On the Complexity of Testing Attainment of the Optimal Value in Nonlinear Optimization

TL;DR

This paper proves that testing whether the optimal value in certain nonlinear optimization problems is attained is computationally hard, and introduces SDP-based conditions for verifying attainment.

Contribution

It establishes NP-hardness results for testing attainment and related properties in nonlinear optimization, and proposes SDP-based criteria for attainment verification.

Findings

Testing attainment is NP-hard for fixed-degree polynomials.

Verifying coercivity and boundedness conditions is NP-hard.

SDP hierarchies can be used to check attainment conditions.

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.