FRINK Linked Data Fragments server

Matches in SemOpenAlex for { <https://semopenalex.org/work/W1967960319> ?p ?o ?g. }

• W1967960319 endingPage "499" @default.
• W1967960319 startingPage "499" @default.
• W1967960319 abstract "This series of papers studies a geometric structure underlying Karmarkar’s projective scaling algorithm for solving linear programming problems. A basic feature of the projective scaling algorithm is a vector field depending on the objective function which is defined on the interior of the polytope of feasible solutions of the linear program. The geometric structure studied is the set of trajectories obtained by integrating this vector field, which we call $P$-trajectories. We also study a related vector field, the affine scaling vector field, and its associated trajectories, called $A$-trajectories. The affine scaling vector field is associated to another linear programming algorithm, the affine scaling algorithm. Affine and projective scaling vector fields are each defined for linear programs of a special form, called strict standard form and canonical form, respectively. This paper derives basic properties of $P$-trajectories and $A$-trajectories. It reviews the projective and affine scaling algorithms, defines the projective and affine scaling vector fields, and gives differential equations for $P$-trajectories and $A$-trajectories. It shows that projective transformations map $P$-trajectories into $P$-trajectories. It presents Karmarkar’s interpretation of $A$-trajectories as steepest descent paths of the objective function $left langle {{mathbf {c}},{mathbf {x}}} right rangle$ with respect to the Riemannian geometry $d{s^2} = sum nolimits _{i = 1}^n {d{x_i}d{x_i}/x_i^2}$ restricted to the relative interior of the polytope of feasible solutions. $P$-trajectories of a canonical form linear program are radial projections of $A$-trajectories of an associated standard form linear program. As a consequence there is a polynomial time linear programming algorithm using the affine scaling vector field of this associated linear program: This algorithm is essentially Karmarkar’s algorithm. These trajectories are studied in subsequent papers by two nonlinear changes of variables called Legendre transform coordinates and projective Legendre transform coordinates, respectively. It will be shown that $P$-trajectories have an algebraic and a geometric interpretation. They are algebraic curves, and they are geodesics (actually distinguished chords) of a geometry isometric to a Hubert geometry on a polytope combinatorially dual to the polytope of feasible solutions. The $A$-trajectories of strict standard form linear programs have similar interpretations: They are algebraic curves, and are geodesics of a geometry isometric to Euclidean geometry." @default.
• W1967960319 created "2016-06-24" @default.
• W1967960319 creator A5005612207 @default.
• W1967960319 creator A5087231176 @default.
• W1967960319 date "1989-02-01" @default.
• W1967960319 modified "2023-10-18" @default.
• W1967960319 title "The nonlinear geometry of linear programming. I. Affine and projective scaling trajectories" @default.
• W1967960319 cites W1039203514 @default.
• W1967960319 cites W1494436834 @default.
• W1967960319 cites W1528546869 @default.
• W1967960319 cites W1533868172 @default.
• W1967960319 cites W1559225673 @default.
• W1967960319 cites W1569339351 @default.
• W1967960319 cites W1573409602 @default.
• W1967960319 cites W1598917891 @default.
• W1967960319 cites W161930422 @default.
• W1967960319 cites W1680579736 @default.
• W1967960319 cites W1964599748 @default.
• W1967960319 cites W1968425795 @default.
• W1967960319 cites W1972505767 @default.
• W1967960319 cites W1973117121 @default.
