4 Subsmooth Case
This chapter presents convergence results for boundedly compact subsmooth moving sets.
4.1 Subsmooth Sets
Let \(S \subset H\) be a closed set. We say that \(S\) is subsmooth at \(x_0 \in S\) if for every \(\varepsilon {\gt} 0\) there exists \(\delta {\gt} 0\) such that
whenever \(x_1, x_2 \in B[x_0, \delta ] \cap S\) and \(\xi _i \in N(S; x_i) \cap \mathbb {B}\) for \(i \in \{ 1,2\} \).
A family \((S(t))_{t\in I}\) of closed sets is called equi-uniformly subsmooth if for all \(\varepsilon {\gt} 0\), there exists \(\delta {\gt} 0\) such that for all \(t \in I\), the subsmoothness inequality holds uniformly.
The class of subsmooth sets includes both convex and uniformly prox-regular sets, making it a very general framework for the sweeping process.
4.2 Ball Compactness
A set \(S \subset H\) is boundedly compact if \(S \cap r\mathbb {B}\) is compact for all \(r {\gt} 0\).
Bounded compactness is weaker than compactness but sufficient for many existence results in infinite-dimensional Hilbert spaces.
4.3 Stability Result
Let \(\mathcal{C} = \{ C_n\} _{n\in \mathbb {N}} \cup \{ C\} \) be a family of nonempty, closed, and equi-uniformly subsmooth sets. Under appropriate convergence assumptions, certain stability properties hold for the subdifferentials of distance functions.
4.4 Convergence Result
For boundedly compact subsmooth moving sets, the catching-up algorithm with approximate projections converges to a solution of the sweeping process.