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36
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 530 (23 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.
Solution of General Linear Complementarity Problems via Nondifferentiable Concave Minimization
 Acta Mathematica Vietnamica
, 1997
"... Finite termination, at point satisfying the minimum principle necessary optimality condition, is established for a stepless (no line search) successive linearization algorithm (SLA) for minimizing a nondifferentiable concave function on a polyhedral set. The SLA is then applied to the general linear ..."
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Cited by 26 (11 self)
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Finite termination, at point satisfying the minimum principle necessary optimality condition, is established for a stepless (no line search) successive linearization algorithm (SLA) for minimizing a nondifferentiable concave function on a polyhedral set. The SLA is then applied to the general linear complementarity problem (LCP), formulated as minimizing a piecewiselinear concave error function on the usual polyhedral feasible region defining the LCP. When the feasible region is nonempty, the concave error function always has a global minimum at a vertex, and the minimum is zero if and only if the LCP is solvable. The SLA terminates at a solution or stationary point of the problem in a finite number of steps. A special case of the proposed algorithm [8] solved without failure 80 consecutive cases of the LCP formulation of the knapsack feasibilty problem, ranging in size between 10 and 3000. 1 Introduction We consider the classical linear complementarity problem (LCP) [4, 12, 5] 0 x ?...
Modified ProjectionType Methods For Monotone Variational Inequalities
 SIAM Journal on Control and Optimization
, 1996
"... . We propose new methods for solving the variational inequality problem where the underlying function F is monotone. These methods may be viewed as projectiontype methods in which the projection direction is modified by a strongly monotone mapping of the form I \Gamma ffF or, if F is affine with un ..."
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Cited by 25 (9 self)
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. We propose new methods for solving the variational inequality problem where the underlying function F is monotone. These methods may be viewed as projectiontype methods in which the projection direction is modified by a strongly monotone mapping of the form I \Gamma ffF or, if F is affine with underlying matrix M , of the form I + ffM T , with ff 2 (0; 1). We show that these methods are globally convergent and, if in addition a certain error bound based on the natural residual holds locally, the convergence is linear. Computational experience with the new methods is also reported. Key words. Monotone variational inequalities, projectiontype methods, error bound, linear convergence. AMS subject classifications. 49M45, 90C25, 90C33 1. Introduction. We consider the monotone variational inequality problem of finding an x 2 X satisfying F (x ) T (x \Gamma x ) 0 8x 2 X; (1) where X is a closed convex set in ! n and F is a monotone and continuous function from ! n to ...
New Improved Error Bounds for the Linear Complementarity Problem
 Mathematical Programming
, 1994
"... New local and global error bounds are given for both nonmonotone and monotone linear complementarity problems. Comparisons of various residuals used in these error bounds are given. A possible candidate for a "best" error bound emerges from our comparisons as the sum of two natural residuals. Key Wo ..."
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Cited by 23 (5 self)
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New local and global error bounds are given for both nonmonotone and monotone linear complementarity problems. Comparisons of various residuals used in these error bounds are given. A possible candidate for a "best" error bound emerges from our comparisons as the sum of two natural residuals. Key Words: Linear complementarity, error bounds. Abbreviated Title: LCP Error Bounds. 1 Introduction We consider the classical linear complementarity problem [1, 8] of finding an x in the ndimensional real space IR n such that Mx+ q 0; x 0; x(Mx + q) = 0; (1) where M 2 IR n\Thetan and q 2 IR n . We denote this problem by LCP(M, q) for short. Let the solution set be ¯ X := fx j Mx+ q 0; x 0; x(Mx + q) = 0g: (2) We assume that ¯ X is nonempty. Define two natural residuals [3, 4, 9]: r(x) := kx \Gamma (x \Gamma Mx \Gamma q) + k (3) and s(x) := k(\GammaM x \Gamma q; \Gammax; x(Mx + q)) + k; (4) where k \Delta k is some norm. Note that x 2 ¯ X if and only if r(x) = 0 or s(x) = 0. Thes...
An implementable activeset algorithm for computing a Bstationary point of a mathematical program with linear complementarity constraints
 SIAM J. Optim
"... Abstract. In [3], an ɛactive set algorithm was proposed for solving a mathematical program with a smooth objective function and linear inequality/complementarity constraints. It is asserted therein that, under a uniform LICQ on the ɛfeasible set, this algorithm generates iterates whose cluster poi ..."
