A Modified Penalty Functions Method For Inequality Constraints Minimization

We present and analyze a modified penalty functions (MPF) method for solving constrained optimization problems with inequality constraints. This method combines the modified penalty and the Lagrangian methods. It is based on MPF, which treats inequality constraints with a modified penalty term. The MPF method alternatively minimizes the MPF and updates the Lagrange multipliers. The MPF method avoids the in-differentiability of the max {x, 0}. For a large enough penalty parameter the MPF method is shown to converge linearly under the standard second order optimality conditions. Super-linear convergence can be achieved by increasing the penalty parameter after each Lagrange multiplier update.This paper is organized as follows. In Chapter 1, the background and the main results of this paper are given. The general nonlinear programming problem and the basic assumptions under which our convergence results hold are introduced in Chapter 2. In Chapter 3, we give the definition of the MPF which our method is based on, the MPF method and the trust region algorithm. The convergence results for the MPF method, some aspects concerning the practical implementation and some concluded remarks of the method are discussed in Chapter 4.