We study search problems that can be solved by performing Gradient Descent on a bounded convex polytopal domain and show that this class is equal to the intersection of two well-known classes: PPAD and PLS. As our main underlying technical contribution, we show that computing a Karush-Kuhn-Tucker (KKT) point of a continuously differentiable function over the domain $[0,1]^2$ is PPAD $\cap$ PLS-complete. This is the first non-artificial problem to be shown complete for this class. Our results also imply that the class CLS (Continuous Local Search) - which was defined by Daskalakis and Papadimitriou as a more "natural" counterpart to PPAD $\cap$ PLS and contains many interesting problems - is itself equal to PPAD $\cap$ PLS.

翻译：我们研究通过在有界凸多面体域上执行梯度下降可解决的搜索问题，并证明该问题类等价于两个著名问题类：PPAD与PLS的交集。作为主要技术贡献，我们证明了在域$[0,1]^2$上计算连续可微函数的Karush-Kuhn-Tucker（KKT）点是PPAD ∩ PLS完全的。这是首个被证明属于该问题类的非人工构造问题。我们的结论还表明，由Daskalakis和Papadimitriou定义为PPAD ∩ PLS更"自然"对应物、并包含许多有趣问题的CLS（连续局部搜索）类本身即等于PPAD ∩ PLS。