Cubic Regularization Method based on Mixed Factorizations for Unconstrained Minimization

Newton’s method for unconstrained optimization, subject to proper regularization or special trust-region procedures, finds first-order stationary points with precision $\varepsilon$ employing, at most, $O(\varepsilon^{-3/2})$ functional and derivative evaluations. However, the computer work per iteration of the best-known implementations may need several factorizations per iteration or may use rather expensive matrix decompositions. In this paper, we … Read more

Cubic-regularization counterpart of a variable-norm trust-region method for unconstrained minimization

In a recent paper we introduced a trust-region method with variable norms for unconstrained minimization and we proved standard asymptotic convergence results. Here we will show that, with a simple modification with respect to the sufficient descent condition and replacing the trust-region approach with a suitable cubic regularization, the complexity of this method for finding … Read more

Algebraic rules for quadratic regularization of Newton’s method

In this work we propose a class of quasi-Newton methods to minimize a twice differentiable function with Lipschitz continuous Hessian. These methods are based on the quadratic regularization of Newton’s method, with algebraic explicit rules for computing the regularizing parameter. The convergence properties of this class of methods are analysed. We show that if the … Read more