Geodesic
Dual multiplicative algorithms for an entropy-linear programming problem
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 49 (2009) no. 3, pp. 453-464

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A multiplicative-barrier generalization of the Cauchy gradient descent method is proposed and studied. The technique is used to search for dual variables in the entropy maximization problem with affine constraints, which arises, for example, in the simulation of equilibria in macroscopic systems. For this class of problems, the dual variables can be used to effectively determine the primal ones. The global convergence of the iterative algorithms proposed is proved. 
@article{ZVMMF_2009_49_3_a5,
     author = {E. V. Gasnikova},
     title = {Dual multiplicative algorithms for an entropy-linear programming problem},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {453--464},
     publisher = {mathdoc},
     volume = {49},
     number = {3},
     year = {2009},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2009_49_3_a5/}
}
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E. V. Gasnikova. Dual multiplicative algorithms for an entropy-linear programming problem. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 49 (2009) no. 3, pp. 453-464. http://geodesic.mathdoc.fr/item/ZVMMF_2009_49_3_a5/