This paper considers an inexact primal-dual algorithm for semi-infinite programming (SIP) for which it provides general error bounds. We create a new prox function for nonnegative measures for the dual update, and it turns out to be a generalization of the Kullback-Leibler divergence. We show that, with a tolerance for small errors (approximation and regularization error), this algorithm achieves an \({\mathcal {O}}(1/\sqrt{K})\) rate of convergence in terms of the optimality gap and constraint violation, where K is the total number of iterations. We then use our general error bounds to analyze the convergence and sample complexity of a specific primal-dual SIP algorithm based on Monte Carlo sampling. Finally, we provide numerical experiments to demonstrate the performance of this algorithm.

中文翻译：

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半无限编程的不精确本原对偶算法

本文考虑了一种用于半无限编程（SIP）的不精确的原始对偶算法，该算法为其提供了一般的误差范围。我们为双重更新的非负度量创建了一个新的代理函数，结果证明是Kullback-Leibler散度的推广。我们证明，在容许小的误差（逼近度和正则化误差）的情况下，该算法在最优间隙方面实现了\（{\ mathcal {O}}（1 / \ sqrt {K}）\）收敛速度和约束违反，其中K是迭代的总数。然后，我们使用通用误差范围来分析基于蒙特卡洛采样的特定原始对偶SIP算法的收敛性和样本复杂度。最后，我们提供了数值实验来证明该算法的性能。