An Exact Penalty Function Method For Nonlinear Constrained Optimization Problem

In this paper, we study the penalty function method for solving nonlinear constrained programming problems. First of all, we introduce the development of penalty function methods briefly, especially some typical exact ones. Then with the knowledge of penalty functions which have been studied, we propose a new penalty function, which is an exact one. Through analysis we find that this exact penalty function is a piecewise differentiable one. When the current iterative point x^k lies in the external or internal part of the feasible region, the exact penalty function is differentiable. Therefore we can use the traditional nonlinear programming methods to minimize the penalty function in that area. But if the point of iteration lies near the boundary of the feasible region, this penalty function is non-differentiable, so we could not use the traditional nonlinear programming methods directly.Using the properties of piecewise differentiable function, together with the idea in T.F.Coleman and A.R.Conn (1982), we design the following numerical algorithm: letεbe a small positive number used to identify theε-active region. If the current iterative point x^k lies outside theε- active region, we can use quasi-Newton method directly (where the approximate Hessian B_k is updated using BFGS equation). When x^k lies in theε- active region, we use the L₁ exact penalty function instead. Then we can separate the L₁ exact penalty function into two parts: the differentiable part and the non-differentiable part. A useable descent direction is then derived by attempting to decrease the differentiable part while trying to maintain the value of the non-differentiable part.The process is repeated until the terminal condition satisfied.It is shown that this approach has global convergence properties under certain conditions. Numerical results of the approach show that it is suitable for nonlinear constrained problems and has good numerical stability.