Global Convergence of Semismooth Newton Methods for Quadratic Problems

Abstract

This paper investigates the semismooth Newton method and its application to solving constrained quadratic optimization problems. We begin by reviewing various generalized concepts of differentiability, such as Clarke's generalized Jacobian, slanting functions, and Newton differentiability, providing a comparative analysis to clarify their differences. We then establish the equivalence between the semismooth Newton method (SSN) and the primal-dual active set algorithm (PDASA), and key global convergence results are summarized. Our study focuses on the cycle behavior of small-dimensional quadratic problems. For the case $n=2$, we prove global convergence for arbitrary quadratic problems, demonstrating the robustness of the method in this setting. A necessary condition for cycles of certain lengths is derived and used to identify possible cycling patterns, when $n=3$. While only two of the cycling patterns were observed in randomly generated examples, the other remain a theoretical possibility, suggesting further exploration of the method's behavior.