Reproduce “Global optimality under amenable symmetry constraints” on OpenResearch

About this paper

Peter Orbanz

The paper develops a framework for symmetrizing convex optimization under amenable group actions. It replaces infinite-group averaging with Følner averages over increasingly representative sets inside a candidate’s closed convex invariant orbitope. When that orbitope is compact, an invariant limit point exists with no worse objective for convex lower-semicontinuous invariant objectives; convergence is generally guaranteed only along an unspecified subsequence.

Key insights

• Amenable groups provide the necessary structure to generalize averaging principles, allowing the construction of invariant objects as limits of Følner averages, even for infinite groups.
• Orbitopes serve as the minimal geometric structures encoding the interplay between convexity, group action, and the vector space, making them central to understanding global optimality under symmetry constraints.
• The 'cocycle trick' extends the framework beyond strict invariance to more generalized symmetry properties (e.g., equivariance), broadening the applicability of the toolkit.

Summary by alphaXiv.