{"paper":{"title":"Fast minimization of structured convex quartics","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OC","authors_text":"Brian Bullins","submitted_at":"2018-12-26T15:54:35Z","abstract_excerpt":"We propose faster methods for unconstrained optimization of \\emph{structured convex quartics}, which are convex functions of the form \\begin{equation*} f(x) = c^\\top x + x^\\top \\mathbf{G} x + \\mathbf{T}[x,x,x] + \\frac{1}{24} \\mathopen\\| \\mathbf{A} x \\mathclose\\|_4^4 \\end{equation*} for $c \\in \\mathbb{R}^d$, $\\mathbf{G} \\in \\mathbb{R}^{d \\times d}$, $\\mathbf{T} \\in \\mathbb{R}^{d \\times d \\times d}$, and $\\mathbf{A} \\in \\mathbb{R}^{n \\times d}$ such that $\\mathbf{A}^\\top \\mathbf{A} \\succ 0$. In particular, we show how to achieve an $\\epsilon$-optimal minimizer for such functions with only $O(n^{"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1812.10349","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}