发表机构
School of Industrial Engineering, Purdue University; National University of Singapore（普渡大学工业工程学院; 新加坡国立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过度量扁平化重参数化S，证明镜像下降法在非凸问题中可收敛到KKT点，还将该框架应用于Shannon熵等示例，为后续扩展Bregman型方法奠定基础。
AI 中文摘要
我们证明了对于非凸问题，镜像下降法会收敛到KKT点，且不排除边界极限。该结果在可验证条件下成立，这些条件将目标函数、Legendre核与可行几何结构耦合在一起。建立收敛性的关键是一个可定义边界扩展的度量扁平化重参数化S。将KL论证应用于重参数化后的目标函数，可得到S(x_k)的收敛性，再结合S⁻¹的连续性，即可恢复原序列到KKT点的收敛性。我们还将该通用框架应用于Shannon熵、Fermi-Dirac熵和幂核等具体示例。未来工作可考虑更一般的约束几何结构、真正非可分的核，并将镜像下降法扩展到更广泛的Bregman型方法，如Bregman近端点算法、Bregman ADMM及其不精确变体。
英文摘要
Sequence convergence to a boundary Karush--Kuhn--Tucker (KKT) point has long remained unclear for nonconvex mirror descent with Legendre kernels. The difficulty arises from the blow-up of the gradient of the Legendre kernel at the boundary. Recent work~\cite{dingtoh2026nonkkt} shows that mirror descent can accumulate at non-KKT boundary points despite decreasing objective values, precluding a convergence guarantee to KKT points in general. Despite this negative result, mirror descent remains effective in many real applications. Motivated by this contrast, we address the boundary difficulty directly and establish KKT convergence of mirror descent for a broad class of structured nonconvex problems. We analyze mirror descent in reparameterized variables, where the Hessian metric is flattened and remains nondegenerate as the boundary is approached. Under extension and definability conditions jointly coupling the objective, the Legendre kernel, and the feasible region, the reparameterized sequence has finite length and converges, thereby recovering convergence to a KKT point of the original sequence. Our general framework applies to some concrete instances: Shannon entropy, Fermi--Dirac entropy, and power kernels on polyhedron.
Comments23 pages
