Stability Study For A Class Of Conic Optimization And Generalized Equations

Conic optimization and generalized equation,especially nonpolyhedral matrix conic optimization and generalized equation,have a very wide range of applications in many fields such as statistics,control and system identification,signal and image processing,machine learning,and so on.The Aubin property,isolated calmness and strong calmness of set-valued mappings is not only the core of perturbation analysis of optimization,but also is the key to the convergence rate analysis of numerical algorithms for optimization.This thesis is concerned with these Lipschitz-type properties of the solution mapping to the C2-cone reducible canonically perturbed conic optimization and parameterized generalized equation.For the C2-cone reducible canonically perturbed conic optimization,Chapter 3 of the thesis focuses on the Aubin property,isolated calmness and strong calmness of its KKT solution mapping,stationary point mapping and multiplier set mapping,and achieves the following main conclusions:(1)the Aubin property of the multiplier set mapping at the reference point implies that of the KKT solution mapping at the corresponding point,while the latter is equivalent to that of the stationary point mapping at the reference point and the constraint nondegeneracy of this point;(2)the isolated calmness of the KKT solution mapping at the reference point is equivalent to one of the following three groups of conditions:the isolated calmness of the multiplier set mapping at the corresponding point and the noncriticality of the multipliers,the strict Robinson constraint qualification and the noncriticality of the multipliers,the strict Robinson constraint qualification and the isolated calmness of the stationary point mapping at the corresponding point;(3)the strong calmness of the KKT solution mapping is equivalent to both the local error bound of the KKT system and the pseudo-isolated calmness of the stationary point mapping along with the calmness of the multiplier set mapping.Among others,the noncriticality of the multipliers implies the pseudo-isolated calmness of the stationary point mapping.For the C2-cone reducible parameterized generalized equation,Chapter 4 character-izes the graphical derivative of the normal cone mapping to the conic constraint set and establish the low estimation for its regular coderivative and the upper estimation for its coderivative without the constraint nondegeneracy of the reference point,and first obtain the isolated calmness characterization and the Aubin property of the solution mapping to the generalized equation without requiring the nondegeneracy of the reference point.Chapter 5 is devoted to the application of isolated calmness in deriving exact recov-ery conditions for low-rank and sparse optimization.We construct the solution mappings to the corresponding affine constrained nuclear norm optimization problems from the primal and dual angle,respectively,derive the exact recovery condition by characterizing the isolated calmness of these mappings,and provide the lower bound for the number of samples of the stochastic sampling operator.In particular,the isolated-calmness type exact recovery conditions derived from the dual angle is equivalent to the strict Robinson constraint qualification of the dual problems,while the deterministic exact recovery con-ditions provided by Candes and Recht[12]and Chandrasekaran et al.[15]are stronger than the constraint nondegeneracy of the dual problems.