Banjac et al. (J Optim Theory Appl 183(2):490–519, 2019) recently showed that the Douglas–Rachford algorithm provides certificates of infeasibility for a class of convex optimization problems. In particular, they showed that the difference between consecutive iterates generated by the algorithm converges to certificates of primal and dual strong infeasibility. Their result was shown in a finite-dimensional Euclidean setting and for a particular structure of the constraint set. In this paper, we extend the result to real Hilbert spaces and a general nonempty closed convex set. Moreover, we show that the proximal-point algorithm applied to the set of optimality conditions of the problem generates similar infeasibility certificates.

巴尼亚克等人。（J Optim Theory Appl 183(2):490–519, 2019）最近表明，Douglas-Rachford 算法为一类凸优化问题提供了不可行性证明。特别是，他们表明算法生成的连续迭代之间的差异收敛到原始和对偶强不可行性的证明。他们的结果显示在有限维欧几里得设置和约束集的特定结构中。在本文中，我们将结果扩展到实 Hilbert 空间和一般非空闭凸集。此外，我们表明，应用于问题的最优条件集的近端算法会生成类似的不可行性证明。