A Nonlinear Lagrangians For Nonconvex Semidefinite Programming

Classical Lagrangians in which the multiplier vectors and the constraint mappings are involved in linear ways, play an important role in studies on the duality theories of convex programming. In nonconvex programming, the primal problems and the duality problems which are based on classical lagrangians have duality gaps. So many scholars are become more and more interested in studying on the varies of classical lagrangians. Nonlinear Lagrangians are variants of the classical lagrangians, in which the multiplier vectors or constraint functions are involved in nonlinear ways. Nonlinear lagrange methods are dual methods based on nonlinear lagrangians for solving optimization problems. This paper proposes a class of nonlinear lagrangians for solving nonconvex semidefinite programming. The convergence analysis shows that the algorithms are locally convergent when the penalty parameter is less than a threshold. Then we establish the error estimate formula. The main results are summarized as follows:1. Chapter2 is devoted to summarizing the optimal conditions of NCSDP. This chapter at first introduces the optimal conditions of abstract constraint optimization, and then utilizes those conditions to the NCSDP.2. Chapter3 proposes a class of nonlinear lagrangians for solving nonconvex semidefinite programming. When the NCSDP satisfies the strict complemnentarity condition and the second order sufficient condition, the convergence theorem indicates 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.