We show that computing even very coarse approximations of critical points is intractable for simple classes of nonconvex functions. More concretely, we prove that if there exists a polynomial-time algorithm that takes as input a polynomial in $n$ variables of constant degree (as low as three) and outputs a point whose gradient has Euclidean norm at most $2^n$ whenever the polynomial has a critical point, then P=NP. The algorithm is permitted to return an arbitrary point when no critical point exists. We also prove hardness results for approximate computation of critical points under additional structural assumptions, including settings in which existence and uniqueness of a critical point are guaranteed, the function is lower bounded, and approximation is measured in terms of distance to a critical point. Overall, our results stand in contrast to the commonly-held belief that, in nonconvex optimization, approximate computation of critical points is a tractable task.

翻译：我们证明，即使对于简单类别的非凸函数，计算临界点的非常粗略近似也是难以处理的。具体而言，我们证明：如果存在一个多项式时间算法，该算法以$n$个变量的常数阶（低至三阶）多项式作为输入，并在多项式存在临界点时输出一个梯度欧几里得范数不超过$2^n$的点，那么P=NP。当不存在临界点时，该算法可以返回任意点。我们还证明了在附加结构假设下近似计算临界点的困难性，包括临界点存在性和唯一性得到保证、函数具有下界以及近似性通过到临界点的距离来度量的情况。总体而言，我们的结果与普遍观点形成鲜明对比，即非凸优化中临界点的近似计算是一项可处理的任务。