On the Iterate Convergence of Bregman Projected Gradient Method

The iterate convergence of \textit{Bregman projected gradient method} (BPGM) has remained a long-standing open problem, especially for the widely adopted Shannon entropy kernel. Existing convergence results are often limited, relying on Lipschitz continuity of the kernel's gradient or restrictive conditions on the objective function. In this paper, we develop a novel convergence analysis framework for the BPGM with the Shannon entropy kernel, yielding strong convergence results for a broad class of objective functions under linear constraints. The cornerstone of our framework is a new concept called \textit{scaled Kurdyka-\L{}ojasiewicz} (SK\L{}) property, which captures the local growth behavior of a function under the Bregman geometry. We show that the SK\L{} property ensures the iterate convergence of BPGM and holds for all continuous subanalytical functions. Furthermore, we prove that the BPGM sequence exhibits linear convergence if the problem possesses an SK\L\ exponent of $1/2$. We then furnish the examples of functions with the SK\L\ exponent $1/2$ by proving that the SK\L\ exponent $1/2$ is implied by the K\L{} exponent $1/2$ under strict complementarity and local Lipschitz continuity of the objective's gradient. Building on these novel results, our work takes a first step towards resolving the open problem of BPGM iterate convergence.