ ,Q) where ^a and ⁱ stand for the asymptotic and invariant -algebras, respectively. We obtain a factorizationL (G , ^a ,Q) L (G , ⁱ ,Q)L (C) whereC is a cyclic group whose order (finite or infinite) coincides with the period of the Markov shift and is determined by the asymptotic behaviour of the convolution powers ⁿ.     

Fourier series criteria for operator decomposability

Let U be an invertible operator on a Banach space Y. U is said to betrigonometrically well-bounded provided the sequence {Uⁿ} _n =– is the Fourier-Stieltjes transform of a suitable projection-valued function E(·): [0, 2](Y). This class of operators is known to apply naturally to a variety of classical phenomena which exclude the presence of spectral measures. In the case Y reflexive we use the Cesáro means _n(U, t) of the trigonometric series _k0 k^–e^iktU^k, whichformally transfers the discrete Hilbert transform to Y, in order to give three separate necessary and sufficient conditions for U to be trigonometrically well-bounded. One of these conditions is sup {_n(U,t): n 1, t [0,2]} <      

Rational approximants to symmetric formal Laurent series

Rational approximants, in the Padé sense, to a given formal Laurent series,F(z)= _– c _k z ^k , have been considered by several authors (see [3] for a survey about the different kinds of approximants which can be defined). In this paper, we shall be concerned with symmetric series, that is, when the complex coefficients {c _k } _– ⁺ satisfyc _–k=c _k,k=0, 1,....Making use of Brezinski's approach [1], for Padé-type approximation to a formal power series, rational approximants toF(z) with prescribed poles are obtained, and their algebraic properties considered. These results will allow us to give an alternative approach for the Padé-Chebyshev approximants.     

Tauberian Theorems for Weighted Means of Double Sequences

Let p := {p _j } _j=0 and q := {q _k } _k–0 be complex (or real) sequences with the property that P _m := _j–0 ^m p _j 0 for all m 0, Q _n := _k–0 ⁿ q _k 0 for all n 0, and both of {P _m } _m=0 and {Q _n } _n=0 are varying away from 1. Assume that {s _mn } is a double sequence in C(or one of R, a Banach space, and an ordered linear space), which is (N¯,p,q; ,) summable to a finite limit, where (,) =(1,1), (1,0), or (0,1). We give necessary and sufficient conditions under which {s _mn } converges in Pringsheim's sense. These conditions are weaker than the two-dimensional analogues of Landau's condition and Schmidt's slow decrease condition. Our results generalize and extend [1 4, 12 15]. We also solve the problems posed in [3, 13, 14].     

Vergleich zweierS ∞-Methoden zur Lösung der Neutronentransportgleichung

Zusammenfassung Es wird eine semi-analytische Methode (S _/0) zur Lösung der monoenergetischen Transportgleichung entwickelt. Diese Methode wird mit der bekanntenS ¹ -Methode verglichen. Bei Anwendung letzterer auf Reaktorabschirmungen können negative Flüsse vorkommen. Dieser unphysikalische Effekt wird durch die neue Methode vollständig vermieden. Numerische Ergebnisse zeigen die Begrenzungen der Anwendbarkeit vonS ¹ und bestätigen die numerische Stabilität vonS ⁰ .

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A semianalytic method (S ⁰ ) is presented for solving the monoenergetic multi-region transport equation. This method is compared with results fromS ¹ -theory given in the literature. Application ofS ¹ -theory to reactor shields may lead to negative neutron fluxes and to flux oscillations. These unphysical effects are completely avoided by the new method. Numerical results demonstrate the limitation ofS ¹ and confirm the numerical stability ofS ⁰ .