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28
A PathFollowing InfeasibleInteriorPoint Algorithm for Linear Complementarity Problems
 Optimization Methods and Software
, 1993
"... We describe an infeasibleinteriorpoint algorithm for monotone linear complementarity problems that has polynomial complexity, global linear convergence, and local superlinear convergence with a Qorder of 2. Only one matrix factorization is required per iteration, and the analysis assumes only tha ..."
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Cited by 50 (10 self)
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We describe an infeasibleinteriorpoint algorithm for monotone linear complementarity problems that has polynomial complexity, global linear convergence, and local superlinear convergence with a Qorder of 2. Only one matrix factorization is required per iteration, and the analysis assumes only that a strictly complementary solution exists. 1 Introduction The monotone linear complementarity problem is to find a vector pair (x; y) 2 IR n \Theta IR n such that y = Mx+ h; (x; y) (0; 0); x T y = 0; (1) where h 2 IR n and M is an n \Theta n positive semidefinite matrix. A vector pair (x ; y ) is called a strictly complementary solution of (1) if it satisfies the three conditions in (1) and, in addition, x i + y i ? 0 for each component i = 1; 2; \Delta \Delta \Delta ; n. We denote the solution set for (1) by S and the set of strictly complementary solutions by S c . A number of interior point methods have been proposed for (1). Among recent papers are the predictor...
A Superlinearly Convergent PrimalDual InfeasibleInteriorPoint Algorithm for Semidefinite Programming
 Department of Mathematics, The University of Iowa, Iowa City, IA
, 1995
"... . A primaldual infeasibleinteriorpoint pathfollowing algorithm is proposed for solving semidefinite programming (SDP) problems. If the problem has a solution, then the algorithm is globally convergent. If the starting point is feasible or close to being feasible, the algorithms finds an optimal ..."
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Cited by 48 (9 self)
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. A primaldual infeasibleinteriorpoint pathfollowing algorithm is proposed for solving semidefinite programming (SDP) problems. If the problem has a solution, then the algorithm is globally convergent. If the starting point is feasible or close to being feasible, the algorithms finds an optimal solution in at most O( p nL) iterations, where n is the size of the problem and L is the logarithm of the ratio of the initial error and the tolerance. If the starting point is large enough then the algorithm terminates in at most O(nL) steps either by finding a solution or by determining that the primaldual problem has no solution of norm less than a given number. Moreover, we propose a sufficient condition for the superlinear convergence of the algorithm. In addition, we give two special cases of SDP for which the algorithm is quadratically convergent. Key words. semidefinite programming, pathfollowing, infeasibleinteriorpoint algorithm, polynomiality, superlinear convergence. AMS ...
A superlinearly convergent predictorcorrector method for degenerate LCP in a wide neighborhood of the central path with O (√n L)iteration complexity
, 2006
"... ..."
Local Convergence of InteriorPoint Algorithms for Degenerate Monotone LCP
 Computational Optimization and Applications
, 1993
"... Most asymptotic convergence analysis of interiorpoint algorithms for monotone linear complementarity problems assumes that the problem is nondegenerate, that is, the solution set contains a strictly complementary solution. We investigate the behavior of these algorithms when this assumption is remo ..."
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Cited by 34 (4 self)
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Most asymptotic convergence analysis of interiorpoint algorithms for monotone linear complementarity problems assumes that the problem is nondegenerate, that is, the solution set contains a strictly complementary solution. We investigate the behavior of these algorithms when this assumption is removed. 1 Introduction In the monotone linear complementarity problem (LCP), we seek a vector pair (x; y) 2 IR n \Theta IR n that satisfies the conditions y = Mx+ q; x 0; y 0; x T y = 0; (1) where q 2 IR n , and M 2 IR n\Thetan is positive semidefinite. We use S to denote the solution set of (1). An assumption that is frequently made in order to prove superlinear convergence of interiorpoint algorithms for (1) is the nondegeneracy assumption: Assumption 1 There is an (x ; y ) 2 S such that x i + y i ? 0 for all i = 1; \Delta \Delta \Delta ; n. In general, we can define three subsets B, N , and J of the index set f1; \Delta \Delta \Delta ; ng by B = fi = 1; \Delta ...
Interior Point Methods For Optimal Control Of DiscreteTime Systems
 Journal of Optimization Theory and Applications
, 1993
"... . We show that recently developed interior point methods for quadratic programming and linear complementarity problems can be put to use in solving discretetime optimal control problems, with general pointwise constraints on states and controls. We describe interior point algorithms for a discrete ..."
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Cited by 31 (5 self)
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. We show that recently developed interior point methods for quadratic programming and linear complementarity problems can be put to use in solving discretetime optimal control problems, with general pointwise constraints on states and controls. We describe interior point algorithms for a discrete time linearquadratic regulator problem with mixed state/control constraints, and show how it can be efficiently incorporated into an inexact sequential quadratic programming algorithm for nonlinear problems. The key to the efficiency of the interiorpoint method is the narrowbanded structure of the coefficient matrix which is factorized at each iteration. Key words. interior point algorithms, optimal control, banded linear systems. 1. Introduction. The problem of optimal control of an initial value ordinary differential equation, with Bolza objectives and mixed constraints, is min x;u Z T 0 L(x(t); u(t); t) dt + OE f (x(T )); x(t) = f(x(t); u(t); t); x(0) = x init ; (1.1) g(x(t); u(...
