TY - JOUR

T1 - On the complexity of testing attainment of the optimal value in nonlinear optimization

AU - Ahmadi, Amir Ali

AU - Zhang, Jeffrey

N1 - Funding Information:
We are grateful to Georgina Hall and two anonymous referees for their careful reading of this manuscript and very constructive feedback. We also thank Etienne de Klerk for offering a simplification of the construction in the proof of Theorem 2.1.
Funding Information:
This work was partially supported by the DARPA Young Faculty Award, the CAREER Award of the NSF, the Innovation Award of the School of Engineering and Applied Sciences at Princeton University, the MURI award of the AFOSR, the Google Faculty Award, and the Sloan Fellowship.
Publisher Copyright:
© 2019, Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society.

PY - 2020/11/1

Y1 - 2020/11/1

N2 - We prove that unless P = NP , there exists no polynomial time (or even pseudo-polynomial time) algorithm that can test whether the optimal value of a nonlinear optimization problem where the objective and constraints are given by low-degree polynomials is attained. If the degrees of these polynomials are fixed, our results along with previously-known “Frank–Wolfe type” theorems imply that exactly one of two cases can occur: either the optimal value is attained on every instance, or it is strongly NP-hard to distinguish attainment from non-attainment. We also show that testing for some well-known sufficient conditions for attainment of the optimal value, such as coercivity of the objective function and closedness and boundedness of the feasible set, is strongly NP-hard. As a byproduct, our proofs imply that testing the Archimedean property of a quadratic module is strongly NP-hard, a property that is of independent interest to the convergence of the Lasserre hierarchy. Finally, we give semidefinite programming (SDP)-based sufficient conditions for attainment of the optimal value, in particular a new characterization of coercive polynomials that lends itself to an SDP hierarchy.

AB - We prove that unless P = NP , there exists no polynomial time (or even pseudo-polynomial time) algorithm that can test whether the optimal value of a nonlinear optimization problem where the objective and constraints are given by low-degree polynomials is attained. If the degrees of these polynomials are fixed, our results along with previously-known “Frank–Wolfe type” theorems imply that exactly one of two cases can occur: either the optimal value is attained on every instance, or it is strongly NP-hard to distinguish attainment from non-attainment. We also show that testing for some well-known sufficient conditions for attainment of the optimal value, such as coercivity of the objective function and closedness and boundedness of the feasible set, is strongly NP-hard. As a byproduct, our proofs imply that testing the Archimedean property of a quadratic module is strongly NP-hard, a property that is of independent interest to the convergence of the Lasserre hierarchy. Finally, we give semidefinite programming (SDP)-based sufficient conditions for attainment of the optimal value, in particular a new characterization of coercive polynomials that lends itself to an SDP hierarchy.

KW - Archimedean quadratic modules

KW - Coercive polynomials

KW - Computational complexity

KW - Existence of solutions in mathematical programs

KW - Frank–Wolfe type theorems

KW - Semidefinite programming

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U2 - 10.1007/s10107-019-01411-1

DO - 10.1007/s10107-019-01411-1

M3 - Article

AN - SCOPUS:85068829886

VL - 184

SP - 221

EP - 241

JO - Mathematical Programming

JF - Mathematical Programming

SN - 0025-5610

IS - 1-2

ER -