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On the Maximization of Real Sequences

Assalé Adjé

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.

On the Maximization of Real Sequences

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 and scalars in . Then, we can associate integers with these pairs using the inverse of the functions on . 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.

Paper Structure

This paper contains 27 sections, 23 theorems, 53 equations, 1 figure, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminary discussions
  3. Basic results on the supremum of a real sequence
  4. An overview of the methodology and tools developed in this paper
  5. The methodology
  6. Related works
  7. Integer programming approaches
  8. On discrete-time peak computation problems
  9. Norm-based peak computation problems on linear systems
  10. Piecewise linear peak computation problems
  11. Polynomial-based peak computation problems
  12. Resolution of Problem \ref{['mainpb']} from an upper approximation of the analyzed sequence
  13. The set of eventually decreasing upper bound sequences
  14. Existence of useful upper bound sequences
  15. Main properties of the upper bound functional $\mathfrak{F}_u$
  16. ...and 12 more sections

Key Result

Proposition 1

If $u\notin \Lambda$ then $\Delta^s_u=\emptyset$ and ${\mathrm R}_{u}^s=\emptyset$. If $u\in\Lambda$, then:

Figures (1)

  • Figure 1: An overview of the methodology and tools employed to solve Problem \ref{['mainpb']}

Theorems & Definitions (52)

  • Proposition 1
  • Definition 1: Upper bound Functional
  • Definition 2: Useful sequence of strictly increasing continuous functions
  • Proposition 2
  • proof
  • Remark 1
  • Example 1
  • Remark 2
  • Definition 3: Usefully decreasing
  • Proposition 3
  • ...and 42 more