A Nonlinear Lagrange Method For Solving Nonconvex Semidefinite Programming Problems

Classical Lagrangians in which the multiplier vectors and the constraint mappings areinvloved in linear ways, play an important role in studies on the duality theories of con-vex programmings,especially that of linear programmings and quadratic programmingswhich should be express through Classical Lagrangians. But in nonconvex programmings,the primal problems and the duality problems which are based on Classical Lagrangianshave duality gaps. So many sholars are become more and more interested in studyingon the varies of Classical Larangians.Nonlinear Lagrangians are variants of the classicalLagrangian, in which the multiplier vectors or constraint functions are involved in nonlin-ear ways. Nonlinear Lagrange methods are dual methods based on nonlinear Lagrangiansfor solving optimization problems.As dual methods usually require no restrictions on thefeasibility of primal variables, nonlinear Lagrange methods are playing important rolesin solving constrained optimization problems. This desseration attempts to extend thismethod to solving NCSDP for the proficiency of nonlinear lagrangeian alogrithm solv-ing nonlinear programming and extensive utilization of NCSDP in practical problems.we construct a dual alogrithm basing on a nonlinear lagrangian for solving NCSDP andthen prove its local convergence.This is to say,the sequence producted by the alogrithmconverge to KKT points of primal problem. And at last we establish the error estimateformula, the main results,obtained in this disseration,may be summerized as follows:1.Chapter2 is devoted to summarizing the optimal conditions of NCSDP which areindispensable to the next chapter.This chapter at first introduces the optimal conditionsof abstract constraint optimization.And then utilizes those conditions to the NCSDP.2.Chapter3 is devoted to bringing forward a nonlinear Lagrange algorithm for NCSDPprogramming and proving its convergence.Namely, under some mild assumptions,it is provedthat there exists a threshold of the penalty parameter such that these sequences gener-ated by the nonlinear algorithm locally converge to the KKT point of NCSDP when thepenalty parameter is less than the threshold. Moreover, we establish the error bound ofthe solutions with the penalty parameter.