• W1967960319 cites W1978561325 @default.
• W1967960319 cites W1979152594 @default.
• W1967960319 cites W1981053712 @default.
• W1967960319 cites W1985978629 @default.
• W1967960319 cites W1993982015 @default.
• W1967960319 cites W1996591064 @default.
• W1967960319 cites W1997422357 @default.
• W1967960319 cites W2014746566 @default.
• W1967960319 cites W2019987083 @default.
• W1967960319 cites W2022182590 @default.
• W1967960319 cites W2025655081 @default.
• W1967960319 cites W2026021734 @default.
• W1967960319 cites W2052895085 @default.
• W1967960319 cites W2086920040 @default.
• W1967960319 cites W21055737 @default.
• W1967960319 cites W2110786726 @default.
• W1967960319 cites W2110845347 @default.
• W1967960319 cites W2116800096 @default.
• W1967960319 cites W2126479355 @default.
• W1967960319 cites W2137723938 @default.
• W1967960319 cites W2611147814 @default.
• W1967960319 cites W3141205388 @default.
• W1967960319 cites W71992527 @default.
• W1967960319 doi "https://doi.org/10.1090/s0002-9947-1989-1005525-6" @default.
• W1967960319 hasPublicationYear "1989" @default.
• W1967960319 type Work @default.
• W1967960319 sameAs 1967960319 @default.
• W1967960319 citedByCount "65" @default.
• W1967960319 countsByYear W19679603192012 @default.
• W1967960319 countsByYear W19679603192013 @default.
• W1967960319 countsByYear W19679603192014 @default.
• W1967960319 countsByYear W19679603192015 @default.
• W1967960319 countsByYear W19679603192018 @default.
• W1967960319 countsByYear W19679603192019 @default.
• W1967960319 countsByYear W19679603192020 @default.
• W1967960319 countsByYear W19679603192021 @default.
• W1967960319 crossrefType "journal-article" @default.
• W1967960319 hasAuthorship W1967960319A5005612207 @default.
• W1967960319 hasAuthorship W1967960319A5087231176 @default.
• W1967960319 hasBestOaLocation W19679603191 @default.
• W1967960319 hasConcept C11413529 @default.
• W1967960319 hasConcept C145691206 @default.
• W1967960319 hasConcept C202444582 @default.
• W1967960319 hasConcept C2524010 @default.
• W1967960319 hasConcept C2779219049 @default.
• W1967960319 hasConcept C33923547 @default.
• W1967960319 hasConcept C41045048 @default.
• W1967960319 hasConcept C91188154 @default.
• W1967960319 hasConcept C92757383 @default.
• W1967960319 hasConcept C9652623 @default.
• W1967960319 hasConcept C99844830 @default.
• W1967960319 hasConceptScore W1967960319C11413529 @default.
• W1967960319 hasConceptScore W1967960319C145691206 @default.
• W1967960319 hasConceptScore W1967960319C202444582 @default.
• W1967960319 hasConceptScore W1967960319C2524010 @default.
• W1967960319 hasConceptScore W1967960319C2779219049 @default.
• W1967960319 hasConceptScore W1967960319C33923547 @default.
• W1967960319 hasConceptScore W1967960319C41045048 @default.
• W1967960319 hasConceptScore W1967960319C91188154 @default.
• W1967960319 hasConceptScore W1967960319C92757383 @default.
• W1967960319 hasConceptScore W1967960319C9652623 @default.
• W1967960319 hasConceptScore W1967960319C99844830 @default.
• W1967960319 hasIssue "2" @default.
• W1967960319 hasLocation W19679603191 @default.
• W1967960319 hasOpenAccess W1967960319 @default.
• W1967960319 hasPrimaryLocation W19679603191 @default.
• W1967960319 hasRelatedWork W16136492 @default.
• W1967960319 hasRelatedWork W2001123062 @default.
• W1967960319 hasRelatedWork W2005933292 @default.
• W1967960319 hasRelatedWork W2082565038 @default.
• W1967960319 hasRelatedWork W2114342468 @default.
• W1967960319 hasRelatedWork W2123493177 @default.
• W1967960319 hasRelatedWork W2150122870 @default.
• W1967960319 hasRelatedWork W2371322041 @default.
• W1967960319 hasRelatedWork W2618042834 @default.
• W1967960319 hasRelatedWork W2803769118 @default.
• W1967960319 hasVolume "314" @default.

• next