A New Filter Nonmonotone Adaptive Trust Region Method for Unconstrained Optimization

Abstract

In this paper, a new filter nonmonotone adaptive trust region with fixed step length for unconstrained optimization is proposed. The trust region radius adopts a new adaptive strategy to overcome additional computational costs at each iteration. A new nonmonotone trust region ratio is introduced. When a trial step is not successful, a multidimensional filter is employed to increase the possibility of the trial step being accepted. If the trial step is still not accepted by the filter set, it is possible to find a new iteration point along the trial step and the step length is computed by a fixed formula. The positive definite symmetric matrix of the approximate Hessian matrix is updated using the MBFGS method. The global convergence and superlinear convergence of the proposed algorithm is proven by some classical assumptions. The efficiency of the algorithm is tested by numerical results.

A Filter and Nonmonotone Adaptive Trust Region Line Search Method for Unconstrained Optimization

“…The problem has widely used in many applications based on medical science, optimal control, and functional approximation, etc. As we all know, there are many methods for solving unconstrained optimization problems, such as the conjugate gradient method [1][2][3], the Newton method [4,5], and the trust region method [6][7][8]. Constrained optimization problems can also be solved by processing constraint conditions and transforming them into unconstrained optimization problems.…”

“…In order to avoid computing the inverse of the matrix and the Euclidean norm ofB −1 k at each iteration point x k , Zhou et al [19] proposed an adaptive trust region radius as follows: ∆ k = c p d k−1 y k−1 g k , where y k−1 = g k − g k−1 , and c and p are parameters. Prompted by the adaptive technique, Wang et al [8] proposed a new adaptive trust region radius as follows: ∆ k = c k g k γ , which reduces the related workload and calculation time. Based on this fact, other authors also proposed modified adaptive trust region methods [20][21][22].…”

“…Based on this situation, Nocedal and Yuan [9] proposed an algorithm in 1998, combining the trust region method and the line search method for the first time. Inspired by this, Michael et al, Li et al, and Zhang et al proposed a trust region method with the line search strategy ([10-12], respectively).As can be seen in other works [4,7,8] monotone techniques are distinguished from nonmonotone techniques in that the value of the function needs to be reduced at each iteration; at the same time, the use of nonmonotone techniques can not only guarantee finding the global optimal solution effectively, but also improve the convergence rate of the algorithm. The watchdog technique was presented by Chamberlain et al [13] in 1982 to overcome the Maratos effect of constrained optimization problems.…”

“…As can be seen in other works [4,7,8] monotone techniques are distinguished from nonmonotone techniques in that the value of the function needs to be reduced at each iteration; at the same time, the use of nonmonotone techniques can not only guarantee finding the global optimal solution effectively, but also improve the convergence rate of the algorithm. The watchdog technique was presented by Chamberlain et al [13] in 1982 to overcome the Maratos effect of constrained optimization problems.…”

A new nonmonotone adaptive trust region line search method for unconstrained optimization

“…. Inspired by these facts, some modified versions of adaptive trust region methods have been proposed in [7][8][9][10][11][12][13][14].…”