A sequential quadratic programming algorithm with non-monotone line search

Today, many practical smooth nonlinear programming problems are routinely solved by sequential quadratic programming (SQP) methods stabilized by a monotone line search procedure subject to a suitable merit function. In case of computational errors as for example caused by inaccurate function or gradient evaluations, however, the approach is unstable and often terminates with an error message. To reduce the number of false terminations, a non-monotone line search is proposed which allows the acceptance of a step length even with an increased merit function value. Thus, the subsequent step may become larger than in case of a monotone line search and the whole iteration process is stabilized. Convergence of the new SQP algorithm is proved assuming exact arithmetic, and numerical results are included. As expected, no significant improvements are observed if function values are computed within machine accuracy. To model more realistic and more difficult, situations, we add randomly generated errors to function values and show that a drastic improvement of the performance is achieved compared to monotone line search. This situation is very typical for complex simulation programs producing inaccurate function values and where, even worse, derivatives are nevertheless computed by forward differences.