Globalized piecewise Levenberg–Marquardt method with a procedure for avoiding convergence to nonstationary points

The modern version of the Levenberg–Marquardt method for constrained equations possess strong properties of local superlinear convergence, allowing for possibly nonisolated solutions and possibly nonsmooth equations. A related globally convergent variant of the algorithm for the piecewise-smooth case, based on linesearch for the squared Euclidian norm residual, has recently been developed. Global convergence of this algorithm to stationary points for some active smooth selections has been shown, and examples demonstrate that no any stronger global convergence properties can be established for this algorithm without further modifications. In this paper, we develop such a modification of the globalized piecewise Levenberg–Marquardt method, that avoids undesirable accumulation points, thus achieving the intended property of B-stationarity of accumulation points for the problem of minimization of the squared Euclidian norm residual of the original equation over the constraint set. The construction consists of identifying smooth selections active at potential accumulation points by means of an appropriate error bound for an active smooth selection employed at the current iteration, and then switching to a more promising identified selection when needed. Global convergence to B-stationary points and asymptotic superlinear convergence rate are established, the latter again relying on an appropriate error bound property, but this time for the solutions of the original constrained equation.