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22
Convergence of a class of inexact interiorpoint algorithms for linear programs,
 Math. Oper. Res.,
, 1999
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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 19 (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...
Interiorpoint methods for optimization
, 2008
"... This article describes the current state of the art of interiorpoint methods (IPMs) for convex, conic, and general nonlinear optimization. We discuss the theory, outline the algorithms, and comment on the applicability of this class of methods, which have revolutionized the field over the last twen ..."
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Cited by 17 (0 self)
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This article describes the current state of the art of interiorpoint methods (IPMs) for convex, conic, and general nonlinear optimization. We discuss the theory, outline the algorithms, and comment on the applicability of this class of methods, which have revolutionized the field over the last twenty years.
Approximate Farkas Lemmas and Stopping Rules for Iterative InfeasiblePoint Algorithms for Linear Programming
, 1996
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A fullNewton step O(n) infeasible interiorpoint algorithm for linear optimization
, 2005
"... We present a primaldual infeasible interiorpoint algorithm. As usual, the algorithm decreases the duality gap and the feasibility residuals at the same rate. Assuming that an optimal solution exists it is shown that at most O(n) iterations suffice to reduce the duality gap and the residuals by the ..."
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Cited by 15 (7 self)
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We present a primaldual infeasible interiorpoint algorithm. As usual, the algorithm decreases the duality gap and the feasibility residuals at the same rate. Assuming that an optimal solution exists it is shown that at most O(n) iterations suffice to reduce the duality gap and the residuals by the factor 1/e. This implies an O(nlog(n/ε)) iteration bound for getting an εsolution of the problem at hand, which coincides with the best known bound for infeasible interiorpoint algorithms. The algorithm constructs strictly feasible iterates for a sequence of perturbations of the given problem and its dual problem. A special feature of the algorithm is that it uses only fullNewton steps. Two types of fullNewton steps are used, socalled feasibility steps and usual (centering) steps. Starting at strictly feasible iterates of a perturbed pair, (very) close its central path, feasibility steps serve to generate strictly feasible iterates for the next perturbed pair. By accomplishing a few centering steps for the new perturbed pair we obtain strictly feasible iterates close enough to the central path of the new perturbed pair. The algorithm finds an optimal solution or detects infeasibility or unboundedness of the given problem.
Probabilistic Analysis of an InfeasibleInteriorPoint Algorithm for Linear Programming
, 1998
"... We consider an infeasibleinteriorpoint algorithm, endowed with a finite termination scheme, applied to random linear programs generated according to a model of Todd. Such problems have degenerate optimal solutions, and possess no feasible starting point. We use no information regarding an optimal ..."
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Cited by 12 (3 self)
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We consider an infeasibleinteriorpoint algorithm, endowed with a finite termination scheme, applied to random linear programs generated according to a model of Todd. Such problems have degenerate optimal solutions, and possess no feasible starting point. We use no information regarding an optimal solution in the initialization of the algorithm. Our main result is that the expected number of iterations before termination with an exact optimal solution is O(n ln(n)). Keywords: Linear Programming, AverageCase Behavior, InfeasibleInteriorPoint Algorithm. Running Title: Probabilistic Analysis of an LP Algorithm 1 Dept. of Management Sciences, University of Iowa. Supported by an Interdisciplinary Research Grant from the Center for Advanced Studies, University of Iowa. 2 Dept. of Mathematics, Valdosta State University. Supported by an Interdisciplinary Research Grant from the Center for Advanced Studies, University of Iowa. 3 Dept. of Mathematics, University of Iowa. Supported by ...
An Infeasible Start Predictor Corrector Method for Semidefinite Linear Programming
, 1995
"... In this paper we present an infeasible start path following predictor corrector method for semidefinite linear programming problem. This method does not assume that the dual pair of semidefinite programs have feasible solutions, and, in at most O(jlog( ffl ffi(A;b;C)ae )jn) iterations of the predi ..."
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Cited by 10 (2 self)
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In this paper we present an infeasible start path following predictor corrector method for semidefinite linear programming problem. This method does not assume that the dual pair of semidefinite programs have feasible solutions, and, in at most O(jlog( ffl ffi(A;b;C)ae )jn) iterations of the predictor corrector method, finds either an approximate solution to the dual pair or shows that there is no optimal solution with duality gap zero in a well defined bounded region. The nonexistence of optimal solutions is detected in a finite number of iterations. Here ffl is a measure of nonoptimality and infeasibility of the pair of solutions found, and is generally chosen to be small; ffi (A; b; C) is a function of the data of the problem and ae is a measure of the size of the region searched, and is generally large. The method we present generalizes a method for linear programming developed by Mizuno. We give some preliminary computational experience for this method, and compare its perform...
Detecting Infeasibility in InfeasibleInteriorPoint Methods for Optimization
 Foundations of Computational Mathematics, Minneapolis 2002, London Mathematical Society Lecture Note Series 312
, 2003
"... We study interiorpoint methods for optimization problems in the case of infeasibility or unboundedness. While many such methods are designed to search for optimal solutions even when they do not exist, we show that they can be viewed as implicitly searching for welldefined optimal solutions to rel ..."
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Cited by 8 (1 self)
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We study interiorpoint methods for optimization problems in the case of infeasibility or unboundedness. While many such methods are designed to search for optimal solutions even when they do not exist, we show that they can be viewed as implicitly searching for welldefined optimal solutions to related problems whose optimal solutions give certificates of infeasibility for the original problem or its dual. Our main development is in the context of linear programming, but we also discuss extensions to more general convex programming problems.