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.
Stability Of Linear Equations Solvers In InteriorPoint Methods
 SIAM J. Matrix Anal. Appl
, 1994
"... . Primaldual interiorpoint methods for linear complementarity and linear programming problems solve a linear system of equations to obtain a modified Newton step at each iteration. These linear systems become increasingly illconditioned in the later stages of the algorithm, but the computed steps ..."
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Cited by 17 (2 self)
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. Primaldual interiorpoint methods for linear complementarity and linear programming problems solve a linear system of equations to obtain a modified Newton step at each iteration. These linear systems become increasingly illconditioned in the later stages of the algorithm, but the computed steps are often sufficiently accurate to be useful. We use error analysis techniques tailored to the special structure of these linear systems to explain this observation and examine how theoretically superlinear convergence of a pathfollowing algorithm is affected by the roundoff errors. Key words. primaldual interiorpoint methods, error analysis, stability AMS(MOS) subject classifications. 65G05, 65F05, 90C33 1. Introduction. The monotone linear complementarity problem (LCP) is the problem of finding a vector pair (x; y) 2 R l n \Theta R l n such that y = Mx+ q; (x; y) 0; x T y = 0; (1) where M (a real, n \Theta n positive semidefinite matrix) and q (a real vector with n elements...
A Superquadratic InfeasibleInteriorPoint Method for Linear Complementarity Problems
 Preprint MCSP4180294, Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL 60439
, 1996
"... We consider a modification of a pathfollowing infeasibleinteriorpoint algorithm described by Wright. In the new algorithm, we attempt to improve each major iterate by reusing the coefficient matrix factors from the latest step. We show that the modified algorithm has similar theoretical global co ..."
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Cited by 16 (1 self)
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We consider a modification of a pathfollowing infeasibleinteriorpoint algorithm described by Wright. In the new algorithm, we attempt to improve each major iterate by reusing the coefficient matrix factors from the latest step. We show that the modified algorithm has similar theoretical global convergence properties to those of the earlier algorithm, while its asymptotic convergence rate can be made superquadratic by an appropriate parameter choice. 1 Introduction We describe an algorithm for solving the monotone linear complementarity problem (LCP), in which we aim to find a vector pair (x; y) with y = Mx+ q; (x; y) 0; x T y = 0; (1) where q 2 IR n and M is an n \Theta n positive semidefinite matrix. The solution set to (1) is denoted by S, while the set S c of strictly complementary solutions is defined as S c = f(x ; y ) 2 S j x + y ? 0g: Our algorithm can be viewed as a modified form of Newton's method applied to the 2n \Theta 2n system y = Mx+ q; x i y i...
Superlinear Convergence Of An Algorithm For Monotone Linear Complementarity Problems, When No Strictly Complementary Solution Exists
 Mathematics of Operations Research
, 1996
"... A new predictorcorrector interior point algorithm for solving monotone linear complementarity problems (LCP) is proposed, and it is shown to be superlinearly convergent with at least order 1.5, even if the LCP has no strictly complementary solution. Unlike Mizuno's recent algorithm [16], the fast ..."
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Cited by 12 (2 self)
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A new predictorcorrector interior point algorithm for solving monotone linear complementarity problems (LCP) is proposed, and it is shown to be superlinearly convergent with at least order 1.5, even if the LCP has no strictly complementary solution. Unlike Mizuno's recent algorithm [16], the fast local convergence is attained without any need for estimating the optimal partition. In the special case that a strictly complementary solution does exist, the order of convergence becomes quadratic. The proof relies on an investigation of the asymptotic behavior of first and second order derivatives that are associated with trajectories of weighted centers for LCP. AMS 1991 subject classification: 90C33. Key words. monotone linear complementarity problem, primaldual interior point method, superlinear convergence, central path. 1 1. Introduction Given n \Theta n real matrices Q and R and a real vector b of order n, the horizontal linear complementarity problem (LCP) is the problem of fin...
New Complexity Analysis of the PrimalDual Newton Method for Linear Optimization
, 1998
"... We deal with the primaldual Newton method for linear optimization (LO). Nowadays, this method is the working horse in all efficient interior point algorithms for LO, and its analysis is the basic element in all polynomiality proofs of such algorithms. At present there is still a gap between the pra ..."
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Cited by 11 (7 self)
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We deal with the primaldual Newton method for linear optimization (LO). Nowadays, this method is the working horse in all efficient interior point algorithms for LO, and its analysis is the basic element in all polynomiality proofs of such algorithms. At present there is still a gap between the practical behavior of the algorithms and the theoretical performance results, in favor of the practical behavior. This is especially true for socalled largeupdate methods. We present some new analysis tools, based on a proximity measure introduced by Jansen et al., in 1994, that may help to close this gap. This proximity measure has not been used in the analysis of largeupdate method before. Our new analysis not only provides a unified way for the analysis of both largeupdate and smallupdate methods, but also improves the known iteration bounds.