Physical Sciences and Mathematics Commons^™

√(X2 + Μ) Is The Most Computationally Efficient Smooth Approximation To |X|: A Proof, Carlos Ramirez, Reinaldo Sanchez, Vladik Kreinovich, Miguel Argaez

Departmental Technical Reports (CS)

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.