Nonmonotonic trust region algorithm

Trust-Region-Based Methods for Nonlinear Programming: Recent Advances and Perspectives

“…It was motivated by the popularity of nonmonotone line search techniques for the solution of unconstrained optimization problems by Newton's method. Condition (4), or equivalently (5), is relaxed by replacing the objective value at the kth-iteration by the maximum of the function values at the current and the previous p iterations, for a given positive integer p. This strategy has been proposed by Deng, Xiao & Zhou [21], and it is based on Grippo, Lampariello & Lucidi's ideas [39]. Subsequent related contributions are the works of Mo, Liu & Yan [52], Gu & Mu [41] and Liu & Ma [45]; nevertheless, these three references have employed the average function value of the latest p + 1 iterations instead of the maximum, following the approach of Zhang & Hager [88].…”

A class of nonmonotone trust region algorithm for solving unconstrained nonlinear optimization problems

“…Some theoretical and numerical results of these trust region methods with line search are also interesting. However, all these methods enforce a monotonic decrease in the objective function values at each iteration and this may slow the convergence rate in the minimization process 2 . To overcome the shortcomings, the earliest nonmonotone line search framework was developed by Grippo et al 3 for Newton's method.…”

Some Notes on Orientation Preserving Involutions of Manifold

“…Because of the manifold is closely linked with bordism group A n (2k) [11][12][13][14][15][16][17], this paper mainly introduces a derivation process on involutions of the manifold, since this process is always with bordism group A n (2k) close connection, also can be seen as bordism group A n (2k), from these derivation process, some skills are considered when the tool (i.e. manifold) can be applied.…”