### Abstract

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.

Original language | English (US) |
---|---|

Journal | Mathematical Programming |

DOIs | |

State | Published - Jan 1 2019 |

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### All Science Journal Classification (ASJC) codes

- Software
- Mathematics(all)

### Keywords

- Archimedean quadratic modules
- Coercive polynomials
- Computational complexity
- Existence of solutions in mathematical programs
- Frank–Wolfe type theorems
- Semidefinite programming

### Cite this

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**On the complexity of testing attainment of the optimal value in nonlinear optimization.** / Ahmadi, Amir Ali; Zhang, Jeffrey.

Research output: Contribution to journal › Article

TY - JOUR

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

AU - Ahmadi, Amir Ali

AU - Zhang, Jeffrey

PY - 2019/1/1

Y1 - 2019/1/1

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

UR - http://www.scopus.com/inward/record.url?scp=85068829886&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85068829886&partnerID=8YFLogxK

U2 - https://doi.org/10.1007/s10107-019-01411-1

DO - https://doi.org/10.1007/s10107-019-01411-1

M3 - Article

JO - Mathematical Programming

JF - Mathematical Programming

SN - 0025-5610

ER -