Geodesic
A generalized Gibbs' lemma and a Wardrop equilibrium
Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, Tome 15 (2019) no. 2, pp. 199-211

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In the article, a generalized Gibbs' lemma is stated and proved. A conclusion of this lemma corresponds to a definition of Wardrop equilibrium in transport networks. This allows us to naturally introduce a well known convex programming problem with linear constraints whose solution is a Wardrop equilibrium vector. The complicated definition of the Wardrop equilibrium is analyzed in detail (typical examples are given). The reason of the Braess paradox' appearance is specified. A large example, that illustrates how the Wardrop equilibrium vector changes when a road with zero driving time is added into the transport network, is also given.
Keywords: generalized Gibbs' lemma, Wardrop equilibrium, Braess paradox, convex programming. 
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     author = {V. N. Malozemov and N. A. Solovyeva},
     title = {A generalized {Gibbs'} lemma and a {Wardrop} equilibrium},
     journal = {Vestnik Sankt-Peterburgskogo universiteta. Prikladna\^a matematika, informatika, processy upravleni\^a},
     pages = {199--211},
     publisher = {mathdoc},
     volume = {15},
     number = {2},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VSPUI_2019_15_2_a3/}
}
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V. N. Malozemov; N. A. Solovyeva. A generalized Gibbs' lemma and a Wardrop equilibrium. Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, Tome 15 (2019) no. 2, pp. 199-211. http://geodesic.mathdoc.fr/item/VSPUI_2019_15_2_a3/