The CoMirror algorithm, by Beck et al. (Oper Res Lett 38(6):493–498, 2010), is designed to solve convex optimization problems with one functional constraint. At each iteration, it performs a mirror-descent update using either the subgradient of the objective function or the subgradient of the constraint function, depending on whether or not the constraint violation is below some tolerance. In this paper, we combine the CoMirror algorithm with inexact cut generation to create the SIP-CoM algorithm for solving semi-infinite programming (SIP) problems. First, we provide general error bounds for SIP-CoM. Then, we propose two specific random constraint sampling schemes to approximately solve the cut generation problem for generic SIP. When the objective and constraint functions are generally convex, randomized SIP-CoM achieves an $${\mathcal {O}}(1/\sqrt{N})$$ convergence rate in expectation (in terms of the optimality gap and SIP constraint violation). When the objective and constraint functions are all strongly convex, this rate can be improved to $${\mathcal {O}}(1/N)$$ .

中文翻译：

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凸半无限规划的随机约束采样 CoMirror 算法

贝克等人的 CoMirror 算法。(Oper Res Lett 38(6):493–498, 2010)，旨在解决具有一个功能约束的凸优化问题。在每次迭代中，它使用目标函数的次梯度或约束函数的次梯度执行镜像下降更新，这取决于约束违反是否低于某个容差。在本文中，我们将 CoMirror 算法与不精确切割生成相结合，以创建用于解决半无限编程 (SIP) 问题的 SIP-CoM 算法。首先，我们提供了 SIP-CoM 的一般错误界限。然后，我们提出了两种特定的随机约束采样方案来近似解决通用 SIP 的切割生成问题。当目标函数和约束函数一般是凸函数时，随机的 SIP-CoM 达到了 $${\mathcal {O}}(1/\sqrt{N})$$ 的期望收敛率（在最优性差距和 SIP 约束违反方面）。当目标函数和约束函数都是强凸时，这个比率可以提高到 $${\mathcal {O}}(1/N)$$ 。