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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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Local behavior of an iterative framework for generalized equations with nonisolated solutions
 MATH. PROGRAM., SER. A
, 2002
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Predictorcorrector methods for sufficient linear complementarity problems in a wide neighborhood of the central path
 Optimization Methods and Software
"... Abstract. A higher order correctorpredictor interiorpoint method is proposed for solving sufficient linear complementarity problems. The algorithm produces a sequence of iterates in the N − ∞ neighborhood of the central path. The algorithm does not depend on the handicap κ of the problem. It has O ..."
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Cited by 14 (6 self)
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Abstract. A higher order correctorpredictor interiorpoint method is proposed for solving sufficient linear complementarity problems. The algorithm produces a sequence of iterates in the N − ∞ neighborhood of the central path. The algorithm does not depend on the handicap κ of the problem. It has O((1 + κ) √ nL) iteration complexity and is superlinearly convergent even for degenerate problems. Key words. neighborhood linear complementarity, interiorpoint, pathfollowing, correctorpredictor, wide AMS subject classifications. 90C51, 90C33 1. Introduction. The MTY predictorcorrector algorithm proposed by Mizuno, Todd and Ye [9] is a typical representative of a large class of MTY type predictorcorrector methods, which play a very important role among primaldual interior point methods. It was the first algorithm for linear programming (LP) that had both polynomial complexity and superlinear convergence. This result was extended to monotone
Qsuperlinear convergence of the iterates in primaldual interiorpoint methods
 MATH. PROGRAM., SER. A 91: 99–115 (2001)
, 2001
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A smoothing Newtontype algorithm of stronger convergence for the quadratically constrained convex quadratic programming
 Comput. Optim. Appl
"... In this paper we propose a smoothing Newtontype algorithm for the problem of minimizing a convex quadratic function subject to finitely many convex quadratic inequality constraints. The algorithm is shown to converge globally and possess stronger local superlinear convergence. Preliminary numerical ..."
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Cited by 8 (0 self)
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In this paper we propose a smoothing Newtontype algorithm for the problem of minimizing a convex quadratic function subject to finitely many convex quadratic inequality constraints. The algorithm is shown to converge globally and possess stronger local superlinear convergence. Preliminary numerical results are also reported. Key words Smoothing Newton method, global convergence, superlinear convergence.
Error Bounds For Quadratic Systems
 High Performance Optimization
, 1998
"... In this paper we consider the problem of estimating the distance from a given point to the solution set of a quadratic inequality system. We show, among other things, that a local error bound of order 1=2 holds for a system defined by linear inequalities and a single (nonconvex) quadratic equality. ..."
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Cited by 7 (2 self)
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In this paper we consider the problem of estimating the distance from a given point to the solution set of a quadratic inequality system. We show, among other things, that a local error bound of order 1=2 holds for a system defined by linear inequalities and a single (nonconvex) quadratic equality. We also give a sharpening of Lojasiewicz' error bound for piecewise quadratic functions. In contrast, the early results for this problem further require either a convexity or a nonnegativity assumption. 1 Introduction Consider a set S defined by an inequality system in ! n : S := fx 2 ! n j g 1 (x) 0; g 2 (x) 0; :::; g m (x) 0g (1.1) where each g i : ! n ! ! is a continuous function. We shall denote the vector function (g 1 ; g 2 ; :::; g m ) by g. A set S of the form above is sometimes called a zero set. A residual function r(x) for S is a nonnegative valued vector function with the property that r(x) = 0 if and only if x 2 S. A popular choice of residual function is given by r(x)...
Primaldual affine scaling interior point methods for linear complementarity problems
 SIAM JOURNAL ON OPTIMIZATION
"... A first order affine scaling method and two mth order affine scaling methods for solving monotone linear complementarity problems (LCP) are presented. All three methods produce iterates in a wide neighborhood of the central path. The first order method has O(nL2 (log nL2)(log log nL2)) iteration c ..."
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Cited by 6 (4 self)
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A first order affine scaling method and two mth order affine scaling methods for solving monotone linear complementarity problems (LCP) are presented. All three methods produce iterates in a wide neighborhood of the central path. The first order method has O(nL2 (log nL2)(log log nL2)) iteration complexity. If the LCP admits a strict complementary solution then both the duality gap and the iteration sequence converge superlinearly with Qorder two. If m = Ω(log ( √ nL)), then both higher order methods have O ( √ n)L iteration complexity. The Qorder of convergence of one of the methods is (m + 1) for problems that admit a strict complementarity solution while the Qorder of convergence of the other method is (m + 1)/2 for general monotone LCPs.
On the Rate of Local Convergence of HighOrderInfeasiblePathFollowing Algorithms for P*Linear Complementarity Problems
 Computational Optimization and Applications
, 1997
"... A simple and unified analysis is provided on the rate of local convergence for a class of highorderinfeasiblepathfollowing algorithms for the P linear complementarity problem (P LCP). It is shown that the rate of local convergence of a order algorithm with a centering step is + 1 if there ..."
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Cited by 5 (2 self)
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A simple and unified analysis is provided on the rate of local convergence for a class of highorderinfeasiblepathfollowing algorithms for the P linear complementarity problem (P LCP). It is shown that the rate of local convergence of a order algorithm with a centering step is + 1 if there is a strictly complementary solution and ( + 1)=2 otherwise. For the order algorithm without the centering step the corresponding rates are and =2, respectively. The algorithm without a centering step does not follow the fixed traditional central path. Instead, at each iteration, it follows a new analytic path connecting the current iterate with an optimal solution to generate the next iterate. An advantage of this algorithm is that it does not restrict iterates in a sequence of contracting neighborhoods of the central path.
Underlying paths in interior point methods for the monotone semidefinite linear complementarity problem
 Mathematical Programming, Series A
"... An interior point method (IPM) defines a search direction at each interior point of the feasible region. The search directions at all interior points together form a direction field, which gives rise to a system of ordinary differential equations (ODEs). Given an initial point in the interior of the ..."
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Cited by 4 (3 self)
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An interior point method (IPM) defines a search direction at each interior point of the feasible region. The search directions at all interior points together form a direction field, which gives rise to a system of ordinary differential equations (ODEs). Given an initial point in the interior of the feasible region, the unique solution of the ODE system is a curve passing through the point, with tangents parallel to the search directions along the curve. We call such curves offcentral paths. We study offcentral paths for the monotone semidefinite linear complementarity problem (SDLCP). We show that each offcentral path is a welldefined analytic curve with parameter µ ranging over (0,∞) and any accumulation point of the offcentral path is a solution to SDLCP. Through a simple example we show that the offcentral paths are not analytic as a function of µ and have first derivatives which are unbounded as a function of µ at µ = 0 in general. On the other hand, for the same example, we can find a subset of offcentral paths which are analytic at µ = 0. These “nice ” paths are characterized by some algebraic equations.