Results 1  10
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15
The Extended Linear Complementarity Problem
, 1993
"... We consider an extension of the horizontal linear complementarity problem, which we call the extended linear complementarity problem (XLCP). With the aid of a natural bilinear program, we establish various properties of this extended complementarity problem; these include the convexity of the biline ..."
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Cited by 776 (28 self)
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We consider an extension of the horizontal linear complementarity problem, which we call the extended linear complementarity problem (XLCP). With the aid of a natural bilinear program, we establish various properties of this extended complementarity problem; these include the convexity of the bilinear objective function under a monotonicity assumption, the polyhedrality of the solution set of a monotone XLCP, and an error bound result for a nondegenerate XLCP. We also present a finite, sequential linear programming algorithm for solving the nonmonotone XLCP.
A new algorithm for stateconstrained separated continuous linear programs
 S/AM Journal on control and optimization
, 1999
"... Abstract. During the last few decades, significant progress has been made in solving largescale finitedimensional and semiinfinite linear programming problems. In contrast, little progress has been made in solving linear programs in infinitedimensional spaces despite their importance as models in ..."
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Cited by 17 (2 self)
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Abstract. During the last few decades, significant progress has been made in solving largescale finitedimensional and semiinfinite linear programming problems. In contrast, little progress has been made in solving linear programs in infinitedimensional spaces despite their importance as models in manufacturing and communication systems. Inspired by the research on separated continuous linear programs, we propose a new class of continuous linear programming problems that has a variety of important applications in communications, manufacturing, and urban traffic control. This class of continuous linear programs contains the separated continuous linear programs as a subclass. Using ideas from quadratic programming, we propose an efficient algorithm for solving largescale problems in this new class under mild assumptions on the form of the problem data. We prove algorithmically the absence of a duality gap for this class of problems without any boundedness assumptions on the solution set. We show this class of problems admits piecewise constant optimal control when the optimal solution exists. We give conditions for the existence of an optimal solution. We also report computational results which illustrate that the new algorithm is effective in solving largescale realistic problems (with several hundred continuous variables) arising in manufacturing systems.
On the Extended Linear Complementarity Problem
 MATHEMATICAL PROGRAMMING
, 1996
"... For the extended linear complementarity problem [11], we introduce and characterize columnsufficiency, rowsufficiency, and Pproperties. These properties are then specialized to the vertical, horizontal, and mixed linear complementarity problems. ..."
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Cited by 14 (2 self)
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For the extended linear complementarity problem [11], we introduce and characterize columnsufficiency, rowsufficiency, and Pproperties. These properties are then specialized to the vertical, horizontal, and mixed linear complementarity problems.
An Interior Point Potential Reduction Method for Constrained Equations
, 1995
"... We study the problem of solving a constrained system of nonlinear equations by a combination of the classical damped Newton method for (unconstrained) smooth equations and the recent interior point potential reduction methods for linear programs, linear and nonlinear complementarity problems. In gen ..."
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Cited by 11 (3 self)
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We study the problem of solving a constrained system of nonlinear equations by a combination of the classical damped Newton method for (unconstrained) smooth equations and the recent interior point potential reduction methods for linear programs, linear and nonlinear complementarity problems. In general, constrained equations provide a unified formulation for many mathematical programming problems, including complementarity problems of various kinds and the KarushKuhnTucker systems of variational inequalities and nonlinear programs. Combining ideas from the damped Newton and interior point methods, we present an iterative algorithm for solving a constrained system of equations and investigate its convergence properties. Specialization of the algorithm and its convergence analysis to complementarity problems of various kinds and the KarushKuhnTucker systems of variational inequalities are discussed in detail. We also report the computational results of the implementation of the algo...
Potential reduction algorithms
 Interior Point Methods in Mathematical Programming
, 1996
"... Potential reduction algorithms have a distinguished role in the area of interior point methods for mathematical programming. Karmarkar’s [44] algorithm for linear programming, whose announcement in 1984 initiated a torrent of research into interior point methods, used three key ingredients: a ..."
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Cited by 8 (0 self)
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Potential reduction algorithms have a distinguished role in the area of interior point methods for mathematical programming. Karmarkar’s [44] algorithm for linear programming, whose announcement in 1984 initiated a torrent of research into interior point methods, used three key ingredients: a
On the Solution of the Extended Linear Complementarity Problem
, 1998
"... The extended linear complementarity problem (XLCP) has been introduced in a recent paper by Mangasarian and Pang. In the present research, minimization problems with simple bounds associated to this problem are defined. When the XLCP is solvable, their solutions are global minimizers of the associat ..."
