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

Amir Ali Ahmadi, Jeffrey Zhang

Research output: Contribution to journalArticle

1 Scopus citations

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 languageEnglish (US)
JournalMathematical Programming
DOIs
StatePublished - Jan 1 2019

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

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