Gradient regularization of Newton method with Bregman distances

Abstract In this paper, we propose a first second-order scheme based on arbitrary non-Euclidean norms, incorporated by Bregman distances. They are introduced directly in the Newton iterate with regularization parameter proportional to square root of norm current gradient. For basic scheme, as applied composite convex optimization problem, establish global convergence rate order $$O(k^{-2})$$ O ( k - 2 ) both terms functional residual and subgradients. Our main assumption smooth part objective is Lipschitz continuity its Hessian. uniformly functions degree three, justify linear rate, for strongly function prove local superlinear convergence. approach can be seen relaxation Cubic Regularization method (Nesterov Polyak Math Program 108(1):177–205, 2006) minimization problems. This preserves properties complexities case, while auxiliary subproblem at each iteration simpler. We equip our adaptive search procedure choosing parameter. also an accelerated $$O(k^{-3})$$ 3 , where k counter.