Working Paper Article Version 3 This version is not peer-reviewed

Evaluation of the Komlos Conjecture Using Multi-Objective Optimization

Version 1 : Received: 5 February 2019 / Approved: 6 February 2019 / Online: 6 February 2019 (10:12:50 CET)
Version 2 : Received: 26 June 2019 / Approved: 9 July 2019 / Online: 9 July 2019 (08:41:45 CEST)
Version 3 : Received: 30 October 2019 / Approved: 31 October 2019 / Online: 31 October 2019 (06:09:49 CET)

How to cite: Brahim Belhaouari, S.; AlQudah, R. Evaluation of the Komlos Conjecture Using Multi-Objective Optimization. Preprints 2019, 2019020059 Brahim Belhaouari, S.; AlQudah, R. Evaluation of the Komlos Conjecture Using Multi-Objective Optimization. Preprints 2019, 2019020059

Abstract

Komlos conjecture is about the existing of a constant upper bound over the dimension n of the function $K(n)$ defined by $$K(n)=max_{ \{ \overrightarrow{V_1},...,\overrightarrow{V_n} \} \in \{ \overrightarrow{V_i} \in { R}^n \| \overrightarrow{V_i} \| \leq 1 \} ^n } min_{ \{\epsilon_1,...\epsilon_n\} \in \{-1,+1\}^n } \|\sum_{i=0}^{n} \epsilon_i \overrightarrow{V_i}\}\|_{\infty} $$ In this paper, the function $K(n)$ is evaluated first for lower dimensions, $n\leq 5$, where it found that $K(2)=\sqrt{2}$. For higher dimension, the function $f(n)=\sqrt{ n-\lceil\log_2(2^{n-1}/n) \rceil}$ is found to be a lower bound for the function $K(n)$, from where it is concluded that the Komlos conjecture is false i.e., the universal constant $k=max_{n \in N} K(n) $ does not exist because of $$\lim_{n \rightarrow \infty } K(n) \geq \lim_{n \rightarrow \infty } \sqrt{ \log_2(n)-1 }=+\infty$$

Subject Areas

Komlos Conjecture; optimazation; discrepancy theory

Comments (1)

Comment 1
Received: 31 October 2019
Commenter: Samir Brahim Belhaouari
Commenter's Conflict of Interests: Author
Comment: Adding some new details
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