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Logiciel Année : 2015

PGR

Jean Charles Gilbert

Résumé

The code PGR solves the optimization problem

   infx f(x)
   s.t. Ax=b,
where f is a polyhedral gauge on ℝn, A is an m×n matrix and b ∈ ℝn.

A gauge is a function f: ℝn → ℝ+ ∪ {+∞} that is convex, nonnegative (f(x) ≥ 0 for all x ∈ ℝn), positively homogeneous of degree one (f(tx) = tf(x) for all x ∈ ℝn and t > 0), and that vanishes at the origin (f(0) = 0). The gauge is said to be polyhedral if its epigraph is a polyhedron. The polyhedral gauge is supposed to be defined by its polyhedral 1-sublevel set {x ∈ ℝn: f(x) ≤ 1}, which is the convex hull of the columns of a given matrix C plus the conic (convex) hull of the columns of a given matrix D. It is assumed that zero belongs to that sublevel set, but nothing is done to check this (it would require too much computational effort); hence the answer may not be correct if the assumption is not verified.

The problem is rewritten as a linear optimization problem and solved using the Matlab function Linprog. The solver can detect uniqueness of the computed solution if any.

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