Some improved projected quasi-Newton algorithms and their convergence. I: Methods and global behavior (Q1824993)

The minimization problem min f(x) with nonlinear equality constraints \(c(x)=0\) is considered where \(f: R^ n\to R^ 1\) and \(c: R^ n\to R^ m\) are twice continuously differentiable functions and \(m\leq n\). Furthermore is assumed that the Hessian matrices \(\nabla^ 2f\) and \(\nabla^ 2c_ i\) are Lipschitz continuous. By improving the method of \textit{J. Nocedal} and \textit{M. L. Overton} [SIAM J. Numer. Anal. 22, 821- 850 (1985; Zbl 0593.65043)] the authors obtain a global convergence of the iterative sequence \(\{x_ k\}\) to a Kuhn-Tucker point. The basic strategy for this result is to use the \(l_ 1\)-nonsmooth penalty function and Fletcher's differentiable penalty function or that of \textit{P. T. Boggs} and \textit{J. W. Tolle} [J. Optimization Theory Appl. 31, 17- 26 (1980; Zbl 0416.90059)].