Could log barrier be implicit in barrier based linear programming solvers ?
Résumé
Given a matrix $A\in \mathbb{Z}^{M\times N}$, computing an approximate solution of the unconstrained optimization problem $\underset{v\in \mathbb{R}^M}{\min} \ v^TAA^Tv - \mathbf{1}^T\log(v)$ is a straightforward way to get a solution of the homogeneous linear program $Ax > \mathbf{0}$ (as general as linear programming).
This can be done by newton method as the function is self concordant.
Yet, this method becomes very complex when dealing with matrix $A$ with arbitrary large number: one has to carefully round the internal state at each step, but, such rounding is made difficult as evaluating the function is even not straightforward.
This paper wonder if minimizing directly $v^TAA^Tv$ while maintaining $v>\mathbf{0}$ could be done in polynomial time.
This could allow to remove (or precisely make implicit) the logarithm.
Domaines
Recherche opérationnelle [math.OC]
Origine : Fichiers produits par l'(les) auteur(s)
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