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Hdr Année : 2017

Conic optimization: affine geometry of self-concordant barriers and copositive cones

Optimisation conique : géométrie affine des barrières auto-concordantes et cônes copositifs

Résumé

The thesis consists of two independent parts. The first part is devoted to the relation between affine differential geometry and logarithmically homogeneous self-concordant barriers on regular convex cones. The level surfaces of the barrier are considered as centro-affine immersions into the ambient real space containing the cone. The second and third derivatives of the barrier are put in correspondence to the affine metric and the cubic form of the immersion, respectively. The self-concordance condition is shown to be equivalent to the boundedness of the cubic form. As an application, a universal barrier is constructed on arbitrary regular convex cones, the canonical barrier. Self-scaled barriers are shown to be exactly those for which the cubic form is parallel with respect to the geodesic flow of the affine metric. As an application, the proper affine hyperspheres with parallel cubic form are classified. The second part contains a summary of different results on the extreme rays of the copositive cone. The extreme rays of the cone of order 5 are classified completely. The concept of minimal zero is introduced as a tool to coarsely classify the extreme copositive matrices of arbitrary order. An application to the cone of order 6 is performed. The boundary of the copositive cone is studied and a criterion of extremality in terms of the minimal zeros is given. A large class of extremal copositive matrices with a specific subset of zeros whose supports have a circular symmetry is constructed.
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Dates et versions

tel-01570016 , version 1 (28-07-2017)

Identifiants

  • HAL Id : tel-01570016 , version 1

Citer

Roland Hildebrand. Conic optimization: affine geometry of self-concordant barriers and copositive cones. Optimization and Control [math.OC]. Université Grenoble Alpes, 2017. ⟨tel-01570016⟩
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