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13
Implementation of Interior Point Methods for Large Scale Linear Programming
 in Interior Point Methods in Mathematical Programming
, 1996
"... In the past 10 years the interior point methods (IPM) for linear programming have gained extraordinary interest as an alternative to the sparse simplex based methods. This has initiated a fruitful competition between the two types of algorithms which has lead to very efficient implementations on bot ..."
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Cited by 69 (22 self)
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In the past 10 years the interior point methods (IPM) for linear programming have gained extraordinary interest as an alternative to the sparse simplex based methods. This has initiated a fruitful competition between the two types of algorithms which has lead to very efficient implementations on both sides. The significant difference between interior point and simplex based methods is reflected not only in the theoretical background but also in the practical implementation. In this paper we give an overview of the most important characteristics of advanced implementations of interior point methods. First, we present the infeasibleprimaldual algorithm which is widely considered the most efficient general purpose IPM. Our discussion includes various algorithmic enhancements of the basic algorithm. The only shortcoming of the "traditional" infeasibleprimaldual algorithm is to detect a possible primal or dual infeasibility of the linear program. We discuss how this problem can be solve...
A simplified homogeneous and selfdual linear programming algorithm and its implementation
 Annals of Operations Research
, 1996
"... 1 Introduction Consider the linear programming (LP) problem in the standard form: (LP) minimize cT x ..."
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Cited by 56 (5 self)
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1 Introduction Consider the linear programming (LP) problem in the standard form: (LP) minimize cT x
Duality And SelfDuality For Conic Convex Programming
, 1996
"... This paper considers the problem of minimizing a linear function over the intersection of an affine space with a closed convex cone. In the first half of the paper, we give a detailed study of duality properties of this problem and present examples to illustrate these properties. In particular, we i ..."
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Cited by 20 (7 self)
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This paper considers the problem of minimizing a linear function over the intersection of an affine space with a closed convex cone. In the first half of the paper, we give a detailed study of duality properties of this problem and present examples to illustrate these properties. In particular, we introduce the notions of weak/strong feasibility or infeasibility for a general primaldual pair of conic convex programs, and then establish various relations between these notions and the duality properties of the problem. In the second half of the paper, we propose a selfdual embedding with the following properties: Any weakly centered sequence converging to a complementary pair either induces a sequence converging to a certificate of strong infeasibility, or induces a sequence of primaldual pairs for which the amount of constraint violation converges to zero, and the corresponding objective values are in the limit not worse than the optimal objective value(s). In case of strong duality, ...
Conic Convex Programming And SelfDual Embedding
 Optim. Methods Softw
, 1998
"... How to initialize an algorithm to solve an optimization problem is of great theoretical and practical importance. In the simplex method for linear programming this issue is resolved by either the twophase approach or using the socalled big M technique. In the interior point method, there is a more ..."
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Cited by 18 (2 self)
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How to initialize an algorithm to solve an optimization problem is of great theoretical and practical importance. In the simplex method for linear programming this issue is resolved by either the twophase approach or using the socalled big M technique. In the interior point method, there is a more elegant way to deal with the initialization problem, viz. the selfdual embedding technique proposed by Ye, Todd and Mizuno [30]. For linear programming this technique makes it possible to identify an optimal solution or conclude the problem to be infeasible/unbounded by solving its embedded selfdual problem. The embedded selfdual problem has a trivial initial solution and has the same structure as the original problem. Hence, it eliminates the need to consider the initialization problem at all. In this paper, we extend this approach to solve general conic convex programming, including semidefinite programming. Since a nonlinear conic convex programming problem may lack the socalled stri...
Duality Results For Conic Convex Programming
, 1997
"... This paper presents a unified study of duality properties for the problem of minimizing a linear function over the intersection of an affine space with a convex cone infinite dimension. Existing duality results are carefully surveyed and some new duality properties are established. Examples are give ..."
