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Linear complementarity, linear and nonlinear programming
01 Jan 1988-
About: The article was published on 1988-01-01 and is currently open access. It has received 1012 citations till now. The article focuses on the topics: Mixed complementarity problem & Complementarity theory.
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TL;DR: It is explained how, in two different settings, observations obtained from imitation games complete a circle of ideas, showing that phenomena that had for many years seemed to be distinct are actually superficially different manifestations of a single structure.
18 citations
Cites background or methods from "Linear complementarity, linear and ..."
...The extensive literature on the linear complementarity problem is surveyed in Murty (1988) and Cottle et al. (1992)....
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...This procedure, which came to be known as the Lemke paths algorithm, includes the Lemke-Howson algorithm as the special case that arises when p = m + n and C is derived from (A,B) as above....
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TL;DR: In this paper, necessary and sufficient conditions for the unique solution of the generalized absolute value equation (GAVE) were presented from the optimization field, which cover the fundamental theorem of the unique solutions of the linear system.
Abstract: In this paper, some useful necessary and sufficient conditions for the unique solution of the generalized absolute value equation (GAVE) $$Ax-B|x|=b$$
with $$A, B\in \mathbb {R}^{n\times n}$$
from the optimization field are first presented, which cover the fundamental theorem for the unique solution of the linear system $$Ax=b$$
with $$A\in \mathbb {R}^{n\times n}$$
. Not only that, some new sufficient conditions for the unique solution of the GAVE are obtained, which are weaker than the previous published works.
17 citations
TL;DR: In this article, a new feasible direction algorithm for nonlinear complementarity problems is presented, strictly satisfying the inequality conditions, and generates a sequence of interior points that converges to a solution of the problem.
Abstract: Complementarity problems are involved in mathematical models of several applications in engineering, economy and different branches of physics. We mention contact problems and dynamics of multiple bodies systems in solid mechanics. In this paper we present a new feasible direction algorithm for nonlinear complementarity problems. This one begins at an interior point, strictly satisfying the inequality conditions, and generates a sequence of interior points that converges to a solution of the problem. At each iteration, a feasible direction is obtained and a line search performed, looking for a new interior point with a lower value of an appropriate potential function. We prove global convergence of the present algorithm and present a theoretical study about the asymptotic convergence. Results obtained with several numerical test problems, and also application in mechanics, are described and compared with other well known techniques. All the examples were solved very efficiently with the present algorithm, employing always the same set of parameters.
17 citations
01 Oct 2017
TL;DR: The uniqueness result generalizes an existing uniqueness condition for games of linear best-responses to games with general best-response functions and identifies the classes of agents that are instrumental in the spread of shocks over the network.
Abstract: We study a class of games played on networks with general (non-linear) best-response functions. Specifically, we let each agent's payoff depend on a linearly weighted sum of her neighbors' actions through a non-linear interaction function. We identify conditions on the network structure underlying the game given which (i) the Nash equilibrium of the game is unique, and (ii) the Nash equilibria are stable under perturbations in the model's parameters. We find that both the uniqueness and stability of the Nash equilibria are related to the lowest eigenvalue of suitably defined matrices, which are determined by the network's adjacency matrix, as well as the slopes of the interaction functions. We show that our uniqueness result generalizes an existing uniqueness condition for games of linear best-responses to games with general best-response functions. We further identify the classes of agents that are instrumental in the spread of shocks over the network. In particular, for small shocks, we show that agents that are strictly inactive at a given equilibrium can be precluded from the equilibrium's stability analysis, irrespective of their network position or links.
17 citations
Cites background from "Linear complementarity, linear and ..."
...In particular, a symmetric matrix G is a P-matrix if and only if it is positive definite [20], and it is positive definite if and only if |λmin(G)| 1....
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...We will identify conditions under which this matrix is a P-matrix, which will mean that it is also positive definite [20]....
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...We therefore identify conditions under which Υ is a P-matrix....
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...The work of Naghizadeh and Liu [3] has further used the theory of linear complementarity problems to show that the Nash equilibrium of network games with linear best-responses (with either symmetric and asymmetric adjacency matrices) is unique if and only if its adjacency matrix W is a P-matrix....
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...A symmetric matrix is a P-matrix if and only if it is positive definite [20]....
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TL;DR: In this paper, a mathematical formulation in terms of a linear complementarity problem with regularized friction is introduced for multibody contact problems, where contacts are characterized based on kinematic constraints while the friction forces are simultaneously regularized and incorporated into the formulation.
17 citations