√(x2 + μ) is the Most Computationally Efficient Smooth Approximation to |x|: a Proof

Authors

Carlos Ramirez, The University of Texas at El PasoFollow
Reinaldo Sanchez, The University of Texas at El Paso
Vladik Kreinovich, The University of Texas at El PasoFollow
Miguel Argaez, Department of Mathematical Sciences, The The University of Texas at El PasoFollow

Publication Date

6-2013

Comments

Technical Report: UTEP-CS-13-44

To appear in Journal of Uncertain Systems, 2014, Vol. 8.

Abstract

In many practical situations, we need to minimize an expression of the type |c₁| + ... + |c_n|. The problem is that most efficient optimization techniques use the derivative of the objective function, but the function |x| is not differentiable at 0. To make optimization efficient, it is therefore reasonable to approximate |x| by a smooth function. We show that in some reasonable sense, the most computationally efficient smooth approximation to |x| is the function √(x² + μ), a function which has indeed been successfully used in such optimization.