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A computational comparison of two simplicial decomposition approaches for the separable traffic assignment problems : RSDTA and RSDVI
Draft pel 4th Meeting del Euro Working Group on Transportation (Newcastle 9-11 setembre de 1.996)The class of simplicial decomposition methods has shown to constitute efficient tools for the solution of the variational inequality formulation of the general traffic assignment problem. The paper presents a particular implementation of such an algorithm, called RSDVI, and a restricted simplicial decomposition algorithm, developed adhoc for diagonal, separable, problems named RSDTA. Both computer codes are compared for large scale separable traffic assignment problems. Some meaningful figures are shown for general problems with several levels of asymmetry.Preprin
The convex hull of a finite set
We study -separately convex hulls of finite
sets of points in , as introduced in
\cite{KirchheimMullerSverak2003}. When is considered as a
certain subset of matrices, this notion of convexity corresponds
to rank-one convex convexity . If is identified instead
with a subset of matrices, it actually agrees with the quasiconvex
hull, due to a recent result \cite{HarrisKirchheimLin18}.
We introduce " complexes", which generalize constructions. For a
finite set , a " -complex" is a complex whose extremal points
belong to . The "-complex convex hull of ", , is the union
of all -complexes. We prove that is contained in the
convex hull .
We also consider outer approximations to convexity based in the
locality theorem \cite[4.7]{Kirchheim2003}. Starting with a crude outer
approximation we iteratively chop off "-prisms". For the examples in
\cite{KirchheimMullerSverak2003}, and many others, this procedure reaches a
" -complex" in a finite number of steps, and thus computes the
convex hull.
We show examples of finite sets for which this procedure does not reach the
convex hull in finite time, but we show that a sequence of outer
approximations built with -prisms converges to a -complex. We
conclude that is always a " -complex", which has interesting
consequences
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