Remarks on the Comparative Evaluation of Algorithms for Mathematical Programming Problems

In this Presentation, we consider the minimization of a function f = f(x), where f is a scalar and x is an n-vector whose components are unconstrained. We consider the comparative evaluation of algorithms for unconstrained minimization. We are concerned with the measurement of the computational speed and examine critically the concept of equivalent number of function evaluations N_e, which is defined by1$${{N}_{e}}={{N}_{0}}+n{{N}_{1}}+m{{N}_{2}}.$$ Here, N₀ is the number of function evaluations; N₁ is the number of gradient evaluations; N₂ is the number of Hessian evaluations? n is the dimension of the vector x; and m = n(n+l)/2 is the number of elements of the Hessian matrix above and including the principal diagonal. We ask the following question: Does the use of the quantity N_e constitute a fair way of comparing different algorithms?