Alternating proximal algorithms for linearly constrained variational inequalities: Application to domain decomposition for PDE's
Alternating proximal algorithms for linearly constrained variational inequalities: Application to domain decomposition for PDE's
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DOI：
10.1016/j.na.2011.07.066
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发表时间：
2011-12-01
影响因子：
1.4
通讯作者：
Peypouquet, J.
中科院分区：
文献类型：
--
作者：
Attouch, H.;Cabot, A.;Peypouquet, J.
设X、Y、Z是真实的Hilbert空间，设f：X -> R布尔OR {+无穷大}，g：Y -> R布尔OR {+无穷大}是闭凸函数，设A：X。Z，B：Y。Z是线性连续算子。让我们考虑约束最小化问题（P）min{f（x）+ g（y）：Ax = By}。给定一个序列（gamma（n）），当n -> +无穷大时，它趋向于0，我们研究了以下交替邻近算法（A）{x（n+1）= argmin {gamma（n+1）f（zeta）+ 1/2平行于A zeta - By（n）平行于（2）（Z）+ alpha/2平行于zeta - x（n）平行于（2）（X）;zeta是X的元素}y（n+1）= argmin {gamma（n+1）g（eta）+ 1/2平行于Ax（n+1）- B eta平行于（2）（Z）+ nu/2平行于eta - y（n）平行于（2）（Y）; eta是Y的元素}，其中α和.是积极的参数。证明了如果序列（gamma（n））缓慢地趋向于0，则（A）的迭代弱收敛于（P）的解.将研究推广到极大单调算子的情形，得到了一般的遍历收敛结果。给出了偏微分方程在区域分解中的应用。(C)2011爱思唯尔有限公司保留所有权利。
Let X, Y, Z be real Hilbert spaces, let f : X -> R boolean OR {+infinity}, g : Y -> R boolean OR {+infinity} be closed convex functions and let A : X. Z, B : Y. Z be linear continuous operators. Let us consider the constrained minimization problem(P) min{f (x) + g(y) : Ax = By}.Given a sequence (gamma(n)) which tends toward 0 as n -> +infinity, we study the following alternating proximal algorithm(A) {x(n+1) = argmin {gamma(n+1) f (zeta) + 1/2 parallel to A zeta - By(n) parallel to(2)(Z) + alpha/2 parallel to zeta - x(n)parallel to(2)(X) ;zeta is an element of X}y(n+1) = argmin {gamma(n+1) g(eta) + 1/2 parallel to Ax(n+1) - B eta parallel to(2)(Z) + nu/2 parallel to eta - y(n)parallel to(2)(Y); eta is an element of Y},where alpha and. are positive parameters. It is shown that if the sequence (gamma(n)) tends moderately slowly toward 0, then the iterates of (A) weakly converge toward a solution of (P). The study is extended to the setting of maximal monotone operators, for which a general ergodic convergence result is obtained. Applications are given in the area of domain decomposition for PDE's. (C) 2011 Elsevier Ltd. All rights reserved.