Incorporating nonmonotone strategies into the trust region method for unconstrained optimization

Abstract

This paper concerns a nonmonotone line search technique and its application to the trust region method for unconstrained optimization problems. In our line search technique, the current nonmonotone term is a convex combination of the previous nonmonotone term and the current objective function value, instead of an average of the successive objective function values that was introduced by Zhang and Hager [H. Zhang, W.W. Hager, A nonmonotone line search technique and its application to unconstrained optimization, SIAM J. Optim. 14 (4) (2004) 1043–1056]. We incorporate this nonmonotone scheme into the traditional trust region method such that the new algorithm possesses nonmonotonicity. Unlike the traditional trust region method, our algorithm performs a nonmonotone line search to find a new iteration point if a trial step is not accepted, instead of resolving the subproblem. Under mild conditions, we prove that the algorithm is global and superlinear convergence holds. Primary numerical results are reported.