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

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 polynomial...

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Veröffentlicht in:Mathematical programming Jg. 184; H. 1-2; S. 221 - 241
Hauptverfasser: Ahmadi, Amir Ali, Zhang, Jeffrey
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Berlin/Heidelberg Springer Berlin Heidelberg 01.11.2020
Springer Nature B.V
Schlagworte:
Algorithms
Calculus of Variations and Optimal Control; Optimization
Coercivity
Combinatorics
Full Length Paper
Job shops
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Mathematics of Computing
Numerical Analysis
Optimization
Polynomials
Semidefinite programming
Theoretical
Semidefinite programming
90C30
Frank–Wolfe type theorems
Archimedean quadratic modules
Existence of solutions in mathematical programs
90C60
Computational complexity
Coercive polynomials
ISSN:0025-5610, 1436-4646
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Zusammenfassung: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.
Bibliographie:ObjectType-Article-1
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ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-019-01411-1