Results 1  10
of
16
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.
Monotone Semidefinite Complementarity Problems
, 1996
"... . In this paper, we study some basic properties of the monotone semidefinite nonlinear complementarity problem (SDCP). We show that the trajectory continuously accumulates into the solution set of the SDCP passing through the set of the infeasible but positive definite matrices under certain conditi ..."
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Cited by 11 (1 self)
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. In this paper, we study some basic properties of the monotone semidefinite nonlinear complementarity problem (SDCP). We show that the trajectory continuously accumulates into the solution set of the SDCP passing through the set of the infeasible but positive definite matrices under certain conditions. Especially, for the monotone semidefinite linear complementarity problem, the trajectory converges to an analytic center of the solution set, provided that there exists a strictly complementary solution. Finally, we propose the globally convergent infeasibleinteriorpoint algorithm for the SDCP. Key words Monotone Semidefinite Complementarity Problem, Trajectory, Interior Point Algorithm Research Report B312 on Mathematical and Computing Sciences, Department of Mathematical and Computing Sciences, Tokyo Institute of Technology. 1 Introduction. Let M(n) and S(n) denote the class of n2n real matrices and the class of n2n symmetric real matrices, respectively. Assume that A; B 2 M(n)....
An Analysis of Zero Set and Global Error Bound Properties of a Piecewise Affine Function Via Its Recession Function
, 1996
"... For a piecewise affine function f : R n ! R m , the recession function is defined by f 1 (x) := lim !1 f(x) : In this paper, we study the zero set and error bound properties of f via f 1 . We show, for example, that f has a zero when f 1 has a unique zero (at the origin) with a nonvanis ..."
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Cited by 8 (4 self)
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For a piecewise affine function f : R n ! R m , the recession function is defined by f 1 (x) := lim !1 f(x) : In this paper, we study the zero set and error bound properties of f via f 1 . We show, for example, that f has a zero when f 1 has a unique zero (at the origin) with a nonvanishing index. We also characterize the global error bound property of a piecewise affine function in terms of the recession cones of the zero sets of the function and its recession function.
Centers of Monotone Generalized Complementarity Problems
 Math. Oper. Res
, 1996
"... . Let C be a full dimensional, closed, pointed and convex cone in a finite dimensional real vector space E with an inner product hx; yi of x; y 2 E , and M a maximal monotone subset of E 2 E . This paper studies the existence and continuity of centers of the monotone generalized complementarity prob ..."
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Cited by 8 (4 self)
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. Let C be a full dimensional, closed, pointed and convex cone in a finite dimensional real vector space E with an inner product hx; yi of x; y 2 E , and M a maximal monotone subset of E 2 E . This paper studies the existence and continuity of centers of the monotone generalized complementarity problem associated with C and M: Find (x; y) 2 M " (C 2C 3 ) such that hx; yi = 0. Here C 3 = fy 2 E : hx; yi 0 for all x 2 Cg denotes the dual cone of C. The main result of the paper unifies and extends some results established for monotone complementarity problems in Euclidean space and monotone semidefinite linear complementarity problems in symmetric matrices. Key words Central Trajectory, Path of Centers, Complementarity Problem, Interior Point Algorithm, Linear Program Research Report B303 on Mathematical and Computing Sciences, Department of Mathematical and Computing Sciences, Tokyo Institute of Technology. 1 Introduction. The central trajectory or the path of centers is known ...
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.
On the Connectedness of Solution Sets in Linear Complementarity Problems
, 1996
"... In this paper, we investigate conditions on a square matrix M for which every LCP(M; q) (with q arbitrary) has connected solution set. We show that a matrix with this property is necessarily fully semimonotone. Using degree theory, we show that the solution set of LCP(M; q) corresponding to a P0mat ..."
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Cited by 6 (1 self)
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In this paper, we investigate conditions on a square matrix M for which every LCP(M; q) (with q arbitrary) has connected solution set. We show that a matrix with this property is necessarily fully semimonotone. Using degree theory, we show that the solution set of LCP(M; q) corresponding to a P0matrix is connected if there is a bounded connected component in the solution set. Email: cristy@math.umbc.edu y Research supported by the National Science Foundation Grant CCR9307685 Email: gowda@math.umbc.edu 2 1. INTRODUCTION Given a matrix M 2 R n\Thetan and a vector q 2 R n , the linear complementarity problem, LCP(M; q) [2] is to find a vector x 2 R n such that x 0; Mx+ q 0; and x T (Mx + q) = 0: (1) This problem has become fundamental in optimization, game theory, economics, and engineering, see [10] and [2]. In the LCP theory one studies various classes of matrices such as the class of Pmatrices, Q (Q 0 )matrices, R 0 matrices, column sufficient matrices, etc. Man...
On the Rate of Local Convergence of HighOrderInfeasiblePathFollowing Algorithms for P*Linear Complementarity Problems
 Computational Optimization and Applications
, 1997
"... A simple and unified analysis is provided on the rate of local convergence for a class of highorderinfeasiblepathfollowing algorithms for the P linear complementarity problem (P LCP). It is shown that the rate of local convergence of a order algorithm with a centering step is + 1 if there ..."
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Cited by 5 (2 self)
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A simple and unified analysis is provided on the rate of local convergence for a class of highorderinfeasiblepathfollowing algorithms for the P linear complementarity problem (P LCP). It is shown that the rate of local convergence of a order algorithm with a centering step is + 1 if there is a strictly complementary solution and ( + 1)=2 otherwise. For the order algorithm without the centering step the corresponding rates are and =2, respectively. The algorithm without a centering step does not follow the fixed traditional central path. Instead, at each iteration, it follows a new analytic path connecting the current iterate with an optimal solution to generate the next iterate. An advantage of this algorithm is that it does not restrict iterates in a sequence of contracting neighborhoods of the central path.
Reformulation Of Variational Inequalities On A Simplex And Compactification Of Complementarity Problems
 SIAM Journal on Optimization
, 2000
"... . Many variational inequality problems (VIP) can be reduced, by a compactification procedure, to a VIP on the canonical simplex. Reformulations of this problem are studied, including smooth reformulations with simple constraints and unconstrained reformulations based on the penalized FischerBurmeis ..."
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Cited by 3 (1 self)
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. Many variational inequality problems (VIP) can be reduced, by a compactification procedure, to a VIP on the canonical simplex. Reformulations of this problem are studied, including smooth reformulations with simple constraints and unconstrained reformulations based on the penalized FischerBurmeister function. It is proved that bounded level set results hold for these reformulations, under quite general assumptions on the operator. Therefore, it can be guaranteed that minimization algorithms generate bounded sequences and, under monotonicity conditions, these algorithms necessarily find solutions of the original problem. Some numerical experiments are presented. Key words. Variational inequalities, complementarity, minimization algorithms, reformulation. AMS subject classifications. 90C33, 90C30 1. Introduction. We are interested in reformulations of variational inequality problems (VIP) where the domain is a simplex. The main motivation is that variational inequalities on generaliz...