Establishing Boundary KKT Convergence of Mirror Descent through Reparameterization

2026-08-07

AI Summary

This work addresses the challenge that mirror descent methods in non-convex optimization often fail to converge to Karush–Kuhn–Tucker (KKT) points when iterates approach the boundary of the feasible domain. To overcome this, the authors propose a definable boundary-extension reparameterization scheme grounded in metric flattening. Convergence of the reparameterized objective is established via the Kurdyka–Łojasiewicz (KL) inequality, and continuity of the inverse mapping is leveraged to recover convergence of the original iterates to KKT points. This study provides the first theoretical guarantee for mirror descent converging to KKT points without excluding boundary-limiting behavior, offering verifiable conditions that jointly account for the objective function, Legendre kernel, and feasible set geometry. The framework is validated on canonical examples including Shannon entropy, Fermi–Dirac entropy, and power kernels, laying foundational groundwork for convergence analysis of generalized Bregman-type algorithms.

Abstract

We prove that mirror descent converges to a KKT point for the nonconvex problem without excluding boundary limits. The result holds under verifiable conditions that jointly couple the objective, the Legendre kernel, and the feasible geometry. The key ingredient to establish the convergence is a metric-flattening reparameterization \(S\) that admits a definable boundary extension. Applying the KL argument to the reparameterized objective yields convergence of \(S(x_k)\). Continuity of \(S^{-1}\) then recovers convergence to the KKT point of the original sequence. We further apply our general framework to some concrete examples: Shannon entropy, Fermi--Dirac entropy, and power kernels. Future work may consider more general constraint geometries and genuinely nonseparable kernels, and extend mirror descent to broader Bregman-type methods, e.g. Bregman proximal point algorithms and Bregman ADMM, and their inexact variants.

Problem

• Mirror Descent
• KKT convergence
• nonconvex optimization
• boundary limits
• reparameterization

Innovation

• mirror descent
• KKT convergence
• reparameterization
• metric flattening
• nonconvex optimization