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Local Convergence of PredictorCorrector InfeasibleInteriorPoint Algorithms for SDPs and SDLCPs
 Mathematical Programming
, 1997
"... . An example of SDPs (semidefinite programs) exhibits a substantial difficulty in proving the superlinear convergence of a direct extension of the MizunoToddYe type predictorcorrector primaldual interiorpoint method for LPs (linear programs) to SDPs, and suggests that we need to force the genera ..."
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Cited by 58 (4 self)
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. An example of SDPs (semidefinite programs) exhibits a substantial difficulty in proving the superlinear convergence of a direct extension of the MizunoToddYe type predictorcorrector primaldual interiorpoint method for LPs (linear programs) to SDPs, and suggests that we need to force the generated sequence to converge to a solution tangentially to the central path (or trajectory). A MizunoToddYe type predictorcorrector infeasibleinteriorpoint algorithm incorporating this additional restriction for monotone SDLCPs (semidefinite linear complementarity problems) enjoys superlinear convergence under strict complementarity and nondegeneracy conditions. Key words. Semidefinite Programming, InfeasibleInteriorPoint Method, PredictorCorrectorMethod, Superlinear Convergence, PrimalDual Nondegeneracy Abbreviated Title. InteriorPoint Algorithms for SDPs y Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, 2121 OhOkayama, Meguroku, Tokyo 152, Japa...
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 56 (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 predictorcorrector method for degenerate LCP in a wide neighborhood of the central path with O (√n L)iteration complexity
, 2006
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On a Homogeneous Algorithm for the Monotone Complementarity Problem
 Mathematical Programming
, 1995
"... We present a generalization of a homogeneous selfdual linear programming (LP) algorithm to solving the monotone complementarity problem (MCP). The algorithm does not need to use any "bigM" parameter or twophase method, and it generates either a solution converging towards feasibility an ..."
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Cited by 40 (3 self)
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We present a generalization of a homogeneous selfdual linear programming (LP) algorithm to solving the monotone complementarity problem (MCP). The algorithm does not need to use any "bigM" parameter or twophase method, and it generates either a solution converging towards feasibility and complementarity simultaneously or a certificate proving infeasibility. Moreover, if the MCP is polynomially solvable with an interior feasible starting point, then it can be polynomially solved without using or knowing such information at all. To our knowledge, this is the first interiorpoint and infeasiblestarting algorithm for solving the MCP that possesses these desired features. Preliminary computational results are presented. Key words: Monotone complementarity problem, homogeneous and selfdual, infeasiblestarting algorithm. Running head: A homogeneous algorithm for MCP. Department of Management, Odense University, Campusvej 55, DK5230 Odense M, Denmark, email: eda@busieco.ou.dk. y De...
Superlinear convergence of an interiorpoint method despite dependent constraints
 Preprint ANL/MCSP6221196, Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, Ill
, 1996
"... Abstract. We show that an interiorpoint method for monotone variational inequalities exhibits superlinear convergence provided that all the standard assumptions hold except for the wellknown assumption that the Jacobian of the active constraints has full rank at the solution. We show that superlin ..."
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Cited by 39 (10 self)
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Abstract. We show that an interiorpoint method for monotone variational inequalities exhibits superlinear convergence provided that all the standard assumptions hold except for the wellknown assumption that the Jacobian of the active constraints has full rank at the solution. We show that superlinear convergence occurs even when the constantrank condition on the Jacobian assumed in an earlier work does not hold. AMS(MOS) subject classi cations. 90C33, 90C30, 49M45 1. Introduction. We
A PathFollowing InteriorPoint Algorithm for Linear and Quadratic Problems
 Preprint MCSP4011293, Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL 60439
, 1995
"... We describe an algorithm for the monotone linear complementarity problem (LCP) that converges from any positive, not necessarily feasible, starting point and exhibits polynomial complexity if some additional assumptions are made on the starting point. If the problem has a strictly complementary solu ..."
