TY - JOUR

T1 - On the complexity of detecting convexity over a box

AU - Ahmadi, Amir Ali

AU - Hall, Georgina

N1 - Funding Information:
We are grateful to two anonymous referees whose detailed and constructive feedback has improved this paper significantly.

PY - 2020/7/1

Y1 - 2020/7/1

N2 - 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., a 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.

AB - 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., a 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.

KW - Computational complexity

KW - Convex optimization

KW - Convexity detection

KW - Interval positive semidefiniteness

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U2 - 10.1007/s10107-019-01396-x

DO - 10.1007/s10107-019-01396-x

M3 - Article

AN - SCOPUS:85065290066

VL - 182

SP - 429

EP - 443

JO - Mathematical Programming

JF - Mathematical Programming

SN - 0025-5610

IS - 1-2

ER -