Abstract
A classical method for solving the variational inequality problem is the projection algorithm. We show that existing convergence results for this algorithm follow from one given by Gabay for a splitting algorithm for finding a zero of the sum of two maximal monotone operators. Moreover, we extend the projection algorithm to solveany monotone affine variational inequality problem. When applied to linear complementarity problems, we obtain a matrix splitting algorithm that is simple and, for linear/quadratic programs, massively parallelizable. Unlike existing matrix splitting algorithms, this algorithm converges under no additional assumption on the problem. When applied to generalized linear/quadratic programs, we obtain a decomposition method that, unlike existing decomposition methods, can simultaneously dualize the linear constraints and diagonalize the cost function. This method gives rise to highly parallelizable algorithms for solving a problem of deterministic control in discrete time and for computing the orthogonal projection onto the intersection of convex sets.
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This research is partially supported by the U.S. Army Research Office, contract DAAL03-86-K-0171 (Center for Intelligent Control Systems), and by the National Science Foundation under grant NSF-ECS-8519058.
Thanks are due to Professor J.-S. Pang for his helpful comments.
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Tseng, P. Further applications of a splitting algorithm to decomposition in variational inequalities and convex programming. Mathematical Programming 48, 249–263 (1990). https://doi.org/10.1007/BF01582258
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DOI: https://doi.org/10.1007/BF01582258