On the Complexity of Detecting Convexity over a Box

TL;DR

This paper proves that testing convexity of degree three polynomials over a box is strongly NP-hard, explaining the limitations of convexity detection in nonlinear optimization and linking it to positive semidefinite interval matrix problems.

Contribution

It establishes the strong NP-hardness of convexity testing over boxes for degree three polynomials, extending previous results from degree four.

Findings

Convexity testing over a box is strongly NP-hard for degree three polynomials.

The problem of positive semidefinite interval matrix testing is also strongly NP-hard.

Supports the practical limitation to quadratic or structured functions in convexity detection.

Abstract

It has recently been shown that the problem of testing global convexity of polynomials of degree four is {strongly} NP-hard, answering an open question of N.Z. Shor. This result is minimal in the degree of the polynomial when global convexity is of concern. In a number of applications however, one is interested in testing convexity only over a compact region, most commonly a box (i.e., hyper-rectangle). In this paper, we show that this problem is also strongly NP-hard, in fact for polynomials of degree as low as three. This result is minimal in the degree of the polynomial and in some sense justifies why convexity detection in nonlinear optimization solvers is limited to quadratic functions or functions with special structure. As a byproduct, our proof shows that the problem of testing whether all matrices in an interval family are positive semidefinite is strongly NP-hard. This problem, which was previously shown to be (weakly) NP-hard by Nemirovski, is of independent interest in the theory of robust control.