The symmetry measure of a convex body \(S\subset\mathbb{R}^n\) is given by: \[ \mathrm{sym}(S):=\max\{\alpha\ge0:\text{ there exists }x\in S\text{ such that } -\alpha(S-x)\subseteq S-x\}\,, \]where such an \(x\) is called a Minkowski center. We prove that every convex body \(S\) admits a \(\sqrt{\frac{n}{\mathrm{sym}(S)}}\)-rounding of \(S\), namely, there exists an origin-centered ellipsoid \(E\) and a center \(c\) such … Read more
In this paper, we develop a systematic framework for deriving explicit error bounds for semidefinite feasibility problems without assuming strict feasibility (Slater’s condition), a setting in which existing results are limited. Our main technical contribution is the introduction of radial-type H\”{o}lder error bounds, where the error bound constant depends explicitly on the norm of the … Read more
In this paper, we propose a unified framework for accelerated gradient methods, dubbed UGM, which subsumes a wide range of accelerated and conventional gradient-type methods designed for minimizing $L$-smooth and $\mu$-strongly convex functions. We demonstrate that the iteration update of the proposed framework can be intrinsically interpreted as a hybrid combination of the heavy-ball method … Read more
We classify the facial exposedness of completely positive cones over symmetric cones in terms of the rank of the associated Euclidean Jordan algebras. The completely positive cones are facially exposed when the rank is at most $2$, but are not facially exposed when the rank is at least $5$. Facial exposedness is not completely determined … Read more
We give a simple example that disproves the following 2015 conjecture of Lacoste-Julien and Jaggi concerning the pyramidal width (aka facial distance): The pyramidal width of a set of vertices is non-increasing when another vertex is added (assuming that all previous points remain vertices). In contrast to the recent example by Zhao (arXiv:2607.29555), our counterexample … Read more
Inspired by the progressive decoupling of linkages methodology for optimization and variational inequalities, we propose an algorithm for solving quasi-variational inequalities as a sequence of variational inequalities. Our method is shown to converge locally under some regularity conditions and globally when such conditions hold throughout the entire domain. Separately, under other type of assumptions, global … Read more
Singularity degree is an important measure of semidefinite programming (SDP) degeneracy, but it is generally unavailable a priori from the problem data. We augment the Shor relaxation of binary sets \(\{x\in\{0,1\}^n:Ax=b\}\) with the first-level Reformulation–Linearization Technique (RLT) equations generated by the defining linear equalities. For the resulting equality-generated SDP–RLT relaxation, we determine the exact worst-case … Read more
We study decentralized minimization of an average of strongly convex, smooth local objectives over an intersection of agent-private closed convex sets, where each agent knows only its own objective and its own set and agents communicate over a gossip network. We show that the complexity is controlled by a single geometric scalar, which we call … Read more
This work considers the design of first-order convex optimization algorithms and convergence proofs. In particular, we consider nonsmooth Lipschitz and smooth problems accessed through a subgradient or gradient oracle, respectively. For the general class of fixed-step first-order methods, prior work on Performance Estimation Problems (PEPs) has shown that structured, tight convergence proofs typically exist. Under … Read more
In nonsmooth optimization, identifiable sets describe the local region eventually reached by sequences converging to a prescribed critical point. When such a set is a \(C^2\) manifold on which the objective restricts to a \(C^2\) function, it is called an identifiable manifold. Their appeal lies in what they enable: many first-order methods identify these manifolds … Read more