On Global Convergence of A Trust Region and Affine Scaling Methodfor Nonlinearly Constrained Minimization

Abstract

A nonlinearly constrained optimization problem can be solved by the exact penalty approach involving nondifferentiable functions $\sum_{i} |c_i(x)|$ and $\sum_{i}\max(0,c_i(x))$. In \cite{Li94a}, a trust region affine scaling approach based on a 2-norm subproblem is proposed for solving a nonlinear $l_1$ problem. The (quadratic) approximation and the trust region subproblem are defined using affine scaling techniques. Explicit sufficient decrease conditions are proposed to obtain a limit point satisfying complementarity, dual feasibility, and second order optimality. In this paper, we present the global convergence properties of this new approach.