Hybrid globalization of convergence of the Levenberg-Marquardt method for equality-constrained optimization problems

The Levenberg-Marquardt method possesses local superlinear convergence for general systems of nonlinear equations under weal assumptions allowing for nonisolated solutions. This justifies its application to first-order optimality systems of constrained optimization problems with possibly violated constraint qualifications, the latter giving rise to nonuniqueness of Lagrange multipliers. However, the existing strategies for globalization of convergence of the Levenberg-Marquardt method are not optimization-oriented by nature, i.e., when applied to optimization problems, they are intended not for finding solutions, but rather any stationary points of such problems. In this work, we propose optimization-oriented globalization strategies for the Levenberg-Marquardt method applied to optimization problems with equality constraints. The proposed strategies are hybrid by their character, i.e., they combine a globally convergent optimization outer phase method with asymptotic switching to the Levenberg-Marquardt method. Global convergence properties and superlinear rate of convergence are established. Numerical results are provided, demonstrating that the proposed hybrid algorithms are workable.