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

1986
*
Algorithmica
*

We describe

doi:10.1007/bf01840455
fatcat:fmujkv2sd5fz3p6tnaf5nvtem4
*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. ...##
###
Progress in mathematical programming

1990
*
European Journal of Operational Research
*

, Stanford, CA 94305

doi:10.1016/0377-2217(90)90262-a
fatcat:gs7on6tmo5ahxnmhx4btkbw6fy
*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. ...##
###
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]

1996
*
Applied Optimization
*

Lemma 3.1 (Todd and Burrell Proof: The

doi:10.1007/978-1-4613-3449-1_4
fatcat:6log3alvingdjpdryewgaos24a
*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*...##
###
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

1986
*
Operations Research Letters
*

Subject Classification: A

doi:10.1016/0167-6377(86)90011-8
fatcat:mffi6k63erfstdw2i545h45mo4
*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). ...##
###
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

1988
*
Operations Research Letters
*

We present

doi:10.1016/0167-6377(88)90045-4
fatcat:i3d6xw33jbdkdbhu6khn32k7we
*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. ...##
###
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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