Line search by curve fitting in minimization algorithms

Summary Consider an unconstrained minimization of an uniformly convex functionf(z). Let be an algorithm that, for solving it constructs a sequence {z _i} withz _i+1 =z _i + ⁽ⁱ⁾ h _i ,h _i R ⁿ , ⁽ⁱ⁾ R ⁼ and –h _i ^T f(z _i ) > 0. For any algorithm that converges linearly and that uses parabolic or cubic interpolations for the line search, upper bounds on the number of function evaluations needed to approximate the minimum off(z) within a given accuracy, are calculated. The obtained results allow to compare the line search procedure under investigation.