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Galerkin methods for complementarity problems and variational inequalities. arXiv:1306.4753
, 2013
"... Complementarity problems and variational inequali-ties arise in a wide variety of areas, including ma-chine learning, planning, game theory, and physical simulation. In all of these areas, to handle large-scale problem instances, we need fast approximate solution methods. One promising idea is Galer ..."
Abstract
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Cited by 1 (1 self)
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Complementarity problems and variational inequali-ties arise in a wide variety of areas, including ma-chine learning, planning, game theory, and physical simulation. In all of these areas, to handle large-scale problem instances, we need fast approximate solution methods. One promising idea is Galerkin approxima-tion, 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 itera-tion 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 ba-sis, 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.
