Adaptive Regularization Minimization Algorithms with Non-Smooth Norms and Euclidean Curvature

Serge Gratton, Philippe TOINT

Research output: Working paper

Abstract

A regularization algorithm (AR1pGN) for unconstrained nonlinear minimization is considered, which uses a model consisting of a Taylor expansion of arbitrary degree and regularization term involving a possibly non-smooth norm. It is shown that the non-smoothness of the norm does not affect the O(epsilon_1^{-(p+1)/p}) upper bound on evaluation complexity for finding first-order epsilon_1-approximate minimizers using p derivatives, and that this result does not hinge on the equivalence of norms in Re^n. It is also shown that, if p=2, the bound of O(epsilon_2^{-3}) evaluations for finding second-order
epsilon_2-approximate minimizers still holds for a variant of AR1pGN named AR2GN, despite the possibly non-smooth nature of the regularization term. Moreover, the adaptation of the existing theory for handling the non-smoothness results in an interesting modification of the subproblem termination rules, leading to an even more compact complexity analysis. In particular, it is shown when the Newton's step is acceptable for an adaptive regularization method. The approximate minimization of quadratic polynomials regularized with non-smooth norms is then discussed, and a new approximate second-order necessary
optimality condition is derived for this case. An specialized algorithm is then
proposed to enforce the first- and second-order conditions that are strong enough to ensure the existence of a suitable step in AR1pGN (when p=2) and in AR2GN, and its iteration complexity is analyzed.
Original languageEnglish
PublisherArxiv
Number of pages27
Volume2105.07765
Publication statusPublished - 28 May 2021

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