Efficient Implementation of Third-order Tensor Methods with Adaptive Regularization for Unconstrained Optimization (2501.00404v2)

Abstract: High-order tensor methods that employ local Taylor models of degree $p$ within adaptive regularization frameworks (AR$p$) have recently received significant attention, due to their optimal global and local rates of convergence for both convex and nonconvex optimization problems. However, their numerical performance for general unconstrained optimization problems remains insufficiently explored, which we address by showcasing the numerical performance of standard second- and third-order variants ($p=2,3$) and proposing novel techniques for key algorithmic aspects when $p\geq3$ to improve numerical efficiency. To improve the adaptive choice of the regularization parameter, we extend the interpolation-based updating strategy introduced in (Gould, Porcelli, and Toint, 2012) for $p=2$ to $p\geq3$. We identify fundamental differences between the local minima of regularized subproblems for $p=2$ and $p\geq3$ and their effect on performance. Then, for $p\geq3$, we introduce a novel pre-rejection technique that rejects poor subproblem minimizers (referred to as transient') before any function evaluation, reducing cost and selecting useful (persistent') ones. Numerical studies confirm efficiency improvements in our modified AR$3$ algorithm. We also assess the effect of different subproblem termination conditions and the choice of the initial regularization parameter on overall performance. Finally, we benchmark our best-performing AR$3$ variants, along with those in (Birgin et al., 2020), against second-order ones (AR$2$). Encouraging results on standard test problems confirm that AR$3$ variants can outperform AR$2$ in terms of objective evaluations, derivative evaluations, and subproblem solves. We provide an efficient, extensive, and modular MATLAB software package including various AR$2$ and AR$3$ variants, allowing ease of use and experimentation for interested users.