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 542 (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.
Superlinear convergence of a symmetric primaldual path following algorithm for semidefinite programming
 SIAM J. on Optimization
, 1998
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Convergence of Interior Point Algorithms for the Monotone Linear Complementarity Problem
, 1994
"... The literature on interior point algorithms shows impressive results related to the speed of convergence of the objective values, but very little is known about the convergence of the iterate sequences. This paper studies the horizontal linear complementarity problem, and derives general convergence ..."
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Cited by 23 (4 self)
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The literature on interior point algorithms shows impressive results related to the speed of convergence of the objective values, but very little is known about the convergence of the iterate sequences. This paper studies the horizontal linear complementarity problem, and derives general convergence properties for algorithms based on Newton iterations. This problem provides a simple and general framework for most existing primaldual interior point methods. The conclusion is that most of the published algorithms of this kind generate convergent sequences. In many cases (whenever the convergence is not too fast in a certain sense), the sequences converge to the analytic center of the optimal face.
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 13 (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.
Superlinear convergence of an algorithm for monotone linear complementarity problems, when no strictly complementary solution exists
 Mathematics of Operations Research
, 1999
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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...
On the convergence of the iteration sequence of infeasible path following algorithms for linear complementarity problems (Revised version)
, 1996
"... A generalized class of infeasibleinteriorpoint methods for solving horizontal linear complementarity problem is analyzed and sufficient conditions are given for the convergence of the sequence of iterates produced by methods in this class. In particular it is shown that the largest step path follo ..."
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Cited by 10 (7 self)
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A generalized class of infeasibleinteriorpoint methods for solving horizontal linear complementarity problem is analyzed and sufficient conditions are given for the convergence of the sequence of iterates produced by methods in this class. In particular it is shown that the largest step path following algorithms generates convergent iterates even when starting from infeasible points. The computational complexity of the latter method is discussed in detail and its local convergent rate is analyzed. The primaldual gap of the iterates produced by this method is superlinearly convergent to zero. A variant of the method has quadratic convergence.
Examples of illbehaved central paths in convex optimization
, 2005
"... This paper presents some examples of illbehaved central paths in convex optimization. Some contain infinitely many fixed length central segments; others manifest oscillations with infinite variation. These central paths can be encountered even for infinitely differentiable data. ..."
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Cited by 5 (0 self)
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This paper presents some examples of illbehaved central paths in convex optimization. Some contain infinitely many fixed length central segments; others manifest oscillations with infinite variation. These central paths can be encountered even for infinitely differentiable data.
A Strongly Polynomial Rounding Procedure Yielding a Maximally Complementary Solution for P*(κ) Linear Complementarity Problems
, 1998
"... We deal with Linear Complementarity Problems (LCPs) with P () matrices. First we establish the convergence rate of the complementary variables along the central path. The central path is parameterized by the barrier parameter , as usual. Our elementary proof reproduces the known result that the var ..."
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Cited by 5 (4 self)
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We deal with Linear Complementarity Problems (LCPs) with P () matrices. First we establish the convergence rate of the complementary variables along the central path. The central path is parameterized by the barrier parameter , as usual. Our elementary proof reproduces the known result that the variables on, or close to the central path fall apart in three classes in which these variables are O(1); O() and O( p ), respectively. The constants hidden in these bounds are expressed in, or bounded by, the input data. All this is preparation for our main result: a strongly polynomial rounding procedure. Given a point with sufficiently small complementarity gap and close enough to the central path, the rounding procedure produces a maximally complementary solution in at most O(n³) arithmetic operations. The result implies that Interior Point Methods (IPMs) not only converge to a complementary solution of P () LCPs but, when furnished with our rounding procedure, they can produce a max...
Nondegenerate Solutions and Related Concepts in Affine Variational Inequalities
 SIAM J. ON CONTROL AND OPTIMIZATION
, 1996
"... The notion of a strictly complementary solution for complementarity problems is extended to that of a nondegenerate solution of variational inequalities. Several equivalent formulations of nondegeneracy are given. In the affine case, an existence theorem for a nondegenerate solution is given in ter ..."
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Cited by 4 (1 self)
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The notion of a strictly complementary solution for complementarity problems is extended to that of a nondegenerate solution of variational inequalities. Several equivalent formulations of nondegeneracy are given. In the affine case, an existence theorem for a nondegenerate solution is given in terms of several related concepts which are shown to be equivalent in this context. These include a weak sharp minimum, the minimum principle sufficiency, and error bounds. The gap function associated with the variational inequality plays a central role in this existence theorem.