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Galerkin methods for complementarity problems and variational inequalities. arXiv:1306.4753
, 2013
"... Complementarity problems and variational inequalities arise in a wide variety of areas, including machine learning, planning, game theory, and physical simulation. In all of these areas, to handle largescale problem instances, we need fast approximate solution methods. One promising idea is Galer ..."
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Complementarity problems and variational inequalities arise in a wide variety of areas, including machine learning, planning, game theory, and physical simulation. In all of these areas, to handle largescale problem instances, we need fast approximate solution methods. One promising idea is Galerkin approximation, in which we search for the best answer within the span of a given set of basis functions. Bertsekas [1] proposed one possible Galerkin method for variational inequalities. However, this method can exhibit two problems in practice: its approximation error is worse than might be expected based on the ability of the basis to represent the desired solution, and each iteration requires a projection step that is not always easy to implement efficiently. So, in this paper, we present a new Galerkin method with improved behavior: our new error bounds depend directly on the distance from the true solution to the subspace spanned by our basis, and the only projections we require are onto the feasible region or onto the span of our basis.
A projection algorithm for strictly monotone linear complementarity problems. *
"... Abstract Complementary problems play a central role in equilibrium finding, physical simulation, and optimization. As a consequence, we are interested in understanding how to solve these problems quickly, and this often involves approximation. In this paper we present a method for approximately sol ..."
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Abstract Complementary problems play a central role in equilibrium finding, physical simulation, and optimization. As a consequence, we are interested in understanding how to solve these problems quickly, and this often involves approximation. In this paper we present a method for approximately solving strictly monotone linear complementarity problems with a Galerkin approximation. We also give bounds for the approximate error, and prove novel bounds on perturbation error. These perturbation bounds suggest that a Galerkin approximation may be much less sensitive to noise than the original LCP.