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Cited by 7 (4 self)
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The extended linear complementarity problem (XLCP) has been introduced in a recent paper by Mangasarian and Pang. In the present research, minimization problems with simple bounds associated to this problem are defined. When the XLCP is solvable, their solutions are global minimizers of the associated problems. Sufficient conditions that guarantee that stationary points of the associated problems are solutions of the XLCP will be proved. These theoretical results support the conjecture that local methods for box constrained optimization applied to the associated problems could be efficient tools for solving the XLCP. Keywords. Complementarity, box constrained minimization. AMS: 90C33, 90C30 Department of Computer Science and Statistics, University of the State of S. Paulo (UNESP), C.P. 136, CEP 15054000, S~ao Jos'e do Rio PretoSP, Brazil. This author was supported by FAPESP (Grant 9615520, 96/125030). Email: andreani@nimitz.dcce.ibilce.unesp.br y Department of Mathematics, I...
Some optimization reformulations of the extended linear complementarity problem
 Comput. Optim. Appl
"... Abstract. We consider the extended linear complementarity problem (XLCP) introduced by Mangasarian and Pang [22], of which the horizontal and vertical linear complementarity problems are two special cases. We give some new sufficient conditions for every stationary point of the natural bilinear prog ..."
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Cited by 6 (2 self)
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Abstract. We consider the extended linear complementarity problem (XLCP) introduced by Mangasarian and Pang [22], of which the horizontal and vertical linear complementarity problems are two special cases. We give some new sufficient conditions for every stationary point of the natural bilinear program associated with XLCP to be a solution of XLCP. We further propose some unconstrained and bound constrained reformulations for XLCP, and study the properties of their stationary points under assumptions similar to those for the bilinear program.
A Quadratically Convergent Algorithm for the Generalized Linear Complementarity Problem[J
 International Mathematical Forum
"... In this paper, the generalized linear complementarity problem (GLCP) is reformulated as a system of nonsmooth equations via the Fischer function. Based on this reformulation, the famous LevenbergMarquardt(LM) algorithm is employed for obtaining its solution. Theoretical results that relate the st ..."
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Cited by 2 (0 self)
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In this paper, the generalized linear complementarity problem (GLCP) is reformulated as a system of nonsmooth equations via the Fischer function. Based on this reformulation, the famous LevenbergMarquardt(LM) algorithm is employed for obtaining its solution. Theoretical results that relate the stationary points of the merit function to the solution of the GLCP are presented. We show that the LM algorithm is both globally and Quadratically convergent without nondegenerate solution. Moreover, a method to calculate a generalized Jacobian is also given.
Smoothing Methods in Mathematical Programming
, 1995
"... ... function. By means of this approximation, linear and convex inequalities are converted into smooth, convex unconstrained minimization problems, the solution of which approximates the solution of the original problem to a high degree of accuracy for sufficiently small positive value of the smooth ..."
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Cited by 1 (0 self)
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... function. By means of this approximation, linear and convex inequalities are converted into smooth, convex unconstrained minimization problems, the solution of which approximates the solution of the original problem to a high degree of accuracy for sufficiently small positive value of the smoothing param eter fl. In the special case when a Slater constraint qualification is satisfied, an exact solution can be obtained for finite fl. Speedup over the linear/nonlinear programming package MINOS 5.4 was as high as 1142 times for linear inequali ties of size 2000 x 1000, and 580 times for convex inequalities with 400 variables. Linear complementarity problems(LCPs) were treated by converting them into a system of smooth nonlinear equations and are solved by a quadratically con vergent Newton method. For monotone LCPs with as many as 10,000 variables, the proposed approach was as much as 63 times faster than Lemke's method. Our smooth approach can also be used to solve nonlinear and mixed complemenrarity problems (NCPs and MCPs) by converting them to classes of smooth parametric nonlinear equations. For any solvable NCP or MCP, existence of an arbitrarily accurate solution to the smooth nonlinear equation as well as the NCP or MCP, is established for sufficiently large value of a smoothing parameter c = l. An efficient smooth algorithm, based on the NewtonArmijo approach with an adjusted smoothing parameter, is also given and its global and local quadratic convergence is established. For NCPs, exact solutions of our smooth nonlinear equation for various values of the parameter c, generate an interior path, which is different from the central path for the interior point method. Computational results for 52 test problems compare favorably with those for another Ne...