Research On A Nonlinear Lagrange Method For Second-order Cone Programming

The second-order cone programming problems have wide applications in engineering design,control,finance,continuous layout optimization,robust optimization,combinatorial optimization and so on.Although the second-order cone programming problem can be transformed into semi-definite programming,the computational complexity will be increased,and the transformed semi-definite programming problem no longer has the structural advantages of the original second-order cone programming problem.Therefore,this paper conducts independent research on second-order cone programming problems.Considering the nonlinear Lagrangian method has lots of advantages,such as no requirement on the penalty parameter tending to 0 or?and the initial point being feasible,we propose a nonlinear Lagrangian method for nonconvex second-order cone programming problems.The main works are summarized as follows:(1)We study a nonlinear Lagrangian function for nonconvex second-order cone programming problems based on a L(?)wner operator associated with a potential function for the optimization problems with inequality constraints.With the help of the theoretical knowledge of second-order cone programming,the properties of the L(?)wner operator are discussed in detail.When the constraint non-degeneracy condition,strict complementary condition and the second-order sufficient condition hold,the nonlinear Lagrange function is proven to have good differential properties.(2)Based on the good properties of the nonlinear Lagrangian,a nonlinear Lagrangian method for second-order cone programming problems is established.Under some mild assumptions,the rate of convergence of the nonlinear Lagrangian method is studied when the sub-problem is assumed to be solved exactly.The convergence results show that the sequence of points generated by the proposed method is locally convergent when the penalty parameter is less than a threshold,and the error bound of solution is proportional to the penalty parameter.(3)Considering that the computational cost of finding the exact solution of the sub-problem is too high.We propose a criterion for the approximate solution of the sub-problem,establish the convergence theorem of the nonlinear Lagrangian method when each approximate solution satisfies this criterion and get the same convergence results as the exact solution of the sub-problem.(4)Based on the proposed algorithm,the preliminary numerical experiment is performed.The obtained results are compared with those of the other algorithms,and the numerical results show that the algorithm proposed in this thesis is feasible and effective.