On Approximate Computation of Critical Points

TL;DR

This paper proves that even approximating critical points of simple nonconvex functions is computationally intractable unless P=NP, challenging the common belief about the tractability of nonconvex optimization.

Contribution

It establishes NP-hardness results for approximate critical point computation for polynomial functions, even under structural assumptions.

Findings

Approximate critical point computation is NP-hard for degree-3 polynomials.

Hardness persists under assumptions like existence, uniqueness, and boundedness of critical points.

Contradicts the belief that nonconvex optimization is generally tractable for approximation.

Abstract

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.