Vasant A Ubhaya

L p approximation from nonconvex subsets of special classes of functions

Abstract An existence theorem for a best approximation to a function in L p , 1 ⩽ p ⩽ ∞, by functions from a nonconvex set is established under certain general conditions on the set. The unifying development and results are applicable to approximation from subsets of various classes of functions including quasi-convex, convex, super-additive, star-shaped, monotone, and n -convex functions.

Uniform approximation by quasi-convex and convex functions

Abstract Given a bounded function f defined on a convex subset of R n , the two problems considered are to find a quasi-convex (convex) function which is a best approximation to f under the uniform norm. It is shown that if f is the greatest quasi-convex (convex) minorant of f , then f′ = f + c , for some c ≧ 0 , is the maximal best quasi-convex (convex) approximation to f . Furthermore, the nonlinear operator T defined by T ( f ) = f ′ is a Lipschitzian selection operator with some least constant C ( T ), where C ( T ) ≤ C ( T ′) for all Lipschitzian operators T ′ which map f to one of its best approximations. Thus T is optimal in this sense.

Quasi-convex optimization

Given a bounded real function f defined on a closed bounded real interval I , the problem is to find a quasi-convex function f′ so as to minimize the supremum of | f ( s )- f′ ( s )| for all s in I , over the class of all quasi-convex functions f′ on I . This article obtains optimal solutions to the problem and derives their properties. This problem arises in the context of curve fitting or estimation.

An O(n) algorithm for least squares quasi-convex approximation

This article is concerned with the approximation problem of fitting n data points by a quasi-convex function using the least squares distance function. An algorithm of O(n) worst-case time complexity for computing a best fit is developed. This problem arises in the context of curve fitting or statistical estimation.

Lipschitz condition in minimum norm problems on bounded functions

Abstract Consider the Banach space of bounded functions with uniform norm. Given an element ƒ and a closed convex set in this space, the minimum norm problem is to find an element in the convex set nearest to ƒ. Such a nearest point is not unique in general. For each ƒ in the space, is it possible to select a nearest element ƒ′ so that the selection operator ƒ → ƒ′ satisfies a Lipschitz condition with some constant C ? If so, does there exist an operator for which C is minimum? This article determines the required Lipschitzian selection operators with smallest possible constants for the minimum norm problem in three cases of special interest.

Lipschitzian selections in best approximation by continuous functions

Abstract The problem under consideration is to find a best uniform approximation to a function ƒ from a set K in the space of continuous functions. Conditions are derived on K such that the selection operator mapping ƒ to one of its best approximations is Lipschitzian. Their application is illustrated by approximation problems.

O(n) algorithms for discrete n-point approximation by quasi-convex functions

This article considers the problem of approximating a function defined on a finite set of (n + 1) points by quasi-convex functions defined there and constructs linear time (O(n)) algorithms for computation of optimal solutions.

Lp approximation by quasi-convex and convex functions

Abstract In this article a problem of approximation from nonconvex sets is considered. Let Lp, 1 ⩽ p ⩽ ∞, be the Lebesgue space of extended real functions on a compact real interval. Given a subset P of quasi-convex functions and a function ƒ in Lp, the problem is to find a best Lp-approximation to ƒ from P ∩ Lp. It is shown that if P is closed under pointwise convergence of sequences of functions, then a best approximation exists. Also investigated are properties of norm bounded subsets and convergent sequences of quasi-convex functions. Since convex and monotone functions are quasi-convex, the results are applicable to the problems of approximation from subsets of convex and monotone functions; in particular, the convex problem is analyzed in some detail.

Lipschitzian selections in approximation from nonconvex sets of bounded functions

Abstract Consider the problem of finding a best uniform approximation to a function ƒ from a nonconvex set K in the space of bounded functions. Conditions are developed on K so that the operator mapping ƒ to one of its best approximations ƒ′ is Lipschitzian with some constant C and is optimal Lipschitzian, i.e., has the smallest C among all such operators.

Optimal Lipschitzian selection operator in quasi-convex optimization

Abstract Given a bounded real function f defined on a closed bounded real interval I, the problem is to find a quasi-convex function f′ which minimizes the supremum of ¦f(s) − f′(s)¦ for all s in I, over the class of all quasi-convex functions f′ on I. Such a nearest function f′ is not unique in general. For each f, is it possible to select a nearest f′ so that the selection operator f → f′ satisfies a Lipschitz condition with some constant C? If so, does there exist an operator for which C is minimum? It is shown that there exists a maximal nearest function f so that the operator f → f is such an optimal Lipschitzian selection operator with C = 2.