Cubic Regularized Newton Method for the Saddle Point Models: A Global and Local Convergence Analysis

In this paper, we propose a cubic regularized Newton method for solving the convex-concave minimax saddle point problems. At each iteration, subproblem is constructed and solved, which provides search direction iterate. With properly chosen stepsizes, shown to converge with global linear local superlinear convergence rates, if function gradient Lipschitz strongly-convex-strongly-concave. case that merely convex-concave, homotopy continuation (or path-following) method. Under Lipschitz-type error bound condition, present an iteration complexity of $${\mathcal {O}}\left( \ln \left( 1/\epsilon \right) $$ reach $$\epsilon -solution through approach, becomes ^{\frac{1-\theta }{\theta ^2}}\right) under Hölderian-type condition involving parameter $$\theta ( $$0<\theta <1$$ ).