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Cited by 23 (5 self)
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We describe an algorithm for the monotone linear complementarity problem (LCP) that converges from any positive, not necessarily feasible, starting point and exhibits polynomial complexity if some additional assumptions are made on the starting point. If the problem has a strictly complementary solution, the method converges subquadratically. We show that the algorithm and its convergence properties extend readily to the mixed monotone linear complementarity problem and, hence, to all the usual formulations of the linear programming and convex quadratic programming problems. 1 Introduction The monotone linear complementarityproblem (LCP) is to find a vector pair (x; y) 2 IR n \ThetaIR n such that 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 (p.s.d.) matrix. The mixed monotone linear complementarity problem (MLCP) is to find a vector triple (x; y; z) 2 IR n \Theta IR n \Theta IR m such that " y 0 # = " M 11 M 12 ...
Superlinear and Quadratic Convergence of AffineScaling InteriorPoint Newton Methods for Problems with Simple Bounds without Strict Complementarity Assumption
, 1998
"... A class of affinescaling interiorpoint methods for bound constrained optimization problems is introduced which are locally qsuperlinear or qquadratic convergent. It is assumed that the strong... ..."
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Cited by 22 (3 self)
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A class of affinescaling interiorpoint methods for bound constrained optimization problems is introduced which are locally qsuperlinear or qquadratic convergent. It is assumed that the strong...
An O(nL) infeasibleinteriorpoint algorithm for LCP with quadratic convergence
 Department of Mathematics, The University of Iowa, Iowa City, IA
, 1994
"... The MizunoToddYe predictorcorrector algorithm for linear programming is extended for solving monotone linear complementarity problems from infeasible starting points. The proposed algorithm requires two matrix factorizations and at most three backsolves per iteration. Its computational complexity ..."
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Cited by 21 (10 self)
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The MizunoToddYe predictorcorrector algorithm for linear programming is extended for solving monotone linear complementarity problems from infeasible starting points. The proposed algorithm requires two matrix factorizations and at most three backsolves per iteration. Its computational complexity depends on the quality of the starting point. If the starting points are large enough then the algorithm has O(nL) iteration complexity. If a certain measure of feasibility at the starting point is small enough then the algorithm has O( p nL) iteration complexity. At each iteration both "feasibility' and "optimality" are reduced exactly at the same rate. The algorithm is quadratically convergent for problems having a strictly complementary solution, and therefore its asymptotic efficiency index is p 2. A variant of the algorithm can be used to detect whether solutions with norm less than a given constant exist. . Key Words:linear complementarity problems, predictorcorrector, infeasib...
Superlinear convergence of an algorithm for monotone linear complementarity problems, when no strictly complementary solution exists
 Math. Oper. Res
, 1999
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Predictorcorrector algorithms for solving P*matrix Lcp from arbitrary positive starting points
, 1994
"... A new predictorcorrector algorithm is proposed for solving P ()matrix linear complementarity problems. If the problem is solvable, then the algorithm converges from an arbitrary positive starting point (x 0 ; s 0 ). The computational complexity of the algorithm depends on the quality of the s ..."
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Cited by 17 (10 self)
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A new predictorcorrector algorithm is proposed for solving P ()matrix linear complementarity problems. If the problem is solvable, then the algorithm converges from an arbitrary positive starting point (x 0 ; s 0 ). The computational complexity of the algorithm depends on the quality of the starting point. If the starting point is feasible or close to being feasible, it has O((1+) p n=ae 0 L)iteration complexity, where ae 0 is the ratio of the smallest and average coordinate of X 0 s 0 . With appropriate initialization, a modified version of the algorithm terminates in O((1 + ) 2 (n=ae 0 )L) steps either by finding a solution or by determining that the problem is not solvable. The algorithm is quadratically convergent for problems having a strictly complementary solution. We also propose an extension of a recent algorithm of Mizuno to P ()matrix linear complementarity problems such that it can start from arbitrary positive points and has superlinear convergence withou...