In contrast to most studies comparing the performance of optimization algorithms, this paper focuses on an algorithm’s key feature: its choice of search direction. The rules of the game are simple. An initial point and direction are chosen, a trial step is taken, and the line search continues until the minimizer is found. Each algorithm is then applied to find a new search direction followed by a second line search. The objective function decrease in the second line search relative to the decrease in the initial search is the figure of merit.

The authors compare the performance of two conjugate gradient methods as well as an algorithm based on conic functions. They find that the conjugate gradient method which uses the trial point instead of the starting point whenever its function value is lower generates a better search direction. Their conic function algorithm also compares favorably, but the reader will have to consult [1] and [2] (cited in the paper) to find out what it is.