Book ChapterDOI
An Algorithm for Solving Linear Programming Problems in O(n 3 L) Operations
Clovis C. Gonzaga
- pp 1-28
TLDR
This chapter describes a short-step penalty function algorithm that solves linear programming problems in no more than O(n 0.5 L) iterations and follows the path of optimal solutions for the penalized functions as in a predictor-corrector homotopy algorithm.Abstract:
This chapter describes a short-step penalty function algorithm that solves linear programming problems in no more than O(n 0.5 L) iterations. The total number of arithmetic operations is bounded by O(n 3 L), carried on with the same precision as that in Karmarkar’s algorithm. Each iteration updates a penalty multiplier and solves a Newton-Raphson iteration on the traditional logarithmic barrier function using approximated Hessian matrices. The resulting sequence follows the path of optimal solutions for the penalized functions as in a predictor-corrector homotopy algorithm.read more
Citations
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Book
Introduction to Numerical Continuation Methods
Eugene L. Allgower,Kurt Georg +1 more
TL;DR: The Numerical Continuation Methods for Nonlinear Systems of Equations (NCME) as discussed by the authors is an excellent introduction to numerical continuuation methods for solving nonlinear systems of equations.
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
Path-following methods for linear programming
TL;DR: A unified treatment of algorithms is described for linear programming methods based on the central path, which is a curve along which the cost decreases, and that stays always far from the centre.
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.
References
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Journal ArticleDOI
A new polynomial-time algorithm for linear programming
TL;DR: It is proved that given a polytopeP and a strictly interior point a εP, there is a projective transformation of the space that mapsP, a toP′, a′ having the following property: the ratio of the radius of the smallest sphere with center a′, containingP′ to theradius of the largest sphere withCenter a′ contained inP′ isO(n).
Book
Nonlinear Programming: Sequential Unconstrained Minimization Techniques
TL;DR: This report gives the most comprehensive and detailed treatment to date of some of the most powerful mathematical programming techniques currently known--sequential unconstrained methods for constrained minimization problems in Euclidean n-space--giving many new results not published elsewhere.
Journal ArticleDOI
Pathways to solutions, fixed points, and equilibria
Journal ArticleDOI
A polynomial-time algorithm, based on Newton's method, for linear programming
TL;DR: A new interior method for linear programming is presented and a polynomial time bound for it is proven and it is conceptually simpler than either the ellipsoid algorithm or Karmarkar's algorithm.
Journal ArticleDOI
On projected Newton barrier methods for linear programming and an equivalence to Karmarkar's projective method
TL;DR: This work reviews classical barrier-function methods for nonlinear programming based on applying a logarithmic transformation to inequality constraints and shows a “projected Newton barrier” method to be equivalent to Karmarkar's projective method for a particular choice of the barrier parameter.