Abstract:This paper studies the fundamental properties of the high-order forward-backward splitting mapping (HiFBS) and its associated high-order forward-backward envelope (HiFBE) through the lens of high-order regularization for nonconvex composite functions. Specifically, we (i) establish the boundedness and uniform boundedness of HiFBS, along with the Hölder and Lipschitz continuity of HiFBE; (ii) derive an explicit form for the subdifferentials of HiFBE; and (iii) investigate necessary and sufficient conditions for the differentiability and weak smoothness of HiFBE under suitable assumptions. By leveraging the prox-regularity of $g$ and the concept of $p$-calmness, we further demonstrate the local single-valuedness and continuity of HiFBS, which in turn guarantee the differentiability of HiFBE in neighborhoods of calm points. This paves the way for the development of gradient-based algorithms tailored to nonconvex composite optimization problems.
From: Alireza Kabgani Dr. [view email]
[v1]
Thu, 13 Nov 2025 15:42:55 UTC (367 KB)
[v2]
Mon, 22 Jun 2026 23:23:17 UTC (385 KB)
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