Differentiation Properties Of The Metric Projectors Over The Epigraph Of The Class Of Norm

The metric projector plays a key role in the study of theoretical analysis and algorithmsof many optimization problems. For example, the metric projector has important applicationsin solving the Lagrangian dual problems via the classic proximal point algorithm and theprojected gradient algorithms and carrying out the sensitivity analysis of the relevantoptimization problems. This thesis focuses on the metric projectors over epigraphs of theweighted l1norm, the weighted infinite norm and the weighted k-norm. We derive thedirectional derivatives of these metric projectors and the B-subdifferential of the metricprojection over the epigraph of the weighted infinite norm. The outline of this thesis isorganized as follows:Chapter1introduces the background, the significance and the preliminary results of theprojection operators.Chapter2characterizes the explicit expressions of the projection operators on severalconvex sets, including the generalized closed convex cone, the epigraph of the weightedinfinite norm, the epigraph of the weighted11norm and the epigraph of the weighted k-norm.Based on the achieved results for the projection operators, we calculate the directionalderivative of the projection operators over the epigraph of the weighted infinite norm.Chapter3focuses on the directional derivatives of the projection operators on theepigraphs of the weighted infinite norm and the weighted11norm. Furthermore, we alsocharacterize the B-subdifferential of the projection operator on the epigraph of the weightedinfinite norm.Chapter4derives the directional derivative of the projection operator on the epigraph ofthe weighted k-norm. Two child algorithms have been set up firstly, and then due to the resultson these two algorithms, the projection operators on four kinds of convex polyhedral conesare calculated. Finally, by applying the results of the study of four kinds of convex polyhedralcones, we obtain the directional derivative of the projection operator over the epigraph of theweighted k-norm.