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Discrete Mathematics and Optimization Seminar
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Nov. 9th, 2009
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Positive Polynomials Over Equality Constrained unbounded Sets
Juan Vera
University of Waterloo
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A simple yet powerful algebraic connection between the set of polynomials that are non-negative on a given closed domain and the set of polynomials that are non-negative on the intersection of the same domain and the zero set of a given polynomial is presented. This connection has interesting theoretical as well as algorithmic implications. It yields a succinct derivation of copositive programming reformulations for a big class of programs, generalizing Burer's copositive formulation for mixed non-convex quadratically constrained quadratic programming problems. As corollary of our main theorem we also obtain representation theorems for positive polynomials on closed sets.
To illustrate the results we show how to use Polynomial Programming as a general framework for conic relaxations. This framework is used to re-derive previous relaxation schemes and provide stronger ones for general binary quadratic optimization. In particular, second-order cone relaxations are proposed. Computational tests show that using the second-order cone relaxation with triangle inequalities provides a bound that is competitive with the semidefinite bound strengthened with triangle inequalities but the former relaxation is computationally more efficient.
This is joint work with Javier Pena and Luis Zuluaga, and Bissan Ghaddar and Miguel Anjos.
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