Beyond First-Order Methods for $\ell_p$-Structured Non-Monotone Variational Inequalities
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arXiv
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| Natura: | Preprint |
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2026
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| _version_ | 1866914392566661120 |
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| author | Vyas, Abhijeet Bullins, Brian |
| author_facet | Vyas, Abhijeet Bullins, Brian |
| contents | We propose novel high-order algorithms for a class of $\ell_p$-structured non-monotone variational inequalities. In particular, work by Diakonikolas et al. (2021), which introduced the weak Minty variational inequality (weak-MVI) setting, showed how to find an approximate first-order Euclidean stationary point for a strictly positive range of the weak-MVI parameter $ρ$. However, for the $\ell_p$-norm stationary point setting ($p \neq 2$), their guarantees are limited to $ρ=0$, which recovers the standard MVI setting. In this work, we address this gap by presenting a suite of high-order methods that converge to $\ell_p$-norm stationary points for a suitable range of $ρ> 0$, thereby circumventing previous fundamental challenges in $\ell_p$ settings. We further show convergence for high-order smooth \textit{monotone} operators, generalizing Adil et al. (2022) to the case where $p \geq 2$, and we extend our Euclidean techniques to continuous-time settings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_13491 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Beyond First-Order Methods for $\ell_p$-Structured Non-Monotone Variational Inequalities Vyas, Abhijeet Bullins, Brian Optimization and Control We propose novel high-order algorithms for a class of $\ell_p$-structured non-monotone variational inequalities. In particular, work by Diakonikolas et al. (2021), which introduced the weak Minty variational inequality (weak-MVI) setting, showed how to find an approximate first-order Euclidean stationary point for a strictly positive range of the weak-MVI parameter $ρ$. However, for the $\ell_p$-norm stationary point setting ($p \neq 2$), their guarantees are limited to $ρ=0$, which recovers the standard MVI setting. In this work, we address this gap by presenting a suite of high-order methods that converge to $\ell_p$-norm stationary points for a suitable range of $ρ> 0$, thereby circumventing previous fundamental challenges in $\ell_p$ settings. We further show convergence for high-order smooth \textit{monotone} operators, generalizing Adil et al. (2022) to the case where $p \geq 2$, and we extend our Euclidean techniques to continuous-time settings. |
| title | Beyond First-Order Methods for $\ell_p$-Structured Non-Monotone Variational Inequalities |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2603.13491 |