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A dual version of tardos's algorithm for linear programming

James B Orlin
1986 Operations Research Letters  
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.  ...  Her algorithm focuses on the dual LP and employs an approximation of the cost coefficients. Here we adopt what may be called a 'dual approach' in that it focuses on the primal LP.  ...  As such, we can use Karmarkar's algorithm to identify an optimal primal solution and then use standard techniques to move to an optimal basic solution.  ... 
doi:10.1016/0167-6377(86)90011-8 fatcat:mffi6k63erfstdw2i545h45mo4

Page 1205 of Mathematical Reviews Vol. , Issue 89B [page]

1989 Mathematical Reviews  
Szidarovszky (1-AZ-I) 89b:90128 90C05 Kojima, Masakazu (J-TOKYTE-I) Determining basic variables of optimal solutions in Karmarkar’s new LP algorithm. Algorithmica 1 (1986), no. 4, 499-515.  ...  A numerical test using the condition is incorporated into Karmarkar’s new LP algorithm to determine columns of an optimal basis.  ... 

Page 3692 of Mathematical Reviews Vol. , Issue 90F [page]

1990 Mathematical Reviews  
The paper is devoted to the discussion of traditional (the sim- plex method) and new (Karmarkar-type algorithms) approaches to the solution of large-scale LP problems.  ...  A variant of Karmarkar’s projective algorithm for linear program- ming problems in which some variables are unrestricted is consid- ered.  ... 

Page 688 of Mathematical Reviews Vol. , Issue 90A [page]

1990 Mathematical Reviews  
An algorithm is said to be well-behaved on a class D of LP problems if, for each prob- lem in D, the computed solution, x’, is a slight perturbation of the solution of another problem in D for which the  ...  (GR-THRC-E); Vachtsevanos, G. (1-GAIT-E) A new algorithm for the solution of the linear minimax approximation problem. Optimization 19 (1988), no. 6, 839-860.  ... 

Page 4759 of Mathematical Reviews Vol. , Issue 89H [page]

1989 Mathematical Reviews  
Only one artificial variable is needed to determine if the linear programming problem has a feasible solution in phase one.  ...  restrict the choice of exiting basic variables.  ... 

Page 4146 of Mathematical Reviews Vol. , Issue 89G [page]

1989 Mathematical Reviews  
On the one hand only approximate projections are used in determining di- rections of search and, secondly, variables which are likely to be nonbasic are successively eliminated.  ...  of the book contains codes in BASIC for all methods described in the text.  ... 

Progress in mathematical programming

1990 European Journal of Operational Research  
, and that are violated by the optimal solution of a LP relaxation.  ...  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 5001 of Mathematical Reviews Vol. , Issue 88i [page]

1988 Mathematical Reviews  
In this case, we show the existence of basic feasible solutions and derive a condition for a basic feasible solution to be optimal.” 88i:90121 90C05 Holm, Seren (DK-ODNS) Adding activities to the dual  ...  An improvement in the bound on the complexity of Karmarkar’s algorithm is proved, namely O(n** log 7).  ... 

Page 2722 of Mathematical Reviews Vol. , Issue 88e [page]

1988 Mathematical Reviews  
Then the extremal flat, where the optimal solutions are located, is determined by the optimal value. A feasible solution is optimal if it satisfies the equation of the extremal flat.  ...  The author is concerned with an optimality test for initial basic feasible solutions for the transportation problem.  ... 

Page 1619 of Mathematical Reviews Vol. , Issue 87c [page]

1987 Mathematical Reviews  
Karmarkar’s publication does not provide a uniquely determined computational procedure, and also the average case behavior of the algorithm is not clear.  ...  , where n is the number of variables and L is the number of bits in the input data.  ... 

The basic solution ellipsoid method approach for the efficiency measurement problems

Eligijus Laurinavičius, Daiva Rimkuvienė, Aurelija Sakalauskaitė
2016 Lietuvos matematikos rinkinys  
This paper presents basic solution ellipsoid method approach associated with some problems of initial basic solution and the steps of it.  ...  The DEA is a linear programming (LP) based technique which deals with the basic models (CCR, BCC, SBM, additive) of the efficiency evaluation.  ...  The solution of the problem (1) does not involve any additional variables and no new conditions for variable non-negation are raised, which are often necessary in many of the algorithms for the solution  ... 
doi:10.15388/lmr.a.2016.08 fatcat:ealz47ktpjgjpafgdxjnyxakfm

Introducing Interior-Point Methods for Introductory Operations Research Courses and/or Linear Programming Courses

Goran Lesaja
2009 Open Operational Research Journal  
Consequently, there has been an increasing need to introduce theory and methods of this new area in optimization into the appropriate undergraduate and first year graduate courses such as introductory  ...  Development of these methods has quickly led to the design of new and efficient optimization codes particularly for Linear Programming.  ...  However, as in the case of the Ellipsoid and Karmarkar's algorithms, it can be shown that if the input data are rational numbers, the IPM finds the exact solution of LP in O n L ( ) iterations proving  ... 
doi:10.2174/1874243200903010001 fatcat:qbdkad3c7bepvijs6m7zjhvgvy


Henry De-Graft Acquah, Sarah Acquah
2017 Apstract: Applied Studies in Agribusiness and Commerce  
Results of the interior point algorithm is similar to that of the simplex algorithm.  ...  The Simplex algorithm saves 39 iterations over Interior Point algorithm in solving the farm resource allocation problem.  ...  This is done by determining the entering variable and the outgoing variable. An entering variable is a variable we choose to find new BV from a current basic feasible solution that is not optimal.  ... 
doi:10.19041/apstract/2017/1-2/6 fatcat:qcymk7ttijfqzomp5q7wvukcva

Page 6085 of Mathematical Reviews Vol. , Issue 90J [page]

1990 Mathematical Reviews  
same set of basic variables in an op- timal solution.  ...  [Elaboration of efficient computational setups using Karmarkar’s algorithm] New methods in optimization and their industrial uses (Pau/Paris, 1987), 173-190, Internat. Schriftenreihe Numer.  ... 

A novel method for solving linear programming problems with symmetric trapezoidal fuzzy numbers

Ali Ebrahimnejad, Madjid Tavana
2014 Applied Mathematical Modelling  
Linear programming (LP) is a widely used optimization method for solving real-life problems because of its efficiency.  ...  We propose a new method for solving FLP problems in which the coefficients of the objective function and the values of the right-hand-side are represented by symmetric trapezoidal fuzzy numbers while the  ...  , it also has an optimal fuzzy basic solution.  ... 
doi:10.1016/j.apm.2014.02.024 fatcat:uvxgx44pqjavlmm7z3k4fu326m
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