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Fast solutions to projective monotone linear complementarity problems. arXiv:1212.6958, (2012)

by Geoffrey J Gordon
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Galerkin methods for complementarity problems and variational inequalities. arXiv:1306.4753

by Geoffrey J. Gordon , 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 ..."
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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.
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... way. However, it turns out that we can solve CP(Nx+r,K) much more quickly than a general LCP of the same dimension, due to the special structure of N . In particular, N is projective in the sense of =-=[3]-=-. So, as shown in that paper, we can run the Unified Interior Point (UIP) method of Kojima et al. [4] very quickly: if M ∈ Rn×n and Φ ∈ Rn×k, then each iteration of the UIP method takes time O(nk2), o...

A projection algorithm for strictly monotone linear complementarity problems. *

by Erik Zawadzki , Geoffrey J Gordon , André Platzer
"... 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.
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...ion onto the nullspace of Φ>. Such LCPs are called projective. Given this solution concept, we can ask if there are other algorithms to achieve it, with different or better convergence properties. It turns out that an interior-point method can be highly effective: its convergence is exponentially faster than gradient descent, but each iteration requires solving a system of linear equations, so it is limited to problems whose size or structure lets us solve this system efficiently. In addition, the method provably solves (strict or non-strict) monotone projective LCPs. In particular, our paper [5] gives an algorithm that solves a monotone projective LCP to relative accuracy inO( √ n ln(1/)) iterations, with each iteration requiringO(nk2) flops. This complexity compares favorably with interior-point algorithms for general monotone LCPs: these algorithms also require O( √ n ln(1/)) iterations, but each iteration needs to solve an n × n system of linear equations, a much higher cost than our algorithm when k n. This algorithm works even though the solution to a projective LCP is not restricted to lie in the low-rank subspace spanned by Φ. 2 Complementarity problems A complementarity...

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