A Nonlinear Lagrangian For Constrained Optimization

The nonlinear rescaling principle employs monotone and sufficiently smooth functions to transform the constraints and objective function into an equivalent problem, The classical Lagrangian has important properties on the primal and the dual spaces. The application of The nonlinear rescaling principle to constrained optimization problems leads to a class of modified barrier functions and modified barrier function methods. Being classical Lagrangian for an equivalent problem, the modified barrier function combine the best properties of the classical Lagrangian and classical barrier functions.Due to the excellent modified barrier functions properties, new characteristics of the dual pair for convex programming problems have been found and the duality theory for nonconvex constrained optimization has been developed. In the modern times a lot of scholars have studied the nonlinear constrained optimization problems and a lot of functions have been constructed. These functions combine the best properties of the classical Lagrangian and classical barrier functions, which also have been as the interior augumented Lagrangian.This paper establishes another nonlinear Lagrangian for solving nonlinear programming problems with inequality constraints. The function has excellent properties and there exists a thres-hold such that the sequence of primal-dual iterate points generated by dual algorithm basing on the nonlinear Lagrangian locally converges to a minimizer if the penalty parameter is more than the threshold. It also report that the nonlinear Lagrangian is strongly convex in a neighborhood. Moreover , the paper develops corresponding duality theory and talks about the convexity and smoothes of the duality. Finally ,the paper uses new Lagrangian to solve some problems, such as Polar2,polar3,Wong2 and Wong3 ,and reports some numerical results by using the proposed dual algorithm.