Results 1  10
of
24
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 777 (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.
S.: Superlinear convergence of a symmetric primaldual pathfollowing algorithm for semidefinite programming
 SIAM Journal on Optimization
, 1998
"... Abstract This paper establishes the superlinear convergence of a symmetric primaldual path following algorithm for semidenite programming under the assumptions that the semidenite pro gram has a strictly complementary primaldual optimal solution and that the size of the central path neighborhood te ..."
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Cited by 63 (5 self)
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Abstract This paper establishes the superlinear convergence of a symmetric primaldual path following algorithm for semidenite programming under the assumptions that the semidenite pro gram has a strictly complementary primaldual optimal solution and that the size of the central path neighborhood tends to zero The interior point algorithm considered here closely resembles the MizunoToddYe predictorcorrectormethod for linear programmingwhere it is known to be quadrat ically convergent It is shown that when the iterates are well centered the duality gap is reduced superlinearly after each predictor step Indeed if each predictor step is succeeded by r consecutive corrector steps then the predictor reduces the duality gap superlinearlywith order
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 25 (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.
Superlinear convergence of an algorithm for monotone linear complementarity problems, when no strictly complementary solution exists
 Math. Oper. Res
, 1999
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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.
On the existence and convergence of the central path for convex programming and some duality results
 Computational Optimization and Applications
, 1998
"... This paper gives several equivalent conditions which guarantee the existence of the weighted central paths for a given convex programming problem satisfying some mild conditions. When the objective and constraint functions of the problem are analytic, we also characterize the limiting behavior of t ..."
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Cited by 12 (5 self)
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This paper gives several equivalent conditions which guarantee the existence of the weighted central paths for a given convex programming problem satisfying some mild conditions. When the objective and constraint functions of the problem are analytic, we also characterize the limiting behavior of these paths as they approach the set of optimal solutions. A duality relationship between a certain pair of logarithmic barrier problems is also discussed. 1
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.
R.: Error bounds and limiting behavior of weighted paths associated with the sdp map X 1/2 SX 1/2
 SIAM Journal on Optimization
, 2005
"... Abstract. This paper studies the limiting behavior of weighted infeasible central paths for semidefinite programming (SDP) obtained from centrality equations of the form X1/2SX1/2 = νW, where W is a fixed positive definite matrix and ν> 0 is a parameter, under the assumption that the problem has ..."
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Cited by 10 (2 self)
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Abstract. This paper studies the limiting behavior of weighted infeasible central paths for semidefinite programming (SDP) obtained from centrality equations of the form X1/2SX1/2 = νW, where W is a fixed positive definite matrix and ν> 0 is a parameter, under the assumption that the problem has a strictly complementary primaldual optimal solution. It is shown that a weighted central path as a function of ν can be extended analytically beyond 0 and hence that the path converges as ν ↓ 0. Characterization of the limit points of the path and its normalized firstorder derivatives are also provided. As a consequence, it is shown that a weighted central path can have two types of behavior: it converges either as Θ(ν) or as Θ( ν) depending on whether the matrix W on a certain scaled space is block diagonal or not, respectively. We also derive an error bound on the distance between a point lying in a certain neighborhood of the central path and the set of primaldual optimal solutions. Finally, in light of the results of this paper, we give a characterization of a sufficient condition proposed by Potra and Sheng which guarantees the superlinear convergence of a class of primaldual interiorpoint SDP algorithms.
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 7 (3 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...