On the Maximization of Real Sequences

TL;DR

This paper introduces a novel methodology for maximizing real sequences by using upper approximations derived from pairs of decreasing and increasing functions, applicable to various sequences and systems.

Contribution

It proposes a general approach to solve sequence maximization problems using function pairs, providing bounds and solutions for well-known sequences and linear systems.

Findings

Method effectively computes supremum and maximizers for various sequences.

Applicable to famous sequences like Fibonacci, logistic, and Syracuse.

Provides bounds and solutions for norm-based peak problems in linear systems.

Abstract

In this paper, we study a maximization problem on real sequences. More precisely, for a given sequence, we are interested in computing the supremum of the sequence and an index for which the associated term is maximal. We propose a general methodology to solve this maximization problem. The method is based on upper approximations constructed from pairs of eventually decreasing sequences of strictly increasing continuous functions on $[0,1]$ and scalars in $(0,1)$. Then, we can associate integers with these pairs using the inverse of the functions on $[0,1]$. We prove that such pairs always exist, and one provides the index maximizer. In general, such pairs provide an upper bound for the greatest maximizer of the sequence. Finally, we apply the methodology to concrete examples, including famous sequences such as the logistic, Fibonacci, and Syracuse sequences. We also apply our techniques to norm-based peak computation problems on discrete-time linear systems.