On the Complexity of Detecting Convexity over a Box
This addresses a fundamental computational complexity problem for nonlinear optimization and robust control, showing incremental hardness results building on prior work.
The paper proves that testing convexity of degree-three polynomials over a box is strongly NP-hard, which is minimal in degree and explains limitations in optimization solvers. As a byproduct, it shows strong NP-hardness for testing positive semidefiniteness of interval matrices, relevant to robust control.
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.