Optimality Conditions for Nonconvex Semidefinite Programming and Augmented Lagrangian Method

Abstract: This dissertation is devoted to discussing the optimality conditions and analyzing the augmented Lagrangian method for nonconvex semidefinite programming. The main contents, in this dissertation, may be summarized as follows:1. Chapter2, mainly summarizing the optimality conditions for the nonconvex semidefinite programming, first recalls the concept of transversality condition, proves that transversality condition and linear independence condition are equivalent, and shows that under strict complementarity condition, the transversality condition is a sufficient condition for uniqueness of the Lagrange multipliers. Based on the properties of constraint set of nonconvex semidefinite programming, we derive a concrete expression for constraint set's first-order tangent cone and second-order tangent set, and establish the optimahty conditions for nonconvex semidefinite programming under some assumptions.2. Chapter3, being the main part of this dissertation, converts nonconvex semidefinite programming problem (SDP) to nonlinear programming problem (ESDP), and proves that under transversality condition, strict complementarity condition and the sufficient optimality condition of local solutions, (SDP) and (ESDP) are equivalent locally. Moreover, we present a convergence theorem of the augmented Lagrangian method for the nonlinear programming problem (ESDP). The convergence theorem shows that the augmented Lagrangian method is locally convergent when the penalty parameter c is greater than a threshold and the error bound of primal-dual solutions is proportional to c-1…
Key words: Nonconvex Semidefinite Programming; First-order Tangent Cone; Second-order Tangent Set; Augmented Lagrangian; Transversality Condition

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