To reduce the difficulty and complexity in computing the projection from a real Hilbert space onto a nonempty closed convex subset, researchers have provided a hybrid steepest-descent method for solving VI(F, K) and a subsequent three-step relaxed version of this method. In a previous study, the latter was used to develop a modified and relaxed hybrid steepest-descent (MRHSD) method. However, choosing an efficient and implementable nonexpansive mapping is still a difficult problem. We first establish the strong convergence of the MRHSD method for variational inequalities under different conditions that simplify the proof, which differs from previous studies. Second, we design an efficient implementation of the MRHSD method for a type of variational inequality problem based on the approximate projection contraction method. Finally, we design a set of practical numerical experiments. The results demonstrate that this is an efficient implementation of the MRHSD method.
The time-band approximation model on flight operations recovery problems is constructed based on the time-band network, and the object of its' mathematical representation can capture approximated delay costs and cancellation costs. But the calculation of delay costs associated with the actual flying time in China, the actual flying time should not be a fixed time but a random time around the planned flying time. In this paper, we describe the approximated delay costs considering the random flying time around the planned flying time. Subsequently, we give out the time-band approximation model on flight operations recovery with the random flying time. Finally, we design a set of practical numerical experiments. The results demonstrate that the time-band approximation model on flight operations recovery with the random flying time may be more efficient than the timeband approximation model on flight operations recovery with the planned flying time.
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