Perturbation Analysis Of Optimization Problems Over Symmetric Cones

This dissertation focuses on the study of variational analysis of symmetric cones and perturbation analysis of optimization over symmetric cones. The main results, obtained in this dissertation, may be summarized as follows:1. Chapter 2, based on the theory of Euclidean Jordan algebras, develops the elements in the variational analysis of symmetric cones related to the optimization problem. The formulas of tangent cones, second order tangent sets of symmetric cones and the B-subdifferential of the projection operator onto a symmetric cone are derived. A linear-quadratic function is introduced and the relationship with the supporting function of the second order tangent set is established.2. Chapter 3, based on the variational analysis developed in Chapter 2, studies the perturbation analysis of the nonlinear symmetric conic optimization problem. Firstly, the outer second order regularity of symmetric cones is proved and the no gap optimality conditions are obtained. Secondly, two types of strong second order sufficient optimality conditions for the optimization over symmetric cones are proposed, one of which is described through the linear-quadratic form and the other through the supporting function of the second order tangent set. For the nonconvex conic optimization problems whose two strong second order sufficient conditions coincide, we show that for a locally optimal solution, the following conditions are equivalent: the strong second order sufficient condition and primal constraint non-degeneracy, the strong regularity of the generalized equation corresponding to the KKT system, the nonsingularity of the Clarke's generalized subdifferential of the nonsmooth mapping (KKT mapping for short) corresponding to the KKT conditions at the KKT point, strong stability of the optimal solution and other 4 conditions. Moreover, for a convex conic programming problem, the equivalence among all the listed conditions still holds when the cone is symmetric. Besides, each of these conditions is equivalent to the nonsingularity of B-subdifferential of the KKT mapping. Especially, for a linear conic programming problem, the relationship between the second order sufficient condition and the dual strict constraint qualification is characterized. Moreover, in this case the above 10 equivalent conditions for convex programming are proven to be equivalent to both the primal and dual constraint non-degeneracies.3. In Chapter 4, formulas for second order tangent sets of second order cone, the cone of positively semidefinite symmetric real matrices, the cone of positively semidefinite complex Hermitian matrices, and the cone of positively semidefinite quaternion Hermitian matrices, are developed and in each case the linear-quadratic form is proved to coincide with the supporting function of the second order tangent set. Therefore, when K is represented as a cone isomorphic to a Cartesian product of a finite number of these four types of cones, the two strong second order sufficient conditions are equivalent and all equivalent conditions about perturbation analysis in Chapter 3 are valid when K is of this kind of form.