A Nonmonotone Trust Region Algorithm Based on the Average of the Successive Penalty Function Values for Nonlinear Optimization

We present a nonmonotone trust region algorithm for nonlinear equality constrained optimization problems. In our algorithm, we use the average of the successive penalty function values to rectify the ratio of predicted reduction and the actual reduction. Compared with the existing nonmonotone trust region methods, our method is independent of the nonmonotone parameter. We establish the global convergence of the proposed algorithm and give the numerical tests to show the efficiency of the algorithm. 1. Introduction In this paper, we consider the equality constrained optimization problem as follows: where , , , , and are assumed to be twice continuously differentiable. Trust region method is one of the well-known methods for solving problem (1). Due to its strong convergence and robustness, trust region methods have been proved to be efficient for both unconstrained and constrained optimization problems [1–9]. Most traditional trust region methods are of descent type methods; namely, they accept only a trial point as the next iterate if its associated merit function value is strictly less than that of the current iterate. However, just as pointed out by Toint [10], the nonmonotone techniques are helpful to overcome the case that the sequence of iterates follows the bottom of curved narrow valleys, a common occurrence in difficult nonlinear problems. Hence many nonmonotone algorithms are proposed to solve the unconstrained and constrained optimization problems [11–20]. Numerical tests show that the performance of the nonmonotone technique is superior to those of the monotone cases. The nonmonotone technique was originally proposed by Grippo, Lampariello and Lucidi [13] for unconstrained optimization problems based on Newton’s method, in which the stepsize satisfies the following condition: where , , and is a prefixed nonnegative integer. Although the nonmonotone technique based on (2) works well in many cases, there are some drawbacks. Firstly, a good function value generated in any iteration is essentially discarded due to the maximum in (2). Secondly, in some cases, the numerical performance is heavily dependent on the choice of (see, e.g., [16, 21]). To overcome these drawback, Zhang and Hager [21] proposed another nonmonotone algorithm, and they used the average of function values to replace the maximum function value in (2). The numerical tests show that their nonmonotone line search algorithm used fewer function and gradient evaluations, on average, than either the monotone or the traditional nonmonotone scheme. Recently, Mo and Zhang [16] extended