On Approximate Computation of Critical Points

2026-01-29

AI Summary

This study investigates the computational complexity of approximating critical points of nonconvex functions, such as cubic polynomials. By leveraging computational complexity theory and reduction techniques, it establishes—for the first time—that the problem remains NP-hard even under extremely coarse approximation criteria, for instance, when the gradient norm is merely required to be at most $2^n$. This hardness result holds under several standard assumptions commonly adopted in nonconvex optimization, including uniqueness of the critical point and lower-boundedness of the objective function. The findings challenge the prevailing belief that critical points of nonconvex functions are generally easy to approximate, thereby revealing an intrinsic computational intractability unless P = NP.

Technology Category

Search and Optimization: Non-convex OptimizationKnowledge Representation and Reasoning: Computational Complexity of ReasoningReasoning under Uncertainty: Stochastic Optimization

Application Category

Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphsEconomics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystemsSecurity and Privacy: Large-scale security measurements

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.

Problem

• critical points
• nonconvex optimization
• computational intractability
• approximate computation
• P=NP

Innovation

• critical points
• nonconvex optimization
• computational hardness
• P vs NP
• approximate computation