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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: In this article, a general theoretical framework for the solvability of variational inequalities with possibly non-convex constraints and objectives is proposed, consisting of a generic constrained nonlinear inequality, derived from new topological fixed point theorems for setvalued maps in the absence of convexity.
Abstract: This is the first part of a work on generalized variational inequalities and their applications in optimization. It proposes a general theoretical framework for the solvability of variational inequalities with possibly non-convex constraints and objectives. The framework consists of a generic constrained nonlinear inequality (
$\exists\hat{u}\in\Psi(\hat {u})$
, $\exists \hat{y}\in\Phi(\hat{u})$
, with $\varphi(\hat{u},\hat{y},\hat{u})\leq \varphi(\hat{u},\hat{y},v)$
, $\forall v\in\Psi(\hat{u})$
) derived from new topological fixed point theorems for set-valued maps in the absence of convexity. Simple homotopical and approximation methods are used to extend the Kakutani fixed point theorem to upper semicontinuous compact approachable set-valued maps defined on a large class of non-convex spaces having non-trivial Euler-Poincare characteristic and modeled on locally finite polyhedra. The constrained nonlinear inequality provides an umbrella unifying and extending a number of known results and approaches in the theory of generalized variational inequalities. Various applications to optimization problems will be presented in the second part to this work to be published ulteriorly.
1 citations
TL;DR: In this article, the authors use the theory of degeneracy graphs to study the adjacency of almost complementary feasible bases, some of which may be degenerate, which are of interest in the context of linear complementarity problem.
Abstract: In this paper, we use the theory of degeneracy graphs recently developed by Gal et al to introduce a graph for studying the adjacency of almost complementary feasible bases, some of which may be degenerate, which are of interest in the context of the linear complementarity problem We study the structure of this graph with particular reference to the possibility of cycling and various anticycling rules in the Lemke complementary pivoting algorithm We consider the transition node pivot rule introduced by Geue and show that this rule helps in avoiding cycling in the Lemke complementary pivoting algorithm under a suitable assumption
1 citations
Posted Content•
TL;DR: In this paper, the main purpose is the virtual human for engineering, especially virtual prototyping, and it takes days to a specialist to build such animated sequences, and is not adaptive to any type of modifications.
Abstract: The animation of human avatars seems very successful; the computer graphics industry shows outstanding results in films everyday, the game industry achieves exploits... Nevertheless, the animation and control processes of such manikins are very painful. It takes days to a specialist to build such animated sequences, and it is not adaptive to any type of modifications. Our main purpose is the virtual human for engineering, especially virtual prototyping. As for this domain of activity, such amounts of time are prohibitive.
1 citations
01 Jan 2011
TL;DR: A pivoting heuristic based on tabu search and its integration into an enumerative framework for solving the Linear Complementarity Problem (LCP) is described in this article.
Abstract: This paper describes a pivoting heuristic based on tabu search and its integration into an enumerative framework for solving the Linear Complementarity Problem (LCP). The tabu pivoting heuristic works with basic solutions and performs pivot operations guided by two indicators, one concerned with the satisfaction of the complementarity conditions and the other with the feasibility of the solution. It incorporates the concept of tabu search employing a strategy that avoids the repetition of recent moves. The heuristic ends when a solution to the LCP is found or after a specified number of iterations. In the latter case, an enumerative algorithm is applied which integrates the tabu pivoting heuristic within a branching framework. Computational experience on test problems is reported to highlight the efficiency of the proposed methodology for solving the LCP.
1 citations
Cites background or methods from "Linear complementarity, linear and ..."
...It is well-known that the LCP is NP-hard, although it is polynomially solvable for some classes of matrices M , such as Positive-Semi-Definite (PSD) matrices [Murty, 1988]....
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...Example 1 Consider the LCP with n = 5 and defined by the following matrix M and vector q: M = 9 4 9 3 3 7 2 10 3 4 5 8 6 3 4 2 7 3 8 8 7 8 1 5 8 q = 3 −9 2 10 −5 Since M is a positive matrix, then the LCP has a solution for each vector q (Murty 1988) and, in particular, for this given q....
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...However, there are some cases where such termination cannot occur or only occurs when the feasible set of the LCP is empty (Murty 1988; Cottle et al. 2009)....
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...ISSN: 1841-4311 A number of important optimization problems can be solved by finding a solution of its associated LCP or one of its generalizations [Murty, 1988; Cottle et al., 2009; Júdice, 1994]....
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...Furthermore, Lemke’s algorithm (Lemke 1968) is able to process this LCP (Murty 1988)....
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Posted Content•
TL;DR: In this paper, a computationally efficient method to solve overlapping generations models with asset choice is presented, which is used to study an OLG economy with many cohorts, up to three different assets, stochastic volatility, short-sale constraints, and subject to rather large technology shocks.
Abstract: The paper presents a computationally efficient method to solve overlapping generations models with asset choice. The method is used to study an OLG economy with many cohorts, up to 3 different assets, stochastic volatility, short-sale constraints, and subject to rather large technology shocks. On the methodological side, the main findings are that global projection methods with polynomial approximations of degree 3 are sufficient to provide a very precise solution, even in the case of large shocks. Globally linear approximations, in contrast to local linear approximations, are sufficient to capture the most important financial statistics, including not only the average risk premium, but also the variation of the risk premium over the cycle. However, global linear approximations are not sufficient to reliably pin down asset choices. With a risk aversion parameter of only 4, the model generates a price of risk, measured as the Sharpe ratio, that is about two thirds of that of US stocks. Being subject to three types of shocks, the equilibiurm allocation, even with 3 assets, differs substantially from an allocation under sequentially complete markets. In particular, the oldest cohorts are more more heavily exposed to negative shocks.
1 citations