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45
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...
Superlinear Convergence Of A Symmetric PrimalDual Path Following Algorithm For Semidefinite Programming
 SIAM Journal on Optimization
, 1996
"... This paper establishes the superlinear convergence of a symmetric primaldual path following algorithm for semidefinite programming under the assumptions that the semidefinite program has a strictly complementary primaldual optimal solution and that the size of the central path neighborhood tends t ..."
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Cited by 50 (5 self)
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This paper establishes the superlinear convergence of a symmetric primaldual path following algorithm for semidefinite programming under the assumptions that the semidefinite program 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 MizunoTodd Ye predictorcorrector method for linear programming which is known to be quadratically 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 superlinearly with order 2 1+2 \Gamma2r . The proof relies on a careful analysis of the central path for semidefinite programming. It is shown that under the strict complementarity assumption, the primaldual central path converges to the analytic center of the primald...
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 46 (3 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 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 ...
A Global and Local Superlinear ContinuationSmoothing Method for ... and Monotone NCP
 SIAM J. Optim
, 1997
"... We propose a continuation method for a class of nonlinear complementarity problems(NCPs), including the NCP with a P 0 and R 0 function and the monotone NCP with a feasible interior point. The continuation method is based on a class of ChenMangasarian smooth functions. Unlike many existing continua ..."
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Cited by 24 (6 self)
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We propose a continuation method for a class of nonlinear complementarity problems(NCPs), including the NCP with a P 0 and R 0 function and the monotone NCP with a feasible interior point. The continuation method is based on a class of ChenMangasarian smooth functions. Unlike many existing continuation methods, the method follows the noninterior smoothing paths, and as a result, an initial point can be easily constructed. In addition, we introduce a procedure to dynamically update the neighborhoods associated with the smoothing paths, so that the algorithm is both globally convergent and locally superlinearly convergent under suitable assumptions. Finally, a hybrid continuationsmoothing method is proposed and is shown to have the same convergence properties under weaker conditions. 1 Introduction Let F : R n ! R n be a continuously differentiable function. The nonlinear complementarity problem, denoted by NCP(F ), is to find a vector (x; y) 2 R n \Theta R n such that F (x)...
On HomotopySmoothing Methods for Variational Inequalities
"... A variational inequality problem with a mapping g : ! n ! ! n and lower and upper bounds on variables can be reformulated as a system of nonsmooth equations F (x) = 0 in ! n . Recently, several homotopy methods, such as interiorpoint and smoothing methods, have been employed to solve the prob ..."
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Cited by 23 (5 self)
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A variational inequality problem with a mapping g : ! n ! ! n and lower and upper bounds on variables can be reformulated as a system of nonsmooth equations F (x) = 0 in ! n . Recently, several homotopy methods, such as interiorpoint and smoothing methods, have been employed to solve the problem. All of these methods use parametric functions and construct perturbed equations to approximate the problem. The solution to the perturbed system constitutes a smooth trajectory leading to the solution of the original variational inequality problem. The methods generate iterates to follow the trajectory. Among these methods ChenMangasarian and GabrielMor'e proposed a class of smooth functions to approximate F . In this paper, we study several properties of the trajectory defined by solutions of these smooth systems. We propose a homotopysmoothing method for solving the variational inequality problem, and show that the method converges globally and superlinearly under mild conditions. ...
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.
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 21 (4 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 ...
A Global Linear and Local Quadratic Continuation Smoothing Method for Variational Inequalities with Box Constraints
, 1997
"... In this paper, we propose a continuation method for box constrained variational inequality problems. The continuation method is based on the class of GabrielMor'e smooth functions and has the following attractive features: It can start from any point; It has a simple and natural neighborhood defini ..."
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Cited by 19 (3 self)
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In this paper, we propose a continuation method for box constrained variational inequality problems. The continuation method is based on the class of GabrielMor'e smooth functions and has the following attractive features: It can start from any point; It has a simple and natural neighborhood definition; It solves only one approximate Newton equation at each iteration; It converges globally linearly and locally quadratically under nondegeneracy assumption at the solution point and other suitable assumptions. A hybrid method is also presented, which is shown to preserve the above convergence properties without the nondegeneracy assumption at the solution point. In particular, the hybrid method converges finitely for affine problems. 1 Introduction Let F : R n ! R n be a continuously differentiable function. Let l 2 fR [ \Gamma1g n and u 2 fR [1g n such that l ! u. The variational inequality problem (VIP) with box constraints, denoted by VIP(l; u; F ), is to find x 2 [l; u] such...