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Cited by 15 (10 self)
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This paper presents a unified study of duality properties for the problem of minimizing a linear function over the intersection of an affine space with a convex cone infinite dimension. Existing duality results are carefully surveyed and some new duality properties are established. Examples are given to illustrate these new properties. The topics covered in this paper include GordonStiemke type theorems, Farkas type theorems, perfect duality, Slater condition, regularization, Ramana's duality, and approximate dualities. The dual representations of various convex sets, convex cones and conic convex programs are also discussed.
A Simple Algebraic Proof Of Farkas's Lemma And Related Theorems
, 1998
"... this paper we have given an alternative proof of Farkas's lemma, a proof that is based on a theorem, the main theorem, that relates to the eigenvectors of certain orthogonal matrices. This theorem is believed to be new, and the author is not aware of any similar theorem concerning orthogonal matrice ..."
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Cited by 3 (0 self)
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this paper we have given an alternative proof of Farkas's lemma, a proof that is based on a theorem, the main theorem, that relates to the eigenvectors of certain orthogonal matrices. This theorem is believed to be new, and the author is not aware of any similar theorem concerning orthogonal matrices although he recently proved the weak form of the theorem using Tucker's theorem (see [5]). His proof of the theorem is "completely elementary" (a referee) and requires little more than a knowledge of matrix algebra for its understanding. Once the theorem is established, Tucker's theorem (via the Cayley transform), Farkas's lemma and many other theorems of the alternative follow trivially. Thus the paper establishes a connection between the eigenvectors of orthogonal matrices, duality in linear programming and theorems of the alternative that is not generally appreciated, and this may be of some theoretical interest.
An InfeasibleInteriorPoint PotentialReduction Algorithm for Linear Programming
 Math. Progr
, 1999
"... This paper studies a new potentialfunction and an infeasibleinteriorpoint method based on this function for the solution of linear programming problems. This work is motivated by the apparent gap between the algorithms with the best worstcase complexity and their most successful implementations. ..."
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Cited by 3 (0 self)
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This paper studies a new potentialfunction and an infeasibleinteriorpoint method based on this function for the solution of linear programming problems. This work is motivated by the apparent gap between the algorithms with the best worstcase complexity and their most successful implementations. For example, analyses of the algorithms are usually carried out by imposing several regularity assumptions on the problem, but implementations can solve problems which do not satisfy these assumptions. Furthermore, most stateoftheart implementations incorporate heuristic tricks, but these modifications are rarely addressed in the theoretical analysis of the algorithms. The method described here and its analysis have the flexibility to integrate any heuristic technique for implementation while maintaining the important polynomial complexity feature. Key words: linear programming, potential functions, infeasibleinteriorpoint methods, homogeneity, selfdual. AMS Subject classification: 9...
Global Behaviour of Polynomial Differential Systems in the Positive Orthant
"... We study polynomial ndimensional differential systems defined in the positive orthant. By using tools from positivity and functions that decrease along the trajectories, we give sufficient conditions for a regular global behaviour: that is, all the trajectories either converge towards the equilibri ..."
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We study polynomial ndimensional differential systems defined in the positive orthant. By using tools from positivity and functions that decrease along the trajectories, we give sufficient conditions for a regular global behaviour: that is, all the trajectories either converge towards the equilibria or are unbounded. We also give results on global stability or unstability in some cases.
Sanjaya Kumar Behera et al. / International Journal on Computer Science and Engineering (IJCSE) SOLUTION OF MULTIOBJECTIVE MATHEMATICAL PROGRAMMING PROBLEMS IN FUZZY APPROACH
"... Recent developments in multi objective programming by Geoffrion, Mond and Wolfe [3, 8, 13] show interesting results with convex functions and related scalar objective programs. In this paper we compare the solution of multi objective linear programming problem with this solution obtained in Zimmerma ..."
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Recent developments in multi objective programming by Geoffrion, Mond and Wolfe [3, 8, 13] show interesting results with convex functions and related scalar objective programs. In this paper we compare the solution of multi objective linear programming problem with this solution obtained in Zimmermannâ€™s method. Zimmermann used membership function to solve the multiobjective linear programming problems. We have used Î±cut to solve the multi objective linear programming problems.