A Penalty-free-type Two-step Line Search Algorithm

Penalty-free-type method,which does not use any merit function to gurantee global convergence,has become an active area of research.Among all the penalty-freetype algorithms,filter methods and Ulbrich-Ulbrich nonmonotone trust region method are two class of practical and effective approachs.Inspired by the work of M.Ulbrich and S.Ulbrich,we present here a two-step line search algorithm,which does not use any penalty function,either.At every iteration,we compute a normal direction and a tangential direction,then we find a step length by backtracing technique.To globalize our algorithm,we need the predicted normal reduction,the predicted tangential reduction and the predicted,objective function reduction to satisfy a certain conditon. The cost in computing normal and tangential direction is small.And the algorithm does not need to recompute these two directions when trial steps were rejected.The backtracing technique was used.Finally,preliminary numerical results were reported to verify the efficiency of the algorithm.