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21
Implementation of interior point methods for large scale linear programming
 Interior Point Methods in Mathematical Programming
, 1996
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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 59 (6 self)
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1 Introduction Consider the linear programming (LP) problem in the standard form: (LP) minimize cT x
Conic convex programming and selfdual embedding
 Optim. Methods and Software
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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, ...
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 16 (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.
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 5 (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...
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 ma ..."
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Cited by 4 (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.
Motzkin’s transposition theorem, and the related theorems of Farkas, Gordan and Stiemke
 Encyclopaedia of Mathematics, Supplement III
, 2002
"... Motzkin’s thesis [6], in particular his Transposition Theorem (Theorems 1–2 below), was a milestone in the development of linear inequalities and related areas. For two vectors u = (ui) and v = (vi) of equal dimension we denote by u ≧ v and u> v that the indicated inequality holds componentwise, ..."
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Motzkin’s thesis [6], in particular his Transposition Theorem (Theorems 1–2 below), was a milestone in the development of linear inequalities and related areas. For two vectors u = (ui) and v = (vi) of equal dimension we denote by u ≧ v and u> v that the indicated inequality holds componentwise, and by u � v the fact u ≧ v and u = v. Systems of linear inequalities appear in several forms; the following examples are typical: (a) A x ≦ b (b) A x = b, x ≧ 0 (c) A x ≦ b, Bx < c