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.
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 -