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

Amir Ali Ahmadi (a_a_a***at***princeton.edu)
Jeffrey Zhang (jeffz***at***princeton.edu)

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.

Keywords: Existence of solutions in mathematical programs, Frank-Wolfe type theorems, coercive polynomials, computational complexity, semidefinite programming, Archimedean quadratic modules

Category 1: Nonlinear Optimization

Category 2: Global Optimization (Theory )

Category 3: Linear, Cone and Semidefinite Programming (Semi-definite Programming )

Citation:

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Entry Submitted: 03/20/2018
Entry Accepted: 03/20/2018
Entry Last Modified: 04/29/2019

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