Collinear scaling algorithms proposed initially by W. C. Davidon [SIAM J. Numer. Anal. 17, 268-281 (1980; Zbl 0424.65026)] are natural extensions of quasi-Newton methods in the sense that they are based on normal conic local approximations that extend positive definite local quadratic approximations, and that they interpolate values of both the gradient and the function at the current and new iterates. Line search termination criteria that guarantee the existence of such a normal conic local approximation, which also allow one to prove that the component of the gradient in the normalized search direction tends to zero, are not known.
In this paper, the author proposes such line search termination criteria for an important special case where the function being minimized belongs to a certain class of convex functions.