Topic
Karmarkar's algorithm
About: Karmarkar's algorithm is a research topic. Over the lifetime, 255 publications have been published within this topic receiving 14658 citations.
Papers published on a yearly basis
Papers
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).
Abstract: We present a new polynomial-time algorithm for linear programming. In the worst case, the algorithm requiresO(n
3.5
L) arithmetic operations onO(L) bit numbers, wheren is the number of variables andL is the number of bits in the input. The running-time of this algorithm is better than the ellipsoid algorithm by a factor ofO(n
2.5). We prove that given a polytopeP and a strictly interior point a eP, 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 the radius of the largest sphere with center a′ contained inP′ isO(n). The algorithm consists of repeated application of such projective transformations each followed by optimization over an inscribed sphere to create a sequence of points which converges to the optimal solution in polynomial time.
5,071 citations
TL;DR: This paper provides a theoretical foundation for efficient interior-point algorithms for convex programming problems expressed in conic form, when the cone and its associated barrier are self-scaled, with long-step and symmetric primal-dual methods.
Abstract: This paper provides a theoretical foundation for efficient interior-point algorithms for convex programming problems expressed in conic form, when the cone and its associated barrier are self-scaled. For such problems we devise long-step and symmetric primal-dual methods. Because of the special properties of these cones and barriers, our algorithms can take steps that go typically a large fraction of the way to the boundary of the feasible region, rather than being confined to a ball of unit radius in the local norm defined by the Hessian of the barrier.
702 citations
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.
Abstract: A new interior method for linear programming is presented and a polynomial time bound for it is proven. The proof is substantially different from those given for the ellipsoid algorithm and for Karmarkar's algorithm. Also, the algorithm is conceptually simpler than either of those algorithms.
639 citations
Book•
1 Jan 1997
TL;DR: This chapter discusses duality Theory for Linear Optimization, a Polynomial Algorithm for the Skew-Symmetric Model, and Parametric and Sensitivity Analysis, as well as implementing Interior Point Methods.
Abstract: Partial table of contents: INTRODUCTION: THEORY AND COMPLEXITY. Duality Theory for Linear Optimization. A Polynomial Algorithm for the Skew-Symmetric Model. Solving the Canonical Problem. THE LOGARITHMIC BARRIER APPROACH. The Dual Logarithmic Barrier Method. Initialization. THE TARGET-FOLLOWING APPROACH. The Primal-Dual Newton Method. Application to the Method of Centers. MISCELLANEOUS TOPICS. Karmarkar's Projective Method. More Properties of the Central Path. Partial Updating. High-Order Methods. Parametric and Sensitivity Analysis. Implementing Interior Point Methods. Appendices. Bibliography. Indexes.
630 citations
TL;DR: This work gives a polynomial algorithm for the minimum cost flow and multicommodity flow problems in which the number of arithmetic steps is independent of the size of the costs and capacities.
Abstract: Khachiyan, and recently Karmarkar, gave polynomial algorithms to solve the linear programming problem. These algorithms have a small theoretical drawback; namely, the number of arithmetic steps depends on the size of the input numbers. We present a polynomial linear programming algorithm whose number of arithmetic steps depends only on the size of the numbers in the constraint matrix, but is independent of the size of the numbers in the right-hand side and objective vectors. In particular, it gives a polynomial algorithm for the minimum cost flow and multicommodity flow problems in which the number of arithmetic steps is independent of the size of the costs and capacities. The algorithm makes use of an existing polynomial linear programming algorithm. The problem of whether any algorithm has a running time that is independent even of the size of the numbers in the constraint matrix remains open.
561 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2018 | 1 |
| 2017 | 2 |
| 2015 | 2 |
| 2014 | 2 |
| 2013 | 1 |
| 2012 | 2 |