Stability Analysis Of Several Classes Of Conic Constrained Optimization Problems

Cone constrained optimization problems refer to optimization problems whose constrained mapping belong to a certain closed convex cone.Such problems are widely used in the fields of finance,statistics,machine learning,engineering and so on.It is often difficult to obtain accurate solutions when solving practical problems,so it is necessary to study stability analysis theory of cone constrained optimization problems,which plays an important role in the convergence anal-ysis of numerical algorithm.This dissertation is mainly devoted to the study of stability analysis of three classes of cone constrained optimization problems,including the nonlinear semidefinite programming problems,the second-order cone optimization problems and the C~2-cone reducible problems.The main results of this dissertation can be summarized as follows:1.Chapter 3 studies the stability analysis of the nonlinear semidefinite programming(NLS-DP)problem.First,consider the perturbed problem of NLSDP which is more general than the C~2-smooth parameterization,this perturbed problem does not require the differentiability of pa-rameter vector.By using the implicit function theorem,we prove that when the NLSDP prob-lem satisfies the Jacobian uniqueness conditions at a feasible point,the perturbed problem also satisfies the Jacobian uniqueness conditions at some feasible solution,and this locally optimal solution is continuous with respect to the parameter vector.Secondly,by using the graphical derivative criterion of isolated calmness,we demonstrate that the second order sufficient con-dition and strict Robinson constraint qualification are sufficient for the isolated calmness of the stationary point mapping,and are both sufficient and necessary for the isolated calmness of the Karush-Kuhn-Tucker(KKT)mapping.Thirdly,under the assumption of the KKT point is strong regular,we establish the equivalence between the parabolic second order directional differentia-bility of the normal mapping and the KKT mapping.Finally,by assuming the metric regularity of the locally optimal solution,the qualitative and quantitative stability analysis of the locally optimal solution set and optimal value function of the NLSDP problem is given.2.Chaper 4 focuses on the stability analysis of the second-order cone optimization prob-lems.First,the main conclusion about Jacobian uniqueness conditions of the NLSDP problem is fully extended to the nonlinear second-order cone optimization problem.For the standard linear second-order cone optimization problem,due to its special structure,we prove that the strong second order sufficient condition is equivalent to the dual constraint nondegeneracy,and then three conditions equivalent to the strong regularity of the KKT point are obtained.3.chapter 5 is devoted to the study of the stability analysis of the C~2-cone reducible prob-lems.First,based on the properties of the C~2-cone reducible sets,we prove that the conclusions of the previous two chapters about the Jacobian uniqueness conditions are also valid for the C~2-cone reducible problems.In addition,for linear composite optimization problems,we characterize the Robinson constraint qualification,constraint nondegeneracy,strict Robinson constraint qualifi-cation and first order and second order optimality conditions.Then the sufficient condition for the strong regularity of the KKT point and the sufficient and necessary condition for the isolat-ed calmness of the KKT mapping are given.Finally,when the linear composite optimization problem is convex,the duality theory is used to establish the equivalence between the quadratic growth condition and the calmness of the optimal solution set of the parameter problem.More-over,we show that the KKT point is strong regular if and only if the Aubin property holds at this point.