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Cited by 22 (4 self)
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Abstract. In [3], an ɛactive set algorithm was proposed for solving a mathematical program with a smooth objective function and linear inequality/complementarity constraints. It is asserted therein that, under a uniform LICQ on the ɛfeasible set, this algorithm generates iterates whose cluster points are Bstationary points of the problem. However, the proof has a gap and only shows that each cluster point is an Mstationary point. We discuss this gap and show that Bstationarity can be achieved if the algorithm is modified and an additional error bound condition holds. Key words. MPEC, Bstationary point, ɛactive set, error bound AMS subject classifications. 65K05, 90C30, 90C33
Convergence rates in forwardbackward splitting
, 1989
"... Forwardbackward splitting methods provide a range of approaches to solving largescale optimization problems and variational inequalities in which structure conducive to decomposition can be utilized. Apart from special cases where the forward step is absent and a version of the proximal point alg ..."
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Cited by 21 (3 self)
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Forwardbackward splitting methods provide a range of approaches to solving largescale optimization problems and variational inequalities in which structure conducive to decomposition can be utilized. Apart from special cases where the forward step is absent and a version of the proximal point algorithm comes out, efforts at evaluating the convergence potential of such methods have so far relied on Lipschitz properties and strong monotonicity, or inverse strong monotonicity, of the mapping involved in the forward step, the perspective mainly being that of projection algorithms. Here convergence is analyzed by a technique that allows properties of the mapping in the backward step to be brought in as well. For the first time in such a general setting, global and local contraction rates are derived, moreover in a form making it possible to determine the optimal step size relative to certain constants associated with the given problem. Insights are thereby gained into the effects of shifting strong monotonicity between the forward and backward mappings when a splitting is selected.
The Linear Complementarity Problem as a Separable Bilinear Program
 Journal of Global Optimization
, 1995
"... . The nonmonotone linear complementarity problem (LCP) is formulated as a bilinear program with separable constraints and an objective function that minimizesa natural error residual for the LCP. A linearprogrammingbasedalgorithm applied to the bilinear program terminates in a finite number of ste ..."
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Cited by 18 (4 self)
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. The nonmonotone linear complementarity problem (LCP) is formulated as a bilinear program with separable constraints and an objective function that minimizesa natural error residual for the LCP. A linearprogrammingbasedalgorithm applied to the bilinear program terminates in a finite number of steps at a solution or stationary point of the problem. The bilinear algorithm solved 80 consecutive cases of the LCP formulation of the knapsack feasibility problem ranging in size between 10 and 3000, with almost constant average number of major iterations equal to four. Keywords: linear complementarity, bilinear programming, knapsack 1. Introduction It is well known that the linear complementarity problem [4], [16] 0 x ? Mx+ q 0; (1) for a given n \Theta n real matrix M and a given n \Theta 1 vector q, can be written as the bilinear program min x;w fx 0 wjw = Mx+ q; x 0; w 0g: (2) For the case of a general M , considered here, the objective function of (2) is nonconvex and the cons...
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 11 (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.
New Inexact Parallel Variable Distribution Algorithms
 Computational Optimization and Applications
, 1997
"... Abstract. We consider the recently proposed parallel variable distribution (PVD) algorithm of Ferris and Mangasarian [4] for solving optimization problems in which the variables are distributed among p processors. Each processor has the primary responsibility for updating its block of variables whil ..."
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Cited by 11 (4 self)
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Abstract. We consider the recently proposed parallel variable distribution (PVD) algorithm of Ferris and Mangasarian [4] for solving optimization problems in which the variables are distributed among p processors. Each processor has the primary responsibility for updating its block of variables while allowing the remaining “secondary ” variables to change in a restricted fashion along some easily computable directions. We propose useful generalizations that consist, for the general unconstrained case, of replacing exact global solution of the subproblems by a certain natural sufficient descent condition, and, for the convex case, of inexact subproblem solution in the PVD algorithm. These modifications are the key features of the algorithm that has not been analyzed before. The proposed modified algorithms are more practical and make it easier to achieve good load balancing among the parallel processors. We present a general framework for the analysis of this class of algorithms and derive some new and improved linear convergence results for problems with weak sharp minima of order 2 and strongly convex problems. We also show that nonmonotone synchronization schemes are admissible, which further improves flexibility of PVD approach. Keywords: convergence parallel optimization, asynchronous algorithms, load balancing, unconstrained minimization, linear 1.