Traffic congestion and emissions are two severe and urgent problems in the transportation field. To alleviate congestion and encourage green travel, this paper studies a mixed charging scheme for the multimodal transportation network including conventional vehicles (CV) and new energy vehicles (NEV). We exploit the optimal mixed charging schemes according to the mixed equilibria and system optimum model. The mixed equilibria characterized by the variational inequality system represents that the players follow either the user equilibrium or Cournot–Nash equilibrium principle in routing choice. The proposed mixed scheme comprises a regular tolling plan and a tradable credit plan for the CV and NEV users, respectively, where the tradable credits can be seen as a subsidy to the NEV users since they are issued free by the government. Moreover, the existence of the optimal mixed charging scheme is proved for the multimodal transportation network and demonstrated numerically by a concrete example.

In this article, we propose a randomized Douglas-Rachford(DR) method for linear system. This algorithm is based on the cyclic DR method. We consider a linear system as a feasible problem of finding intersection of hyperplanes. In each iteration, the next iteration point is determined by a random DR operator. We prove the convergence of the iteration points based on expectation. And the variance of the iteration points declines to zero. The numerical experiment shows that the proposed algorithm performs better than the cyclic DR method.

An ideal image is desirable to faithfully represent the real-world scene. However, the observed images from imaging system are typically involved in the illumination of light. As the human visual system (HVS) is capable of perceiving identical color with spatially varying illumination, retinex theory is established to probe the rationale of HVS on perceiving color. In the realm of image processing, the retinex models are devoted to diminishing illumination effects from observed images. In this paper, following the recent work by Ng and Wang (SIAM J. Imaging Sci. 4:345-356, 2011), we develop a parallel operator splitting algorithm tailored for the constrained total-variation retinex model, in which all the resulting subproblems admit closed form solutions or can be tractably solved by some subroutines without any internally nested iterations. The global convergence of the novel algorithm is analysed on the perspective of variational inequality in optimization community. Preliminary numerical simulations demonstrate the promising performance of the proposed algorithm.