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15
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 ...
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 41 (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...
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 and compleme ..."
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Cited by 24 (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...
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 13 (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 11 (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...
Equivalence between Different Formulations of the Linear Complementarity Problem
, 1995
"... One shows that different formulations of the linear complementarity problem (LCP), such as the horizontal LCP, the mixed LCP and the geometric LCP can be transformed into a standard LCP. The P ()property of the corresponding formulations as well as the convergence properties of a large class of in ..."
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Cited by 11 (10 self)
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One shows that different formulations of the linear complementarity problem (LCP), such as the horizontal LCP, the mixed LCP and the geometric LCP can be transformed into a standard LCP. The P ()property of the corresponding formulations as well as the convergence properties of a large class of interiorpoint algorithms are invariant with respect to the transformations. Therefore it is sufficient to study the algorithms only for the standard LCP. Key Words:P matrix, linear complementarity problems, predictorcorrector, infeasibleinterior point algorithm, polynomiality, quadratic convergence. Abbreviated Title: Equivalence of different LCP Department of Mathematics, University of Iowa, Iowa City, IA 52242, USA. y Department of Mathematics, University of Iowa, Iowa City, IA 52242, USA. z Department of Mathematics, University of Iowa, Iowa City, IA 52242, USA. The work of this author was supported in part by NSF, Grant DMS 9305760. 1 Introduction The linear complementarity...
Complementarity Problems
 J. Comput. Appl. Math
, 2000
"... This paper provides an introduction to complementarity problems, with an emphasis on applications and solution algorithms. Various forms of complementarity problems are described along with a few sample applications, which provide a sense of what types of problems can be addressed eectively with ..."
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Cited by 6 (0 self)
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This paper provides an introduction to complementarity problems, with an emphasis on applications and solution algorithms. Various forms of complementarity problems are described along with a few sample applications, which provide a sense of what types of problems can be addressed eectively with complementarity problems. The most important algorithms are presented along with a discussion of when they can be used eectively. We also provide a brief introduction to the study of matrix classes and their relation to linear complementarity problems. Finally, we provide a brief summary of current research trends. Key words: complementarity problems,variational inequalities, matrix classes 1 Introduction The distinguishing feature of a complementarity problem is the set of complementarity conditions. Each of these conditions requires that the product of two or more nonnegative quantities should be zero. (Here, each quantity is either a decision variable, or a function of the decisi...
Simplified Analysis of an O(nL)Iteration Infeasible PredictorCorrector PathFollowing Method for Monotone LCP
, 1994
"... We give a simplified analysis of an infeasible predictorcorrector pathfollowing method for solving monotone linear complementarity problem. This method, like those studied by Mizuno et al. and by Potra and Sheng, (i) requires two factorizations and two backsolves per iteration, (ii) can find a sol ..."
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Cited by 5 (1 self)
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We give a simplified analysis of an infeasible predictorcorrector pathfollowing method for solving monotone linear complementarity problem. This method, like those studied by Mizuno et al. and by Potra and Sheng, (i) requires two factorizations and two backsolves per iteration, (ii) can find a solution in O(√nL) or O(nL) iterations, depending on the quality of the starting point, and (iii) has local quadratic convergence, provided a strictly complementary solution exists. The method decreases the centering parameter and the infeasibility at both predictor step and corrector step, and it is flexible in that either a primalscaling or dualscaling or primaldual scaling can be used for the corrector step without affecting the global and local convergence properties of the method.
Tapia indicators and finite termination of infeasibleinteriorpoint methods for degenerate LCP
, 1995
"... The convergence of the Tapia indicators for infeasibleinteriorpoint methods for solving degenerate linear complementarity problems is investigated. A new estimate of the rate of convergence of the Tapia indicators for the indices where both primal and dual variables vanish in the solution is obt ..."
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Cited by 4 (2 self)
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The convergence of the Tapia indicators for infeasibleinteriorpoint methods for solving degenerate linear complementarity problems is investigated. A new estimate of the rate of convergence of the Tapia indicators for the indices where both primal and dual variables vanish in the solution is obtained, showing that Tapia indicators for these indices converge slower than for other indices. Use of Tapia indicators in a finite termination procedure for infeasibleinteriorpoint algorithms is proposed. Keywords: Interior Point Algorithm, Linear Complementarity Problem, Rate of Convergence, Tapia Indicators. Abbreviated Title: Tapia Indicators Department of Mathematics and Computer Science, Valdosta State University, Valdosta, GA 31698. The work of this author was supported in part by a grant from the center for Faculty Development and Research at Valdosta State University. y Department of Mathematics, The University of Iowa, Iowa City, Iowa 52242. The work of this author was supp...
A PredictorCorrector Method for Solving the P*matrix LCP from Infeasible Starting Points
, 1994
"... A predictorcorrector method for solving the P ()matrix linear complementarity problems from infeasible starting points is analyzed. Two matrix factorizations and two backsolves are performed at each iteration. The algorithm terminates in O \Gamma ( + 1) 2 nL \Delta steps either by finding a ..."
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Cited by 4 (4 self)
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A predictorcorrector method for solving the P ()matrix linear complementarity problems from infeasible starting points is analyzed. Two matrix factorizations and two backsolves are performed at each iteration. The algorithm terminates in O \Gamma ( + 1) 2 nL \Delta steps either by finding a solution or by determining that the problem is not solvable. The computational complexity depends on the quality of the starting points. If the problem is solvable and if a certain measure of feasibility at the starting point is small enough then the algorithm finds a solution in O (( + 1) p nL) iterations. The algorithm is quadratically convergent for problems having a strictly complementary solution. Key Words: linear complementarity problems, P matrices, predictorcorrector, infeasibleinterior point algorithm, polynomiality, superlinear convergence. Abbreviated Title: A predictorcorrector method for LCP. Department of Mathematics and Computer Science, Valdosta State University...