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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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01 Jan 2008
TL;DR: This article presented an overview of the main theoretical explanations for this bias proposed in the literature, and discussed the role of longshot bias in the expected return on longshot bets and the importance of theoretical explanations.
Abstract: In betting markets, the expected return on longshot bets tends to be systematically lower than on favorite bets. This favorite-longshot bias is a widely documented empirical fact, often perceived to be an important deviation from the market efficiency hypothesis. This chapter presents an overview of the main theoretical explanations for this bias proposed in the literature.
102 citations
TL;DR: By using the topological degree the concept of ’’exceptionalfamily of elements‘‘ specifically for continuous functions is introduced, which has important consequences pertaining to the solvability of the explicit, the implicit and the general order complementarity problems.
Abstract: By using the topological degree we introduce the concept of ’’exceptional family of elements‘‘ specifically for continuous functions. This has important consequences pertaining to the solvability of the explicit, the implicit and the general order complementarity problems. In this way a new direction for research in the complementarity theory is now opened.
96 citations
TL;DR: In this article, it is shown that, under certain assumptions, any stationary point of the unconstrained objective function is already a solution of the nonlinear complementarity problem, and conditions for its local quadratic convergence are given.
Abstract: Several methods for solving the nonlinear complementarity problem (NCP) are developed. These methods are generalizations of the recently proposed algorithms of Mangasarian and Solodov (Ref. 1) and are based on an unconstrianed minimization formulation of the nonlinear complementarity problem. It is shown that, under certain assumptions, any stationary point of the unconstrained objective function is already a solution of NCP. In particulr, these assumptions are satisfied by the mangasarian and Soolodov implicit Lagranian functioin. Furthermore, a special Newton-type method is suggested, and conditions for its local quadratic convergence are given. Finally, some preliminary numerical results are presented.
95 citations
TL;DR: In this paper, the authors consider parallel matrix multisplitting methods for solving linear complementarity problem that finds a real vector z ] R n such that Mz + q S 0, z S 0 and z T (Mz+ q ) = 0.
Abstract: We consider parallel matrix multisplitting methods for solving linear complementarity problem that finds a real vector z ] R n such that Mz + q S 0, z S 0 and z T ( Mz + q )=0, where M ] R n 2 n is...
90 citations
TL;DR: In this paper, a homotopy-smoothing method for solving the variational inequality problem is proposed, and the method converges globally and superlinearly under mild conditions.
Abstract: A variational inequality problem with a mapping $g:\Re^n \to \Re^n$ and lower and upper bounds on variables can be reformulated as a system of nonsmooth equations $F(x)=0$ in $\Re^n$. Recently, several homotopy methods, such as interior point and smoothing methods, have been employed to solve the problem. All of these methods use parametric functions and construct perturbed equations to approximate the problem. The solution to the perturbed system constitutes a smooth trajectory leading to the solution of the original variational inequality problem. The methods generate iterates to follow the trajectory. Among these methods Chen--Mangasarian and Gabriel--More proposed a class of smooth functions to approximate $F$. In this paper, we study several properties of the trajectory defined by solutions of these smooth systems. We propose a homotopy-smoothing method for solving the variational inequality problem, and show that the method converges globally and superlinearly under mild conditions. Furthermore, if the involved function $g$ is an affine function, the method finds a solution of the problem in finite steps. Preliminary numerical results indicate that the method is promising.
89 citations