A Smooth Approximation-BFGS Method For Inequality Constrained Nonlinear Programming

In this paper, a smooth approximation-BFGS method for solving inequality constrained nonlinear programming is presented. By using the entropy function approximate to the constrained functions, a smooth approximation problem of the original problem is derived. By using the BFGS correction to approximate the inverse of the Hessian matrix of the Lagrangian function, the subproblem that positive definite quadratic programming with only one linear constraint is formulated, it is always uniquely solveable and the explicit solution is obtained. Thus, the general quadratic programming or the linear equations subproblem which is usually required in the exisiting constrained nonlinear programming methods is avoided and the fast convergence property of the BFGS method is preserved. The algorithms with the merit function of both the nondifferentiable exact penalty function and the smooth exact penalty function are discussed. Numerical results show the effectiveness of the method that the iterative sequence generated by the method is fastly approched the quite small neighborhood of the solution with low requirement of computation.