A General Merit Function-Based Global Convergent Framework for Nonlinear Optimization

In this paper, we revisit the convergence theory of the inexact restoration paradigm for nonlinear optimization. The paper first identifies the basic elements of a globally convergent method based on merit functions. Then, the inexact restoration method that employs a two-phase iteration is introduced as a special case. A specific implementation is presented that is supported in the solution of regularized subproblems. The proposed inexact restoration method includes more freedom in the computation of iterates and a novel procedure that integrates the computation of the penalty parameter with the optimization phase. It also includes an acceleration step based on solving a classic quadratic programming subproblem. The proposed theoretical framework is broad and flexible enough to provide a convergence theory for inexact restoration applications tailored to specific problems. Theoretical results include asymptotic convergence theory as well as a worst-case iteration and evaluation complexity analysis of the introduced methods. The paper concludes with numerical experiments that illustrate the practical performance of two variations of the proposed method for solving general nonlinear programming problems.