Affine Scaling and Logarithmic Barrier Interior-Point Methods

2020 ◽  
pp. 495-514
Author(s):  
Craig A. Tovey
Keyword(s):  
Interior Point ◽  
Interior Point Methods ◽  
Affine Scaling ◽  
Logarithmic Barrier

scholarly journals On the relationship of interior-point methods

1993 ◽  
Vol 16 (3) ◽  
pp. 565-572
Author(s):  
Ruey-Lin Sheu ◽  
Shu-Cherng Fang
Keyword(s):  
Interior Point ◽  
Barrier Function ◽  
Affine Scaling ◽  
Logarithmic Barrier Function ◽  
Scaling Method ◽  
Dual Affine ◽  
Primal Dual ◽  
Affine Scaling Method ◽  
Relationship Of ◽  
Logarithmic Barrier

In this paper, we show that the moving directions of the primal-affine scaling method (with logarithmic barrier function), the dual-affine scaling method (with logarithmic barrier function), and the primal-dual interior point method are merely the Newton directions along three different algebraic “paths” that lead to a solution of the Karush-Kuhn-Tucker conditions of a given linear programming problem. We also derive the missing dual information in the primal-affine scaling method and the missing primal information in the dual-affine scaling method. Basically, the missing information has the same form as the solutions generated by the primal-dual method but with different scaling matrices.

NEWTON FLOW AND INTERIOR POINT METHODS IN LINEAR PROGRAMMING

2005 ◽  
Vol 15 (03) ◽  
pp. 827-839 ◽  
Author(s):  
JEAN-PIERRE DEDIEU ◽  
MIKE SHUB
Keyword(s):  
Linear Programming ◽  
Vector Field ◽  
Interior Point ◽  
Barrier Function ◽  
Interior Point Methods ◽  
Logarithmic Barrier Function ◽  
Programming Theory ◽  
Central Paths ◽  
Logarithmic Barrier

We study the geometry of the central paths of linear programming theory. These paths are the solution curves of the Newton vector field of the logarithmic barrier function. This vector field extends to the boundary of the polytope and we study the main properties of this extension: continuity, analyticity, singularities.

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