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On Approximate Computation of Critical Points

Amir Ali Ahmadi, Georgina Hall

TL;DR

The paper proves that even coarse, $2^n$-level approximations to critical points of simple nonconvex functions are computationally intractable unless $P=NP$, challenging the common belief that approximate critical-point computation is easy. It establishes this via explicit reductions from the Clique problem to carefully constructed low-degree polynomials, including degree-3 and degree-4 instances, whose gradient properties encode clique structure. The results extend to lower-bounded functions and to notions of $\epsilon$-near critical points, showing hardness persists under several structural guarantees and even when no spurious critical points exist. Collectively, these findings delineate a broad hardness border for nonconvex optimization, and motivate identifying function classes where efficient approximation of critical points may still be feasible.

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.

On Approximate Computation of Critical Points

TL;DR

The paper proves that even coarse, -level approximations to critical points of simple nonconvex functions are computationally intractable unless , challenging the common belief that approximate critical-point computation is easy. It establishes this via explicit reductions from the Clique problem to carefully constructed low-degree polynomials, including degree-3 and degree-4 instances, whose gradient properties encode clique structure. The results extend to lower-bounded functions and to notions of -near critical points, showing hardness persists under several structural guarantees and even when no spurious critical points exist. Collectively, these findings delineate a broad hardness border for nonconvex optimization, and motivate identifying function classes where efficient approximation of critical points may still be feasible.

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 variables of constant degree (as low as three) and outputs a point whose gradient has Euclidean norm at most 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.
Paper Structure (9 sections, 12 theorems, 67 equations, 1 figure)

This paper contains 9 sections, 12 theorems, 67 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Outline, contributions, and related literature
  3. Complexity of computing an $\epsilon$-approximate critical point
  4. Complexity of computing an $\epsilon$-approximate critical point of a general function
  5. Complexity of computing an $\epsilon$-approximate critical point of a lower bounded function
  6. Complexity of computing an $\epsilon$-near critical point
  7. Complexity of computing an $\epsilon$-near critical point of a general function
  8. Complexity of computing an $\epsilon$-near critical point of a lower bounded function
  9. Concluding remarks

Key Result

Theorem 3

If there is an algorithm $\mathcal{A}$ that takes as input a cubic polynomial $p$ in $n$ variables, runs in polynomial time, and is such that then $P=NP.$

Figures (1)

  • Figure 1: The difference between near critical and approximate critical points demonstrated in a one-dimensional example: $x^*$ is a critical point of $f$, and for a given $\epsilon>0$, $x_N$ is an $\epsilon$-near critical point while $x_A$ is an $\epsilon$-approximate critical point.

Theorems & Definitions (25)

  • Definition 1: Critical point
  • Definition 2: $\epsilon$-approximate critical point
  • Theorem 3
  • Lemma 4
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • Lemma 7
  • proof
  • ...and 15 more