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An extension of Karmarkar's algorithm for linear programming using dual variables

Michael J. Todd, Bruce P. Burrell
1986 Algorithmica  
We describe an extension of Karmarkar's algorithm for linear programming that handles problems with unknown optimal value and generates primal and dual solutions with objective values converging to the  ...  We also describe an implementation for the dense case and show how extreme point solutions can be obtained naturally, with little extra computation.  ...  The basic method is an iterative technique for solving a linear programming problem of a certain type.  ... 
doi:10.1007/bf01840455 fatcat:fmujkv2sd5fz3p6tnaf5nvtem4

Progress in mathematical programming

1990 European Journal of Operational Research  
, Stanford, CA 94305 An extension of Karmarkar's algorithm and the trust region method is developed for solving quadratic programming and linearly constrained programming.  ...  Karmarkar's projective algorithm for linear programming provides not only primal solutions but dual solutions giving bounds on the optimal value.  ... 
doi:10.1016/0377-2217(90)90262-a fatcat:gs7on6tmo5ahxnmhx4btkbw6fy

Page 2315 of Mathematical Reviews Vol. , Issue 92d [page]

1992 Mathematical Reviews  
Linear Algebra Appl. 152 (1991), 343-363. In this paper the authors propose a method for identifying an op- timal basis for linear programming problems which can be used in interior point algorithms.  ...  The authors consider the directions used in Karmarkar’s projective linear programming algorithm and in its affine variant.  ... 

Potential Reduction Algorithms [chapter]

Kurt M. Anstreicher
1996 Applied Optimization  
Lemma 3.1 (Todd and Burrell Proof: The dual of HLP is: HLD: max z A T y + dz ≤c .  ...  Acknowlegement I would like to thank Rob Freund, Tamas Terlaky, Mike Todd, and Yinyu Ye for their comments on a draft of this article.  ...  Karmarkar's [44] algorithm for linear programming, whose announcement in 1984 initiated a torrent of research into interior point methods, used three key ingredients: a non-standard linear programming  ... 
doi:10.1007/978-1-4613-3449-1_4 fatcat:6log3alvingdjpdryewgaos24a

Page 5231 of Mathematical Reviews Vol. , Issue 93i [page]

1993 Mathematical Reviews  
The author considers Karmarkar’s algorithm for linear program- ming with general linear constraints.  ...  (NL-DELFM) A projective variant of the approximate center method for the dual linear programming problem.  ... 

Page 1773 of Mathematical Reviews Vol. , Issue 91C [page]

1991 Mathematical Reviews  
The novel feature of our algorithm is the use of continuous heuris- tics applied to the dual of the linear programming relaxation to provide lower bounds for a branch and bound algorithm.  ...  This block reduction is an extension of the flow morphism for network flow problems and, at the same time, a re- striction of the reduction on linear programs presented by Harper.  ... 

A dual version of tardos's algorithm for linear programming

James B Orlin
1986 Operations Research Letters  
Subject Classification: A Dual Version of Tardos's Algorithm for Linear Programming.  ...  Abstract Recently, Eva Tardos developed an algorithm for solving the linear program min(cx: Ax = b, x > O) whose solution time is polynomial in the size of A, independent of the sizes of c and b.  ...  linear programs solved by Karmarkar's algorithm has an input size L = O(m log M) and thus the number of arithmetic steps per linear program is 0(m3.5 log M) = O(m 4 .5 log (m + Amax + 1).  ... 
doi:10.1016/0167-6377(86)90011-8 fatcat:mffi6k63erfstdw2i545h45mo4

Page 1635 of Mathematical Reviews Vol. , Issue 88c [page]

1988 Mathematical Reviews  
Millham (1-WAS-C) 88c:90089 90C05 Todd, Michael J. (1-CRNL-O); Burrell, Bruce P. (1-CRNL-O) An extension of Karmarkar’s algorithm for linear programming using dua! variables.  ...  Summary: “A simple algorithm for the solution of a linear program- ming problem is proposed. An elementary proof of the validity of the algorithm is given.” 88c:90091 90C08 15A06 60G42 Lin, T. F.  ... 

Page 4891 of Mathematical Reviews Vol. , Issue 90H [page]

1990 Mathematical Reviews  
Fu An Zhao (Qufu) 90h:90134 90C20 90C30 Ye, Yin Yu (1-IA-MG); Tse, Edison (1-STF-ES) An extension of Karmarkar’s projective algorithm for convex quadratic programming. Math.  ...  This paper presents a way of using Karmarkar’s polynomial-time linear programming (KPLP) algorithm for solving a more gen- 90C Mathematical programming 90n:90135 eral group of optimization problems: convex  ... 

Page 4419 of Mathematical Reviews Vol. , Issue 88h [page]

1988 Mathematical Reviews  
(i-IBM) A variation on Karmarkar’s algorithm for solving linear programming problems. Math. Programming 36 (1986), no. 2, 174-182.  ...  Summary: “Interest in linear programming has been intensified re- cently by Karmarkar’s publication in 1984 of an algorithm that is claimed to be much faster than the simplex method for practical problems  ... 

Page 558 of Mathematical Reviews Vol. , Issue 89A [page]

1989 Mathematical Reviews  
M. (1-SHELLD); Turner, Kathryn (1-RICE) A variable-metric variant of the Karmarkar algorithm for linear programming. Math. Programming 39 (1987), no. 1, 1-20.  ...  The authors address an implementation of the basic Karmarkar linear-programming procedure, with some substantial algorithmic innovations, to give what the authors call the “DMT-Karmarkar algorithm”.  ... 

An analog of Karmarkar's algorithm for inequality constrained linear programs, with a 'new' class of projective transformations for centering a polytope

Robert M Freund
1988 Operations Research Letters  
We present an algorithm, analogous to Karmarkar's algorithm, for the linear programming problem: maximizing c x subject to Ax < b , which works directly in the space of linear inequalities.  ...  The main new idea in this algorithm is the simple construction of a projective transformation of the feasible region that maps the current iterate x to the analytic center of the transformed feasible region  ...  Acknowledgement I would like to acknowledge Michael Todd for stimulating discussions on this topic, and his reading of a draft of this paper.  ... 
doi:10.1016/0167-6377(88)90045-4 fatcat:i3d6xw33jbdkdbhu6khn32k7we

Page 6713 of Mathematical Reviews Vol. , Issue 88m [page]

1988 Mathematical Reviews  
, Andrew J. (1-BELL6) Computational experience with a dual affine variant of Karmarkar’s method for linear programming.  ...  The third chapter, which investigates geometric programming, is presented as an extension of analytical methods that introduces the concept of a dual prob- lem.  ... 

Page 1161 of Mathematical Reviews Vol. , Issue 92b [page]

1992 Mathematical Reviews  
The true worst-case complexity of N. Karmarkar’s (projective) algorithm for linear programming is a problem of considerable theoretical interest.  ...  Azpeitia (1-MAB) 92b:90122 90C05 Anstreicher, Kurt M. (1-YALE) Dual ellipsoids and degeneracy in the projective algorithm for linear programming.  ... 

Page 3480 of Mathematical Reviews Vol. , Issue 91F [page]

1991 Mathematical Reviews  
optimal basic variables in Karmarkar’s polynomial algorithm for linear programming.  ...  An interior point algorithm for linear programming finds an opti- mal solution.  ... 
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