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On the convergence of Newton iterations to non-stationary points. (English) Zbl 1072.90038

The paper deals with the sensitivity of the convergence behavior of Newton’s method from the chosen merit function and the applied step size procedure. This sensitivity occurs if singular non-stationary points exist. Using SVD of the Jacobian of the iterates the single Newton steps are studied in detail and more light is shed upon the local gains in a single step of the method. So the reader can clearly understand the cause of failing.
Turning to unconstrained minimization the authors derive conditions that the iterates are bounded away from singular points. Thus it can ensure that Newton’s method converges to a stationary point. Finally constrained problems are taken into consideration via the log-barrier function. It is shown by two examples that Newton’s method may fail due to the almost singular behavior of the Hessian of the obtained Lagrangian.

MSC:

90C30 Nonlinear programming
90C51 Interior-point methods
65H10 Numerical computation of solutions to systems of equations
90C55 Methods of successive quadratic programming type

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