Adaptive Algorithm For Monotone Variational Inequality Problems And Its Applications

Optimization has a wide application in many fields,such as signal processing,image recovery,matrix com-plete,machine learning,etc.In those applications main problems can be stated as an optimization problem.The well-known KKT condition that the optimal solutions satisfy can be transformed into a variational inequality problem.Especially in case of convex optimization KKT system of a convex optimization corresponds to a monotonic variational inequality problem.Projection algorithm is a class of simple but effective solver for variational inequality problems,and is especially suitable for large scale problems.The motivation of this work is to improve the search sizes in the projection algorithms.The main contributions of the work are as follows(1)By introducing a new metric function for projection algorithms for variational inequality problems,an adaptive step size choice strategy is proposed.It can prove that the modified algorithm has a better descend prop-erty than the original algorithm.The global convergence is established on some suitable conditions,numerical results show that the proposed algorithm outperforms the original one.(2)For monotonic variational inequality problems,an adaptive spectral gradient projection algorithm is proposed,equipped with a BB step size strategy.The global convergence of the algorithm is established on some mild conditions.Numerical experiment reveals that the introduction of BB step size can effectively improve the computation efficiency.