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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 s ..."
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Cited by 60 (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.
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...
Algorithms For Complementarity Problems And Generalized Equations
, 1995
"... Recent improvements in the capabilities of complementarity solvers have led to an increased interest in using the complementarity problem framework to address practical problems arising in mathematical programming, economics, engineering, and the sciences. As a result, increasingly more difficult pr ..."
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Cited by 49 (5 self)
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Recent improvements in the capabilities of complementarity solvers have led to an increased interest in using the complementarity problem framework to address practical problems arising in mathematical programming, economics, engineering, and the sciences. As a result, increasingly more difficult problems are being proposed that exceed the capabilities of even the best algorithms currently available. There is, therefore, an immediate need to improve the capabilities of complementarity solvers. This thesis addresses this need in two significant ways. First, the thesis proposes and develops a proximal perturbation strategy that enhances the robustness of Newtonbased complementarity solvers. This strategy enables algorithms to reliably find solutions even for problems whose natural merit functions have strict local minima that are not solutions. Based upon this strategy, three new algorithms are proposed for solving nonlinear mixed complementarity problems that represent a significant improvement in robustness over previous algorithms. These algorithms have local Qquadratic convergence behavior, yet depend only on a pseudomonotonicity assumption to achieve global convergence from arbitrary starting points. Using the MCPLIB and GAMSLIB test libraries, we perform extensive computational tests that demonstrate the effectiveness of these algorithms on realistic problems. Second, the thesis extends some previously existing algorithms to solve more general problem classes. Specifically, the NE/SQP method of Pang & Gabriel (1993), the semismooth equations approach of De Luca, Facchinei & Kanz...
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 19 (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...
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...
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...
An InfeasibleInteriorPoint PredictorCorrector Algorithm for the P4Geometric LCP
 Department of Mathematics, The University of Iowa, Iowa City, Iowa
, 1994
"... A P Geometric linear complementarity problem (P GP) as a generalization of the monotone geometric linear complementarity problem is introduced. In particular, it contains the monotone standard linear complementarity problem and the horizontal linear complementarity problem. Linear and quadratic p ..."
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Cited by 12 (9 self)
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A P Geometric linear complementarity problem (P GP) as a generalization of the monotone geometric linear complementarity problem is introduced. In particular, it contains the monotone standard linear complementarity problem and the horizontal linear complementarity problem. Linear and quadratic programming problems can be expressed in a "natural" way (i. e. , without any change of variables) as P GP. It is shown that the algorithm of Mizuno et al. [6] can be extended to solve the P GP. The extended algorithm is globally convergent and its computational complexity depends on the quality of the starting points. The algorithm is quadratically convergent for problems having a strictly complementary solution. Key Words:P matrix, linear complementarity problems, predictorcorrector, infeasibleinterior point algorithm, polynomiality, quadratic convergence. Abbreviated Title: Algorithm for P GP Department of Mathematics, University of Iowa, Iowa City, IA 52242, USA. y Department...
Global and Local Convergence of PredictorCorrector InfeasibleInteriorPoint Algorithms for Semidefinite Programs
, 1995
"... . The purpose of this technical report is twofold. The one is to present a globally convergent, predictorcorrector, primaldual, infeasibleinteriorpoint algorithm for SDPs (semidefinite programs). The algorithm is a special case of the generic interiorpoint algorithm (with a minor modification) ..."
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Cited by 12 (1 self)
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. The purpose of this technical report is twofold. The one is to present a globally convergent, predictorcorrector, primaldual, infeasibleinteriorpoint algorithm for SDPs (semidefinite programs). The algorithm is a special case of the generic interiorpoint algorithm (with a minor modification) proposed by Kojima, Shindoh and Hara [10] for SDLCPs (semidefinite linear complementarity problems). The other purpose is to study its local convergence; we show that a variation of the algorithm enjoys the superlinear convergence under a primaldual nondegeneracy condition. Many interiorpoint algorithms [1, 3, 5, 6, 7, 16, 20, 21, etc.] have been already developed so far for SDPs, but the local convergence analysis has not been made for those algorithms. We place our main emphasis on the local convergence analysis. Quite recently, there has been made much progress in the primaldual interiorpoint algorithms for SDPs. Monteiro [15] devised a new formulation of the primaldual search direct...
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.
A Path Following Method for LCP with Superlinearly Convergent Iteration Sequence
 Department of Mathematics, The University of Iowa, Iowa City, IA
, 1995
"... A new algorithm for solving linear complementarity problems with sufficient matrices is proposed. If the problem has a solution the algorithm is superlinearly convergent from any positive starting points, even for degenerate problems. Each iteration requires only one matrix factorization and at most ..."
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Cited by 10 (9 self)
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A new algorithm for solving linear complementarity problems with sufficient matrices is proposed. If the problem has a solution the algorithm is superlinearly convergent from any positive starting points, even for degenerate problems. Each iteration requires only one matrix factorization and at most two backsolves. Only one backsolve is necessary if the problem is known to be nondegenerate. The algorithm generates points in a large neighborhood of the central path and has the lowest iteration complexity obtained so far in the literature. Moreover, the iteration sequence converges superlinearly to a maximal solution with the same Qorder as the complementarity sequence. Key Words: linear complementarity problems, sufficient matrices, P matrices, pathfollowing, infeasibleinteriorpoint algorithm, polynomiality, superlinear convergence. Abbreviated Title: A method for LCP. Department of Mathematics, University of Iowa, Iowa City, IA 52242, USA. The work of this author was supporte...