109. Efficient Adaptive Regularized Tensor Methods
Invited abstract in session WB-2: High-order and tensor methods, stream Nonsmooth and nonconvex optimization.
Wednesday, 10:30-12:30Room: B100/7011
Authors (first author is the speaker)
| 1. | Yang Liu
|
| Mathematical Institute, University of Oxford | |
| 2. | Karl Welzel
|
| Mathematical Institute, University of Oxford | |
| 3. | Coralia Cartis
|
| Mathematical Institute, University of Oxford | |
| 4. | Raphael Hauser
|
| Oxford University Computing Laboratory, Oxford University | |
| 5. | Wenqi Zhu
|
| University of Oxford |
Abstract
High-order tensor methods employing local Taylor approximations have attracted considerable attention for convex and nonconvex optimization. The pth-order adaptive regularization (ARp) approach builds a local model comprising a pth-order Taylor expansion and a (p+1)th-order regularization term, delivering optimal worst-case global and local convergence rates. However, for p≥2, subproblem minimization can yield multiple local minima, and while a global minimizer is recommended for p=2, effectively identifying a suitable local minimum for p≥3 remains elusive. This work extends interpolation-based updating strategies, originally proposed for p=2, to cases where p≥3, allowing the regularization parameter to adapt in response to interpolation models. Additionally, it introduces a new prerejection mechanism to discard unfavorable subproblem minimizers before function evaluations, thus reducing computational costs for p≥3. Numerical experiments, particularly on Chebyshev-Rosenbrock problems with p=3, indicate that the proper use of different minimizers can significantly improve practical performance, offering a promising direction for designing more efficient high-order methods.
Keywords
- Second- and higher-order optimization
- Complexity and efficiency of algorithms
- Optimization software
Status: accepted
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