A Class Of Nonlinear Lagrange Methods For Solving Nonconvex Semidefinite Programming

This dissertation focuses on studying the convergence rate of a class of nonlinear Lagrange methods for solving nonconvex semidefinite programming.The Lagrangians are generated by L(o|¨)wner operators and the constraint functions in the Lagrangians are involved in nonlinear ways.After a brief introduction of the background about semidefinite programming and the development of the nonlinear Lagrange methods,Chapter 2 presents the preliminaries and describes the main results obtained in this dissertation.The preliminaries include the theory of optimality conditions and the differential formulas of L(o|¨)wner operators.The main results, mainly in Chapter 3,Chapter 4 and Chapter 5,are summarized as follows:1.Chapter 3 establishes a theory framework of the class of nonlinear Lagrange methods for solving nonconvex semidefinite programming.First,we provide a series of conditions that the L(o|¨)wner operators should satisfy and assume that our problem satisfies constraint nondegeneracy condition,the strict complementarity condition and the second order sufficient condition.Second,the differential properties of the nonlinear Lagrangians are discussed. Finally we analyze the rate of convergence of the nonlinear Lagrange algorithms with subproblems exactly solved.The convergence analysis shows that the algorithms are locally convergent when the penalty parameter is less than a threshold and the error bound of the primal-dual solutions is proportional to the penalty parameter.Compared to the convergence analysis of the nonlinear Lagrangian method for nonlinear programming,we have to overcome the difficulty caused by the sigma term in the second order sufficient condition of nonconvex semidefinite programming.Some comments ought to be made about the difference between this dissertation and Stingl's. First,a set of equality constraints are included in the constraint of the problem considered. Second,the convergence theorem given in this dissertation is elegant compared to that of Stingl,which completely corresponds to the convergence theorem(e.g.Theorem 4.1 in Polyak) in the nonlinear programming case.Additionally,the conditions for the L(o|¨)wner operators in this dissertation are weaker than the ones in Stingl's work.2.Based on the assumptions in Chapter 3,Chapter 4 presents the convergence analysis of the nonlinear Lagrange algorithm with subproblems inexactly solved.This convergence theorem also shows the algorithms are locally convergent when the penalty parameter is less than a threshold and the error bound of primal-dual solutions is proportional to the penalty parameter.3.Chapter 5 enumerates five specific nonlinear Lagrangians.We prove that the L(o|¨)wner operators in these Lagrangians satisfy the conditions mentioned in Chapter 3,which means that the algorithms constructed by these Lagrangians are locally convergent and the error bound of the primal-dual solutions is proportional to the penalty parameter.