论文标题
凸出可行性问题的规律性和稳定性
Regularity and stability for a convex feasibility problem
论文作者
论文摘要
让我们考虑两个封闭凸的序列$ \ {a_n \} $，$ \ {b_n \} $分别相对于attouch-wets收敛到$ a $ a和$ b $。给定一个起点$ a_0 $，我们考虑通过在“扰动”集上投影获得的积分序列，即，$ \ {a_n \} $和$ \ {b_n \} $由$ b_n = p_ = p_ {b_n}（a_ {n-1} n-1} n-1}） $ a_n = p_ {a_n}（b_n）$。 假设$ a \ cap b $（或合适的替代品，如果$ a \ cap b = \ emptySet $）是有限的，我们证明，如果（有限的）$（a，b）$是（有限的），则夫妇$（a，b）$是$ d $ stable，即每种$ \ \ \ \ {a_n \} $ \ as $ \ al as $ \ a \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ^ beans $ \ mathrm {dist}（a_n，a \ cap b）\ to 0 $和$ \ mathrm {dist}（b_n，a \ cap b）\ to 0 $。
Let us consider two sequences of closed convex sets $\{A_n\}$ and $\{B_n\}$ converging with respect to the Attouch-Wets convergence to $A$ and $B$, respectively. Given a starting point $a_0$, we consider the sequences of points obtained by projecting on the "perturbed" sets, i.e., the sequences $\{a_n\}$ and $\{b_n\}$ defined inductively by $b_n=P_{B_n}(a_{n-1})$ and $a_n=P_{A_n}(b_n)$. Suppose that $A\cap B$ (or a suitable substitute if $A \cap B=\emptyset$) is bounded, we prove that if the couple $(A,B)$ is (boundedly) regular then the couple $(A,B)$ is $d$-stable, i.e., for each $\{a_n\}$ and $\{b_n\}$ as above we have $\mathrm{dist}(a_n,A\cap B)\to 0$ and $\mathrm{dist}(b_n,A\cap B)\to 0$.
