Failure of Whole-Sequence Convergence in Variable-Smoothing Full-Splitting Methods

Abstract

We investigate whole-sequence convergence of the smoothing-based full-splitting proximal subgradient method (S-FSPS) for structured nonconvex and nonsmooth fractional programs, introduced as Algorithm~4.1 by Boţ, Li, and Tao (\emph{SIAM J. Optim.}, 35(4):2623--2653, 2025). The existing convergence theory establishes only the existence of subsequences converging to exact limiting lifted stationary points, leaving open whether the entire generated sequence must converge. We answer this question in the negative by constructing two admissible instances whose primal sequences have cluster set \(\{1\}\times\mathbb S^1\) and infinite-length trajectories, although every cluster point is an exact limiting lifted stationary point. In the first construction, the linear operator \(A\) is nonzero and every generated dual iterate is nonzero. In the second, the feasible set is full-dimensional, \(A\) has full row rank, and both \(f\circ K\) and \(g\circ A\) are nonconstant on the feasible set. The first construction also yields a nonconvergent example for the corresponding variable-smoothing, single-loop, full-splitting method for nonconvex and nonsmooth composite optimization, although a subsequence converges to an exact limiting stationary point. To the best of our knowledge, this is the first explicit construction showing that vanishing nonsummable smoothing, even together with exact limiting stationarity, does not imply whole-sequence convergence for variable-smoothing full-splitting schemes.