Near boundary behavior of primal—dual potential reduction algorithms for linear programming

Ye, Y.; Kortanek, K. O.; Kaliski, John A.; Huang, Shuqiang

doi:10.1007/bf01581269

Cited by 9 publications

(6 citation statements)

References 13 publications

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“…where X = diag(x), S = diag(s), e is a vector of ones of appropriate dimension, and p' is a parameter related to p. Kojima, Mizuno, and Yoshise choose p' = p. Ye et al (1993) generalized their result to the case where p' is not restricted to equal p and can vary from one iteration to the other. If p' is chosen carefully, it is possible to maintain the guarantee of constant reduction in every iteration and to improve the practical performance.…”

Section: Potential Functions For Linear Programmingmentioning

confidence: 99%

A Primal-Dual Variant of the IRI-IMAI Algorithm for Linear Programming

Tütüncü

2000

Mathematics of OR

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A local acceleration method for primal-dual potential-reduction algorithms is introduced. The method developed here uses modified Newton search directions to minimize the Tanabe-Todd-Ye (TTY) potential function, and can be regarded as a primal-dual variant of the Iri-Imai algorithm based on the multiplicative analogue of Karmarkar's potential function. When iterates are close to an optimal solution, the TTY potential function has negative curvature along the generated search directions. Therefore, large reductions in the potential function can be obtained, guaranteeing polynomial and quadratic convergence to nondegenerate solutions.

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Section: Potential Functions For Linear Programmingmentioning

confidence: 99%

A Primal-Dual Variant of the IRI-IMAI Algorithm for Linear Programming

Tütüncü

2000

Mathematics of OR

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“…For any x 2 X and z 2 R n , a) kP(x + z) xk= is a nonincreasing function of > 0, b) kP(x + z) xk= kzk.Before proving the main result (Theorem 4.5), we show that the conditions (9),(10) ensure that the projection is computed exactly when x k is critical (Lemma 4.3) and, in a technical result, show that the algorithm produces descent at a noncritical point (Lemma 4.4).Lemma 4.3. Suppose that (A) holds and that (B) holds at z = x k .…”

mentioning

confidence: 90%

“…We state without proof the following well-known result, which actually applies for any closed convex X R n . From (10) and the fact that C 1 < 1, 1 =2 2 k ( ) 1 C 1 2 > 1 2 0; since 2 0; 1] and > 2. Since k ( ) 0, the inequality (22) can hold only if k ( ) = 0.…”

Section: Introduction We Address the Problem Min X F(x)mentioning

confidence: 97%

“…Step 2: For = p , p = 0; 1; 2; (in sequence) approximately calculate x k ( ) = P(x k g k ), terminating when x k ( ; k ( )) = x kj ( ) and its associated k ( ) = kj ( ) are found that satisfy (9), (10). If the test (11) is passed for this value of , set k = = p , x k+1 = x k ( ; k ( )), k k + 1, and go to the next iteration.…”

Section: Introduction We Address the Problem Min X F(x)mentioning

confidence: 99%

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An Interior-Point Algorithm for Linearly Constrained Optimization

Wright¹

1992

SIAM J. Optim.

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We describe an algorithm for optimization of a smooth function subject to general linear constraints. An algorithm of the gradient projection class is used, with the important feature that the \projection" at each iteration is performed using a primal-dual interior point method for convex quadratic programming. Convergence properties can be maintained even if the projection is done inexactly in a well-de ned way. Higher-order derivative information on the manifold de ned by the apparently active constraints can be used to increase the rate of local convergence.

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“…On voit que λ est borné.On a donc ∆ ∼ y + Y 2 ( b i − λĉ). La droite d'origine x =x + R t y et de direction R∆ passe donc très près de la solution optimalex et est donc une très bonne direction de descente, on voit d'après l'équation(18) que la distortion dueà c diminue lorsque p diminue. Par ailleurs, dans la recherche linéaire, le numérateur ( c, x −α) p est moins proche de 0 si on travaille avec f p , la quantité p log( c, x −α) est moins proche de −∞ si on travaille avec g p , la détermination du pas est donc moins sujetteà erreurs.…”

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Les effets de l'exposant de la fonction barrière multiplicative dans les méthodes de points intérieurs

Coulibaly

Crouzeix

2003

RAIRO-Oper. Res.

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Potential functions in interior point methods are used to determine descent directions and to prove the convergence. They depend on a parameter which is usually taken equal to or greater than the size of the problem. Actually, smaller values give a better conditioning of the method near an optimal solution. This assertion is illustrated by a few numerical experiments. Résumé. Les méthodes de points intérieurs en programmation linéaire connaissent un grand succès depuis l'introduction de l'algorithme de Karmarkar. La convergence de l'algorithme repose sur une fonction potentielle qui, sous sa forme multiplicative, fait apparaître un exposant p. Cet exposant est, de façon générale, choisi supérieur au nombre de variables n du problème. Nous montrons dans cet article que l'on peut utiliser des valeurs de p plus petites que n. Ceci permet d'améliorer le conditionnement de la méthode au voisinage de la solution optimale. Mots Clés. Interior point methods, Karmarkar algorithm, multiplicative and additive potential functions, barrier function.

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Near boundary behavior of primal—dual potential reduction algorithms for linear programming

Cited by 9 publications

References 13 publications

A Primal-Dual Variant of the IRI-IMAI Algorithm for Linear Programming

A Primal-Dual Variant of the IRI-IMAI Algorithm for Linear Programming

An Interior-Point Algorithm for Linearly Constrained Optimization

Les effets de l'exposant de la fonction barrière multiplicative dans les méthodes de points intérieurs

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