A Finite Algorithm for Solving Infinite Dimensional Optimization Problems

We consider the general optimization problem (P) of selecting a continuous function x over a -compact Hausdorff space T to a metric space A, from a feasible region X of such functions, so as to minimize a functional c on X. We require that X consist of a closed equicontinuous family of functions lying in the product (over T) of compact subsets Y _t of A. (An important special case is the optimal control problem of finding a continuous time control function x that minimizes its associated discounted cost c(x) over the infinite horizon.) Relative to the uniform-on-compacta topology on the function space C(T,A) of continuous functions from T to A, the feasible region X is compact. Thus optimal solutions x ^* to (P) exist under the assumption that c is continuous. We wish to approximate such an x ^* by optimal solutions to a net {P _i }, iI, of approximating problems of the form min_{xX i} c i(x) for each iI, where (1) the net of sets {X _i } _I converges to X in the sense of Kuratowski and (2) the net {c _i } _I of functions converges to c uniformly on X. We show that for large i, any optimal solution x ^* _i to the approximating problem (P _i ) arbitrarily well approximates some optimal solution x ^* to (P). It follows that if (P) is well-posed, i.e., limsupX _i ^* is a singleton {x ^*}, then any net {x _i ^*} _I of (P _i )-optimal solutions converges in C(T,A) to x ^*. For this case, we construct a finite algorithm with the following property: given any prespecified error and any compact subset Q of T, our algorithm computes an i in I and an associated x _i ^* in X _i ^* which is within of x ^* on Q. We illustrate the theory and algorithm with a problem in continuous time production control over an infinite horizon.     

On quadratic and sesquilinear functionals

LetX be a leftA-module, whereA is either a complex Banach *-algebra with an identity element or the field of quaternions. The main result of this note is that forQ, anA-quadratic functional defined onX, there exists a sesquilinear functionalB such thatB(x,x)=Q(x) holds for allxX.     

A topological position of the set of continuous maps in the set of upper semicontinuous maps

Let (X, ρ) be a metric space and ↓USCC(X) and ↓CC(X) be the families of the regions below all upper semi-continuous compact-supported maps and below all continuous compact-supported maps from X to I = [0, 1], respectively. With the Hausdorff-metric, they are topological spaces. In this paper, we prove that, if X is an infinite compact metric space with a dense set of isolated points, then (↓USCC(X), ↓CC(X)) ≈ (Q, c0 ∪ (Q \ Σ)), i.e., there is a homeomorphism h :↓USCC(X) → Q such that h(↓CC(X)) = c0 ∪ (Q \ Σ...     

Varieties of strange plane curves

Abstract

Let A be a commutative ring with identity, let X, Y be indeterminates and let F(X,Y), G(X, Y) ∈ A[X, Y] be homogeneous. Then the pair F(X, Y), G(X, Y) is said to be radical preserving with respect to A if Rad((F(x, y), G(x, y))R) = Rad((x,y)R) for each A-algebra R and each pair of elements x, y in R. It is shown that infinite sequences of pairwise radical preserving polynomials can be obtained by homogenizing cyclotomic polynomials, and that under suitable conditions on a ?-graded ring A these can be used to produce an infinite set of homogeneous prime ideals between two given homogeneous prime ideals P ? Q of A such that ht(Q/P) = 2.     

Continuous extension operators and convexity

Given a nonempty closed subset A of a Hilbert space X, we denote by L(A) the space of all bounded Lipschitz mappings from A into X, equipped with the supremum norm. We show that there is a continuous mapping F_c:L(A)?L(X) such that for each g∈L(A), F_c(g)|_A=g, , and . We also prove that the corresponding set-valued extension operator is lower semicontinuous.     

An eigenvalue problem for the numerical range of a bounded linear operator

LetX be a complex Lebesgue space with a unique duality mapJ fromX toX ^*, the conjugate space ofX. LetA be a bounded linear operator onX. In this paper we obtain a non-linear eigenvalue problem for (A)=sup{Re: W(A} whereW(A)={J(x)A(x)) : x=1}, under the assumption that (A) and the convex hull ofW(A) for some linear operatorsA onl _p , 2<p<.     

