Research Reports from the Department of Operations
Document Type
Report
Publication Date
3-1-1976
Abstract
This paper was originally titled "Extremum Problems on the Space of Infinitely Differentiable Functions, I”. A nonnegative, infinitely differentiable function φ defined on the real line is called a Friedrichs mollifier function if it has support in [0,1] and ∫₀¹ φ(t)dt = 1. In this article and a sequel [18], the following problem is considered: Determine Ak = inf ∫₀¹ φ(k)(t)dt, k=1,2,..., where φ(k) denotes the kth derivative of φ and the infimum is taken over the set of all mollifier functions, which is a convex set. This problem has applications to monotone polynomial approximation as shown by this author in [16]. In this article, the structure of the problem of determining Ak, which can be reduced to several other problems, including a continuous time optimal control one, is analyzed in detail, and it is shown that the problem can be solved by solving a nonlinear programming problem involving minimization of a strictly convex function of [(k-1)/2] variables, subject to a simple ordering restriction on the variables. The analytic arguments used involve applications of theory of moments to optimization, nonlinear programming and classical real analysis. The results of this article and those from approximation of functions theory are applied in [18] to derive numerical values of various mathematical entities involved in this article. In particular, it is shown that Ak = k! 2^(k-1) for all k=1,2,...
Keywords
Operations research, Nonlinear programming, Mathematical optimization, Approximation theory, Differentiable functions
Publication Title
Technical Memorandums from the Department of Operations, School of Management, Case Western Reserve University
Issue
Technical memorandum no. 391
Rights
This work is in the public domain and may be freely downloaded for personal or academic use
Recommended Citation
Ubhaya, Vasant A., "Nonlinear Programming and an Optimization Problem on Infinitely Differentiable Functions" (1976). Research Reports from the Department of Operations. 340.
https://commons.case.edu/wsom-ops-reports/340
