Perturbation Analysis Of Optimization Problem Over The Epigraph Of The Weighted L1 Norm

It is known that l₁optimization problem has been widely applied in compressive sensing,image processing and statistical optimization,especially compressive sensing lays an important theoretical support for the acquisition,transmission and storage of vast information and image.This dissertation focuses on the study the perturbation analysis of the optimization problem over the epigraph of the weighted₁lnorm,which is denoted asK₁^w.The main results of this dissertation may be summarized as follows:Chapter 1 introduces the derivation of the optimization problem over the epigraph of the weighted₁l norm and its application in practical engineering.The research process of the theoretical analysis of various types of constrained problems has been mainly reviewed,as well as the results of the analysis of the varational analysis of special structural cones.The research contents and the main conclusions are summarized at the end of the chapter.Chapter 2 clearly derives some important geometical varational properties associated withK₁^w.First,a serial of knowledge about the varational properties on a closed convex cone was given as the basis of the subsequent analysis.The duality ofK₁^wandK₁^ww,which is called the epigraph of the weighted_?lnorm,was proved.Then,the the expression of the tangent cone and its linearity space and the normal cone are all derived due to the structure of K₁^w.Next,the critical cone is described based on the calculation formula of the projector of K₁^wandK₁^ww,its affine cone is also given by the relationship between the critical cone and its affine cone of a closed convex cone.Chapter 3 derives the differential properties of the projector overK₁^wwconsidering that K₁^wandK₁^wware dual cones.This Chapter emphasizes on the derivations of the directional derivative,B-subdifferential and Clarke's generalized Jacobian of the projector overK₁^ww.They are all derived in specific expressions laying a solid foundation for the analysis of the next chapters.Chapter 4 is based on the results of the previous chapters,which states the theoretical results of the perturbation analysis of the optimization problem overK₁^wsystematically.Chapter 4 first introduces three crucial conditions of the optimization problem:the strict Robinson constraint qualication?SRCQ?;the strong second-order sufficient optimality condition?SOSC?and constraint nondegeneracy.Then,under the Robinson constraint qualication?RCQ?,the equivalence of the following conditions is proved:the strong second-order sufficient optimality condition and constraint nondegeneracy,the strong regularity of the Karush-Kuhn-Tucker point and the nonsingularity of Clarke's generalized Jacobian of the Karush-Kuhn-Tucker system.Also,the equivalence between the second order growth condition and the strong second-order sufficiency condition is illustrated.Combining all the results,the main theorem for the optimization problem of the weighted L1 norm is present,namely,under RCQ,there are ten equivalent conditions:strong stability of the optimal solution is equivalent to the Lipschitz homeomorphism of the KKT mapping,which are equivalent to the above four conclusions,and other four conclusions.Chapter 5 provides an important characterization of the robust isolated calmness for the optimization problem overK₁^w,which indicates that under the RCQ,the KKT solution mapping is robustly isolated clam if and only if the strict RCQ and the second order sufficient condition both hold.