Second-order methods for quartically-regularised cubic polynomials, with applications to high-order tensor methods.
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| Title: | Second-order methods for quartically-regularised cubic polynomials, with applications to high-order tensor methods. |
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| Authors: | Cartis, Coralia1 (AUTHOR) cartis@maths.ox.ac.uk, Zhu, Wenqi1 (AUTHOR) wenqi.zhu@maths.ox.ac.uk |
| Source: | Mathematical Programming. Jan2026, Vol. 215 Issue 1/2, p669-715. 47p. |
| Subject Terms: | *NONCONVEX programming, *MATHEMATICAL optimization, *NUMERICAL analysis, *MATHEMATICAL regularization |
| Abstract: | There has been growing interest in high-order tensor methods for nonconvex optimization, with adaptive regularization, as they possess better/optimal worst-case evaluation complexity globally and faster convergence asymptotically. These algorithms crucially rely on repeatedly minimizing nonconvex multivariate Taylor-based polynomial sub-problems, at least locally. Finding efficient techniques for the solution of these sub-problems, beyond the second-order case, has been an open question. This paper proposes a second-order method, Quadratic Quartic Regularisation (QQR), for efficiently minimizing nonconvex quartically-regularized cubic polynomials, such as the ARp sub-problem (Birgin et al. Math Program 163:359–368, 2017) with p = 3 . Inspired by Nesterov (Quartic regularity, 2022), QQR approximates the third-order tensor term by a linear combination of quadratic and quartic terms, yielding (possibly nonconvex) local models that are solvable to global optimality. In order to achieve accuracy ϵ in the first-order criticality of the sub-problem in finitely many iterations, we show that the error in the QQR method decreases either linearly or by at least O (ϵ 4 / 3) for locally convex iterations, while in the nonconvex case, by at least O (ϵ) ; thus improving, on these types of iterations, the general cubic-regularization bound. Preliminary numerical experiments indicate that two QQR variants perform competitively with state-of-the-art approaches such as ARC (also known as ARp with p = 2 ), achieving either lower objective value or iteration counts. [ABSTRACT FROM AUTHOR] |
| Database: | Academic Search Index |
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Nájsť tento článok vo Web of Science | Abstract: | There has been growing interest in high-order tensor methods for nonconvex optimization, with adaptive regularization, as they possess better/optimal worst-case evaluation complexity globally and faster convergence asymptotically. These algorithms crucially rely on repeatedly minimizing nonconvex multivariate Taylor-based polynomial sub-problems, at least locally. Finding efficient techniques for the solution of these sub-problems, beyond the second-order case, has been an open question. This paper proposes a second-order method, Quadratic Quartic Regularisation (QQR), for efficiently minimizing nonconvex quartically-regularized cubic polynomials, such as the ARp sub-problem (Birgin et al. Math Program 163:359–368, 2017) with p = 3 . Inspired by Nesterov (Quartic regularity, 2022), QQR approximates the third-order tensor term by a linear combination of quadratic and quartic terms, yielding (possibly nonconvex) local models that are solvable to global optimality. In order to achieve accuracy ϵ in the first-order criticality of the sub-problem in finitely many iterations, we show that the error in the QQR method decreases either linearly or by at least O (ϵ 4 / 3) for locally convex iterations, while in the nonconvex case, by at least O (ϵ) ; thus improving, on these types of iterations, the general cubic-regularization bound. Preliminary numerical experiments indicate that two QQR variants perform competitively with state-of-the-art approaches such as ARC (also known as ARp with p = 2 ), achieving either lower objective value or iteration counts. [ABSTRACT FROM AUTHOR] |
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| ISSN: | 00255610 |
| DOI: | 10.1007/s10107-025-02235-y |