We consider the problem of minimizing a finite sum of convex functions subject to the set of minimizers of a convex differentiable function. In order to solve the problem, an algorithm combining the incremental proximal gradient method with smooth penalization technique is proposed. We show the convergence of the generated sequence of iterates to an optimal solution of the optimization problems, provided that a condition expressed via the Fenchel conjugate of the constraint function is fulfilled. Finally, the functionality of the method is illustrated by some numerical experiments addressing image inpainting problems and generalized Heron problems with least squares constraints.

中文翻译：

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带有约束的复合凸优化问题的惩罚的增量近端梯度方案

我们考虑在凸可微函数的最小化器集合下最小化凸函数的有限和的问题。针对该问题，提出了一种增量近端梯度法与平滑惩罚技术相结合的算法。我们展示了生成的迭代序列收敛到优化问题的最优解，前提是满足通过约束函数的 Fenchel 共轭表达的条件。最后，通过一些解决图像修复问题和具有最小二乘约束的广义 Heron 问题的数值实验来说明该方法的功能。