Counterexamples to Whole-Sequence Convergence of Variable-Smoothing Full-Splitting Methods

Abstract

We study whole-sequence convergence of the smoothing-based full-splitting proximal subgradient method (S-FSPS) for structured nonconvex and nonsmooth fractional programs, introduced by Bo\c{t}, Li, and Tao (SIAM J. Optim., 35(4):2623--2653, 2025) as Algorithm~4.1. Existing theory guarantees only the existence of a subsequence converging to a limiting lifted stationary point. We show that this guarantee is sharp by constructing two admissible instances whose corresponding primal sequences both have cluster set $\{1\}\times\mathbb S^1$ and infinite length, although every cluster point is a limiting lifted stationary point. The construction prescribes a slowly rotating spiral and realizes it exactly through a compatible first-order jet and a $C^{1,1}$ Whitney extension. In the first instance, $A$ has rank one and every smoothing-dual iterate is nonzero. In the second, the feasible set is full-dimensional, $A$ has full row rank, and each of $f\circ K$, $g\circ A$, and the numerator $g\circ A+h$ is nonconstant on the feasible set. The first instance also yields a nonconvergent example for the corresponding variable-smoothing, single-loop, full-splitting method for nonconvex and nonsmooth composite optimization, although that method still admits a subsequence converging to an exact stationary point. Thus, a vanishing but nonsummable smoothing schedule does not imply whole-sequence convergence.