Resolvent Ellipsoid for Minty Set Inclusions

원문 초록

We study Minty set inclusion on a compact convex set $\mathcal{X} \subseteq \mathbb{R}^d$ with a set-valued operator $A : \mathcal{X} \rightrightarrows \mathbb{R}^d$. Under the Minty promise, the goal is to compute $x \in \mathcal{X}$ and $u \in A(x)$ such that $\sup_{y \in \mathcal{X}} \langle u, x - y \rangle \le \varepsilon$. Our approach is geometric and we work in the fixed-scale constrained-resolvent oracle model, where a query at anchor $a$ is assumed to return a resolvent point $x^+$ together with the Yosida vector $g_{\eta}(a) = (a - x^+)/\eta$. We prove that each oracle response yields a sharp dichotomy: either the returned point provides an $\varepsilon$-SVI certificate, or the Yosida vector defines a strict separating normal for the Minty set. Thus, a single black-box resolvent query supplies either a certified candidate solution or a valid geometric cut. Combining this separation principle with the ellipsoid method, we obtain RESOLVENT-ELLIPSOID, an algorithm that computes an $\varepsilon$-SVI in $\mathcal{O}\!\left(\operatorname{poly}(d, \log(1/\varepsilon))\right)$ black-box oracle calls to the fixed-scale constrained resolvent. For monotone inclusions, our algorithm gives, to our knowledge, the first $\mathcal{O}\!\left(\mathrm{poly}(d, \log(1/\varepsilon))\right)$ black-box guarantee in the fixed-scale resolvent model.