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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: A Complementarity Problem approach to solve the Economic Dispatch Problem (EDP) is presented and is more compact and needs less CPU time compared with other methods and algorithms presented in the literature.
Abstract: In this paper, a Complementarity Problem approach to solve the Economic Dispatch Problem (EDP) is presented. The problem is formulated as, Linear Complementarity Problem (LCP) when the transmission losses are neglected and Nonlinear Complementarity Problem (NCP) if the transmission losses are included in the power balance equation. The presented approach is more compact and needs less CPU time compared with other methods and algorithms presented in the literature. The main ideas of this work is presented via numerical examples.
4 citations
Cites methods from "Linear complementarity, linear and ..."
...The formulated LCP can be solved using iterative methods, Newton methods, and pivoting methods [20]....
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TL;DR: The algorithm has applications in wireless networks, such as positioning a given number of mobile stations in a given region, one goal of which is to cover the largest area possible.
Abstract: This paper presents an algorithm for positioning circles in a given region to maximise the covered area. Our algorithm has applications in wireless networks, such as positioning a given number of mobile stations in a given region, one goal of which is to cover the largest area possible. Although the evaluation of the function value, i.e., the total covered area, is difficult, we bypass this difficulty by calculating the gradient of the total covered area directly. As long as nodes continuously move in directions that guarantee increasing coverage, a configuration of node positions corresponding to a maximal covered area can eventually be identified.
4 citations
Cites methods from "Linear complementarity, linear and ..."
...The details and proofs for each of those methods can be found in Murty (1988), for example, and they are not repeated here....
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10 Jun 2012
TL;DR: It is shown that the diode state determination problem can be formulated as a complementarity problem and Discontinuous jumps in the state variables are allowed after a switching event.
Abstract: A new event-driven method is proposed for the numerical simulation of power electronic circuits with ideal switches and diodes, and their accompanying control systems The power electronic circuit is modeled as a complementarity system and the control system is modeled separately It is shown that the diode state determination problem can be formulated as a complementarity problem Discontinuous jumps in the state variables are allowed after a switching event The method is applied to two examples and the numerical accuracy with respect to the step size is compared with time-stepping methods The importance of the event localization accuracy is discussed
4 citations
Cites background from "Linear complementarity, linear and ..."
...The problem classes are well known from literature [9], [19], [20] and will be used throughout this paper....
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...The LCP can be infeasible without solutions, or feasible with a unique solution or multiple solutions, dependent on the structure of M [19]....
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...When M is positive-semidefinite, which will be the case for the circuit equations encountered in this paper, the LCP can be reformulated as a convex Quadratic Programming (QP) optimization problem [19]:...
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01 Jan 2010
TL;DR: It is claimed that the critical line algorithm received scant attention around the time of its introduction, and an argument is elaborate for the repeatedly made assertion that Markowitz’scritical line algorithm and Philip Wolfe's simplex method for quadratic programming are equivalent.
Abstract: Despite his fame as the father of modern portfolio selection theory, Harry Markowitz’s pioneering efforts in the methodology of quadratic programming are surprisingly obscure. This article is primarily about Markowitz’s critical line algorithm as a contribution to the early history of quadratic programming (as distinct from the more specialized portfolio selection problem). After documenting our claim that the critical line algorithm received scant attention around the time of its introduction, we discuss some factors that may have led to this state of affairs. We then elaborate an argument for the repeatedly made assertion that Markowitz’s critical line algorithm and Philip Wolfe’s simplex method for quadratic programming are equivalent. We do this by relating both of them to the parametric principal pivoting method of quadratic programming and linear complementarity theory.
4 citations
TL;DR: The convergence of the proposed new preconditioned GAOR method for solving the linear complementarity problem with an M-matrix is analyzed, and the comparison results are obtained to show it accelerates the convergence of its predecessor.
Abstract: Recently, based on the Hadjidimos preconditioner, a preconditioned GAOR method was proposed for solving the linear complementarity problem (Liu and Li in East Asian J. Appl. Math. 2:94–107, 2012). In this paper, we propose a new preconditioned GAOR method for solving the linear complementarity problem with an M-matrix. The convergence of the proposed method is analyzed, and the comparison results are obtained to show it accelerates the convergence of the original GAOR method and the preconditioned GAOR method in (Liu and Li in East Asian J. Appl. Math. 2:94–107, 2012). Numerical examples verify the theoretical analysis.
4 citations
Cites background from "Linear complementarity, linear and ..."
...The LCP of the form (1) arises in many scientific computing and engineering applications, for example, the Nash equilibrium point of a bimatrix game, the contact problem, and the free boundary problem for journal bearings; see [5, 16] and the references therein....
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...As is known, LCP (1) possesses a unique solution if and only if A ∈ Rn×n is a P-matrix, namely a matrix whose all principal submatrices have positive determinants; see [4, 5, 16]....
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