Journal ArticleDOI
Polynomiality of infeasible-interior-point algorithms for linear programming
TLDR
This paper shows that a modification of the Kojima—Megiddo—Mizuno algorithm “solves” the pair of problems in polynomial time without assuming the existence of the LP solution, and develops anO(nL)-iteration complexity result for a variant of the algorithm.Abstract:
Kojima, Megiddo, and Mizuno investigate an infeasible-interior-point algorithm for solving a primal--dual pair of linear programming problems and they demonstrate its global convergence. Their algorithm finds approximate optimal solutions of the pair if both problems have interior points, and they detect infeasibility when the sequence of iterates diverges. Zhang proves polynomial-time convergence of an infeasible-interior-point algorithm under the assumption that both primal and dual problems have feasible points. In this paper, we show that a modification of the Kojima--Megiddo--Mizuno algorithm "solves" the pair of problems in polynomial time without assuming the existence of the LP solution. Furthermore, we develop anO(nL)-iteration complexity result for a variant of the algorithm.read more
Citations
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Journal ArticleDOI
Interior-point methods for optimization
TL;DR: The current state of the art of interior point methods (IPMs) for convex, conic, and general nonlinear optimization is described in this paper, where the authors discuss the theory, outline the algorithms, and comment on the applicability of this class of methods.
Journal ArticleDOI
A Full-Newton Step O ( n ) Infeasible Interior-Point Algorithm for Linear Optimization
TL;DR: It is shown that at most at most $O(n)$ iterations suffice to reduce the duality gap and the residuals by the factor $1/{e}$, which implies an $O (n\log(n/\varepsilon)$ iteration bound for getting an $\varePSilon-solution of the problem at hand, which coincides with the best known bound for infeasible interior-point algorithms.
Journal ArticleDOI
Full Nesterov–Todd step infeasible interior-point method for symmetric optimization
TL;DR: In this paper, the authors generalize the full-Newton step infeasible interior-point method for linear optimization of Roos [Roos, C., 2006] to symmetric optimization and unify the analysis for linear, second-order cone and semidefinite optimizations.
Journal ArticleDOI
Global and polynomial-time convergence of an infeasible-interior-point algorithm using inexact computation
Shinji Mizuno,Florian Jarre +1 more
TL;DR: An infeasible-interior-point algorithm for solving a primal-dual linear programming problem and it is shown that the algorithm finds an approximate solution of the linear program whenever the primal- duallinear programming problem is feasible.
Journal ArticleDOI
Convergence of a Class of Inexact Interior-Point Algorithms for Linear Programs
TL;DR: This work presents a convergence analysis for a class of inexact infeasible-interior-point methods for solving linear programs and allows that these linear systems are only solved to a moderate accuracy in the residual, but no assumptions are made on the accuracy of the search direction in the search space.
References
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Book ChapterDOI
Pathways to the optimal set in linear programming
TL;DR: In this article, the authors present continuous paths leading to the set of optimal solutions of a linear programming problem, which are derived from the weighted logarithmic barrier function and have nice primal-dual symmetry properties.
Book ChapterDOI
A primal-dual interior point algorithm for linear programming
TL;DR: In this article, the authors present an algorithm that works simultaneously on primal and dual linear programming problems and generates a sequence of pairs of their interior feasible solutions along the sequence generated, the duality gap converges to zero at least linearly with a global convergence ratio (1 − η/n).
Journal ArticleDOI
Interior path following primal-dual algorithms. Part I: Linear programming
Renato D. C. Monteiro,Ilan Adler +1 more
TL;DR: A primal-dual interior point algorithm for linear programming problems which requires a total of O(n L) number of iterations to find the Newton direction associated with the Karush-Kuhn-Tucker system of equations which characterizes a solution of the logarithmic barrier function problem.
Journal ArticleDOI
On Adaptive-Step Primal-Dual Interior-Point Algorithms for Linear Programming
TL;DR: Heuristic reasoning is provided for expecting that the algorithms will perform much better in practice than guaranteed by the worst-case estimates, based on an analysis using a nonrigorous probabilistic assumption.
Journal ArticleDOI
A polynomial-time algorithm for a class of linear complementary problems
TL;DR: An algorithm is presented that solves the problem of finding n-dimensional vectors in O(n3L) arithmetic operations by tracing the path of centers by identifying the centers of centers of the feasible region.