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Certificates of polynomial nonnegativity can be used to obtain tight dual bounds for polynomial optimization problems. We consider Sums of Nonnegative Circuit (SONC) polynomials certificates, which are well suited for sparse problems since the computational cost depends only on the number of terms in the polynomials and does not depend on the degrees of the polynomials. This work is a first step to integrating SONC-based relaxations of polynomial problems into a branch-and-bound algorithm. To this end, the SONC relaxation for constrained optimization problems is extended in order to better utilize variable bounds, since this property is key for the success of a relaxation in the context of branch-and-bound. Computational experiments show that the proposed extension is crucial for making the SONC relaxations applicable to most constrained polynomial optimization problems and for integrating the two approaches.

Nonnegativity certificates can be used to obtain tight dual bounds for polynomial optimization problems. Hierarchies of certificate-based relaxations ensure convergence to the global optimum, but higher levels of such hierarchies can become very computationally expensive, and the well-known sums of squares hierarchies scale poorly with the degree of the polynomials. This has motivated research into alternative certificates and approaches to global optimization. We consider sums of nonnegative circuit polynomials (SONC) certificates, which are well-suited for sparse problems since the computational cost depends on the number of terms in the polynomials and does not depend on the degrees of the polynomials. We propose a method that guarantees that given finite variable domains, a SONC relaxation will yield a finite dual bound. This method opens up a new approach to utilizing variable bounds in SONC-based methods, which is particularly crucial for integrating SONC relaxations into branch-and-bound algorithms. We report on computational experiments with incorporating SONC relaxations into the spatial branch-and-bound algorithm of the mixed-integer nonlinear programming framework SCIP. Applying our strengthening method increases the number of instances where the SONC relaxation of the root node yielded a finite dual bound from 9 to 330 out of 349 instances in the test set.