An Inertial-type CG Projection Method with Restart for Pseudo-monotone Costs with Application to Traffic Assignment

In practical computational applications, many models can be transformed into systems of nonlinear equations for resolution. In this paper, we introduce an inertial-type conjugate gradient projection method that incorporates a restart procedure to solve it. Within the restart procedure, we suggest a new two-term composite restart direction with a flexible non-zero vector and also introduce a tuning parameter, the spectral parameter, as part of the restart direction. Moreover, we design a new self-adaptive line search, which is well-defined. The suggested search direction, which integrates the restart procedure, possesses the sufficient descent and trust region properties, eliminating the need for additional conditions. To achieve the theoretical convergence for the introduced method, we discuss a specific scenario involving pseudo-monotone costs, namely, the system of nonlinear pseudo-monotone equations, without requiring the Lipschitz continuity and monotonicity assumptions. To evaluate the effectiveness of the introduced method, we conduct comparative tests against existing methods on nonlinear equations. Finally, the practicality of the introduced method is demonstrated through its application to the traffic assignment problem.