Viscosity methods for zeroes of accretive operators

This paper deals with the general iteration method , for calculating a particular zero of A, an m-accretive operator in a Banach space X, T_n being a sequence of nonexpansive self-mappings in X. Under suitable conditions on the parameters and X, we state strong and weak convergence results of (x_n). We also show how to compute a common zero of two m-accretive operators in X.     

A Reuniformization for Contractive Mappings in Uniform Spaces

Let X be a complete uniform HAUSDORFF space with a uniformity generated by a saturated family of pseudometrics ?? = {?_α(x, y): α ? A} and let T: X → X be a continuous mapping. The paper contains necessary and sufficient conditions for the existence of a new family of pseudometrics ??*={?*(x, y): α*?A*} generated the same topology such that T is contractive with respect to ??*.     

Hamilton-Jacobi semi-groups in infinite dimensional spaces

Let (X,ρ) be a Polish space endowed with a probability measure μ. Assume that we can do Malliavin Calculus on (X,μ). Let be a pseudo-distance. Consider QtF(x)=infy∈X{F(y)+d²(x,y)/2t}. We shall prove that QtF satisfies the Hamilton-Jacobi inequality under suitable conditions. This result will be applied to establish transportation cost inequalities on path groups and loop groups in the spirit of Bobkov, Gentil and Ledoux.     

Solution of augmented linear systems using orthogonal factorizations

We consider solvingx+Ay=b andA ^T x=c for givenb, c andm ×n A of rankn. This is called the augmented system formulation (ASF) of two standard optimization problems, which include as special cases the minimum 2-norm of a linear underdetermined system (b=0) and the linear least squares problem (c=0), as well as more general problems. We examine the numerical stability of methods (for the ASF) based on the QR factorization ofA, whether by Householder transformations, Givens rotations, or the modified Gram-Schmidt (MGS) algorithm, and consider methods which useQ andR, or onlyR. We discuss the meaning of stability of algorithms for the ASF in terms of stability of algorithms for the underlying optimization problems.We prove the backward stability of several methods for the ASF which useQ andR, inclusing a new one based on MGS, and also show under what circumstances they may be regarded as strongly stable. We show why previous methods usingQ from MGS were not backward stable, but illustrate that some of these methods may be acceptable-error stable. We point out that the numerical accuracy of methods that do not useQ does not depend to any significant extent on which of of the above three QR factorizations is used. We then show that the standard methods which do not useQ are not backward stable or even acceptable-error stable for the general ASF problem, and discuss how iterative refinement can be used to counteract these deficiencies.Dedicated to Carl-Eric Fröberg on the occasion of his 75th birthdayThis research was partially supported by NSERC of Canada Grant No. A9236.     

Random symmetric polynomials

Let {Q⁽ⁿ⁾(x₁,...,x_n)} be a sequence of symmetric polynomials having a fixed degree equal to k. Let {X_n1,...,X_nn}, n k, be some sequence of series of random variables (r.v.). We form the sequence of r.v. Y_n=Q⁽ⁿ⁾(X_n1, ... X_nn), n k One obtains limit theorems for the sequence Y_n, under very general assumptions.Translated from Veroyatnostnye Raspredeleniya i Matematicheskaya Statistika, pp. 170–188, 1986.     

Asymptotic stability of solitons of the gKdV equations with general nonlinearity

We consider the generalized Korteweg-de Vries equation (gKdV)

Numerical decomposition of a convex function

Given then×p orthogonal matrixA and the convex functionf:R ⁿR, we find two orthogonal matricesP andQ such thatf is almost constant on the convex hull of ± the columns ofP, f is sufficiently nonconstant on the column space ofQ, and the column spaces ofP andQ provide an orthogonal direct sum decomposition of the column space ofA. This provides a numerically stable algorithm for calculating the cone of directions of constancy, at a pointx, of a convex function. Applications to convex programming are discussed.This work was supported by the National Science and Engineering Research Council of Canada (Grant No. A3388 and Summer Grant).     

