A Nonlinear Lagrangian Based On NCP Function

It is very interesting that the well-known proximal augmented Lagrangian is justgenerated by minimum NCP function. In the same way, the famous Fischer-Burmeister NCPfunction can also generate a nonlinear Lagrangian, which has well convergence properties.This dissertation proposes a nonlinear Lagrangian based on a modified Fischer-BurmeisterNCP function for solving nonlinear optimization problem with inequality constraints..1. Chapter1mainly discusses a nonlinear Lagrangian generated by integral operatorbased on a modified Fischer-Burmeister NCP function.Under certain assumptions, theproperties of the function at the Kuhn-Tucker point(x*, u*)are analyzed. Furthermore,the convergence of the nonlinear Lagrange method based on the NCP function isdiscussed. The convergence theorem shows that the sequence of points generated bythis nonlinear Lagrange algorithm is locally convergent when the penalty parameter isless than a threshold under a set of suitable conditions on problem functions, and theerror bound of dual solution depending on the penalty parameter is also established.Finally, the condition number of the nonlinear Lagrangian Hessian at the optimalsolution is discussed. It is shown that the condition number of the nonlinear LagrangianHessian at the optimal solution is proportional to the controlling penalty parametert1.It suggests that t should not be too small in the actual calculation. Otherwise，we willencounter some numerical difficulties in solving mx∈IiRnn F(x,u,t)when we apply somemethods depended on the condition number of the nonlinear Lagrangian Hessian forsolving unconstrained optimization problems.2. Chapter2develops the dual approach based on the dual functiond (u,t)=F(x(u,t),u,t). Firstly, under the assumptions, the duality theorem isdemonstrated. The optimal function value of primal problem equals to the optimalfunction value of dual function. Furthermore, the dual algorithm associated with thisnonlinear Lagrangian is developed.