We consider rather a general class of multi-level optimization problems, where a convex objective function is to be minimized, subject to constraints to optima of a nested convex optimization problem. As a special case, we consider a trilevel optimization problem, where the objective of the two lower layers consists of a sum of a smooth and a non-smooth term. Based on fixed-point theory and related arguments, we present a natural first-order algorithm and analyze its convergence and rates of convergence in several regimes of parameters.

翻译：我们所考虑的是一个多层次优化问题的一般类别,在这个类别中,要尽量减少一个分流目标功能,但受嵌套的分流优化问题所制约的制约。 作为一个特例,我们所考虑的是一个三层优化问题,即两个下层的目标包含一个平滑和非平坦的术语的总和。 根据固定点理论和相关论点,我们提出一个自然的第一阶算法,并分析其在若干参数制度中的趋同和趋同率。