Fubini theorem for anticipating stochastic integrals in Hilbert space

Let (X,l,) be a measure space, letW be a cylindrical Hilbert-Wiener process, and let be an anticipating integrable process-valued function onX. We prove, under natural assumptions on, that there exists a measurable version Y_x,x X, of the anticipating integral of(x) such that the integral _x Y_x(dx) is a version of the anticipating integral of _X (x)(dx). We apply this anticipating Fubini theorem to study solutions of a class of stochastic evolution equations in Hilbert space.     

A separation theorem for semicontinuous functions on preordered topological spaces

Suppose that X is a topological space with preorder , and that –g, f are bounded upper semicontinuous functions on X such that g(x) f(y) whenever x y. We consider the question whether there exists a bounded increasing continuous function h on X such that g h f, and obtain an existence theorem that gives necessary and sufficient conditions. This result leads to an extension theorem giving conditions that allow a bounded increasing continuous function defined on an open subset of X to be extended to a function of the same type on X. The application of these results to extremally disconnected locally compact spaces is studied.Received: 26 May 2004     

Parametrization of the orbits of cubic surfaces

The aim of the paper is the study of the orbits of the action of PGL₄ on the space ³ of the cubic surfaces of ³, i.e., the classification of cubic surfaces up to projective motions. A varietyQ¹⁹ is explicitely constructed as the union of 22 disjoint irreducible components which are either points or open subsets of linear spaces. More precisely, each orbit of the above action intersects one componentX ofQ in a finite number of points and the action of PGL₄ restricted on each componentX is equivalent to the action of a finite groupG _X onX which can be explicitely computed. Finally the cubic surfaces of each component ofQ are studied in details by determining their stabilizers, their rational representations and whether they can be expressed as the determinant of a 3×3 matrix of linear forms.The results are obtained with computational techniques and with the aid of some computer algebra systems like CoCoA, Macaulay and Maple.Partially supported by MURSTPartially supported by MURST and CNR     

Characterisations of Chebyshev sets in c0

A subset MX of a normed linear space X is a Chebyshev set if, for every xX, the set of all nearest points from M to x is a singleton. We obtain a geometrical characterisation of approximatively compact Chebyshev sets in c₀. Also, given an approximatively compact Chebyshev set M in c₀ and a coordinate affine subspace Hc₀ of finite codimension, if M∩H≠, then M∩H is a Chebyshev set in H, where the norm on H is induced from c₀.     

Local Lipschitz-constant Functions and Maximal Subdifferentials

It is shown that if k(x) is an upper semicontinuous and quasi lower semicontinuous function on a Banach space X, then k(x)B _X* is the Clarke subdifferential of some locally Lipschitz function on X. Related results for approximate subdifferentials are also given. Moreover, on smooth Banach spaces, for every locally Lipschitz function with minimal Clarke subdifferential, one can obtain a maximal Clarke subdifferential map via its local Lipschitz-constant function. Finally, some results concerning the characterization and calculus of local Lipschitz-constant functions are developed.     

Extending an operator from a Hilbert space to a larger Hilbert space,so as to reduce its spectrum

In our earlier paper [1] we showed that given any elementx of a commutative unital Banach algebraA, there is an extensionA′ ofA such that the spectrum ofx inA′ is precisely the essential spectrum ofx inA. In [2], we showed further that ifT is a continuous linear operator on a Banach spaceX, then there is an extensionY ofX such thatT extends continuously to an operatorT ⁻ onY, and the spectrum ofT ⁻ is precisely the approximate point spectrum ofT. In this paper we take the second of these results, and show further that ifX is a Hilbert space then we can ensure thatY is also a Hilbert space; so any operatorT on a Hilbert spaceX is the restriction to one copy ofX of an operatorT ⁻ onX ⊕X, whose spectrum is precisely the approximate point spectrum ofT. This result is “best possible” in the sense that if isany extension to a larger Banach space of an operatorT, it is a standard exercise that the approximate point spectrum ofT is contained in the spectrum of .     

Construction of Separately Continuous Functions with Given Restriction

We solve the problem of the construction of separately continuous functions on a product of two topological spaces with given restriction. It is shown, in particular, that, for an arbitrary topological space X and a function g: X R of the first Baire class, there exists a separately continuous function f: X × X R such that f(x, x) = g(x) for every x X.