A United Probabilistic Approach to Minimising the Sum of Absolute Values
Keywords:
median, probability distribution , minimum valueAbstract
Aim: Introduce a novel method for minimizing functions in the form of a sum of absolute values.Β
Methodology: The sum of absolute values can be standardized so that the sum of the coefficients equals 1. In this case, the sum of absolute values takes the form πΈπΈ|ππβππ|, where ππ is a random variable.
Results: Any median of ππ is a minimiser of the function πΈπΈ|ππβππ|. To minimise the function, it suffices to find any median of ππ.
Implications and recommendations: The method introduced in this paper can be applied to minimise a large family of functions.
Originality/value: Our work uses the probabilistic method to solve optimization problems.
Downloads
References
Bogomolny, A. (2021, March 20). Sum of Absolute Values. Cut the Knot. https://www.cut-theknot.org/m/Algebra/MinimumWithAbsoluteValue.shtml
Koenker, R., & Bassett, G. (1978). Regression Quantiles. Econometrica, 46, 33β50.
Lehmann, E. L. & Casella, G. (1998). Theory of Point Estimation. 2nd edition. Springer.
Nahin, P. J. (2004). When Least Is Best. Princeton University Press.
Shao, J. (2005). Mathematical Statistics: Exercises and Solutions. Springer.
Xu, J. (2012). Lecture Notes on Mathematical Olympiad Courses. For senior section, vol. 1. World Scientific.
Downloads
Published
Issue
Section
Categories
License
Copyright (c) 2024 Changyong Feng, Honghong Liu, Ethan Poon, Ge Feng
This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.
