TY - JOUR
T1 - Complexity aspects of local minima and related notions
AU - Ahmadi, Amir Ali
AU - Zhang, Jeffrey
N1 - Funding Information:
This work was supported partially by an AFOSR MURI award, the DARPA Young Faculty Award, the Princeton SEAS Innovation Award, the NSF CAREER Award, the Google Faculty Award, and the Sloan Fellowship .
Publisher Copyright:
© 2021 Elsevier Inc.
PY - 2022/3/5
Y1 - 2022/3/5
N2 - We consider the notions of (i) critical points, (ii) second-order points, (iii) local minima, and (iv) strict local minima for multivariate polynomials. For each type of point, and as a function of the degree of the polynomial, we study the complexity of deciding (1) if a given point is of that type, and (2) if a polynomial has a point of that type. Our results characterize the complexity of these two questions for all degrees left open by prior literature. Our main contributions reveal that many of these questions turn out to be tractable for cubic polynomials. In particular, we present an efficiently-checkable necessary and sufficient condition for local minimality of a point for a cubic polynomial. We also show that a local minimum of a cubic polynomial can be efficiently found by solving semidefinite programs of size linear in the number of variables. By contrast, we show that it is strongly NP-hard to decide if a cubic polynomial has a critical point. We also prove that the set of second-order points of any cubic polynomial is a spectrahedron, and conversely that any spectrahedron is the projection of the set of second-order points of a cubic polynomial. In our final section, we briefly present a potential application of finding local minima of cubic polynomials to the design of a third-order Newton method.
AB - We consider the notions of (i) critical points, (ii) second-order points, (iii) local minima, and (iv) strict local minima for multivariate polynomials. For each type of point, and as a function of the degree of the polynomial, we study the complexity of deciding (1) if a given point is of that type, and (2) if a polynomial has a point of that type. Our results characterize the complexity of these two questions for all degrees left open by prior literature. Our main contributions reveal that many of these questions turn out to be tractable for cubic polynomials. In particular, we present an efficiently-checkable necessary and sufficient condition for local minimality of a point for a cubic polynomial. We also show that a local minimum of a cubic polynomial can be efficiently found by solving semidefinite programs of size linear in the number of variables. By contrast, we show that it is strongly NP-hard to decide if a cubic polynomial has a critical point. We also prove that the set of second-order points of any cubic polynomial is a spectrahedron, and conversely that any spectrahedron is the projection of the set of second-order points of a cubic polynomial. In our final section, we briefly present a potential application of finding local minima of cubic polynomials to the design of a third-order Newton method.
KW - Computational complexity
KW - Critical and second-order points
KW - Local minima
KW - Polynomial optimization
KW - Semidefinite programming
KW - Sum of squares polynomials
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U2 - 10.1016/j.aim.2021.108119
DO - 10.1016/j.aim.2021.108119
M3 - Article
AN - SCOPUS:85119441981
SN - 0001-8708
VL - 397
JO - Advances in Mathematics
JF - Advances in Mathematics
M1 - 108119
ER -