Exploiting Equalities in Polynomial Programming ∗ Javier F. Pe˜ na † 1 , Juan C. Vera ‡ 2 , and Luis F. Zuluaga § 3 1 Tepper School of Business, Carnegie Mellon University 2 Mellon College of Science, Carnegie Mellon University 3 Faculty of Business Administration, University of New Brunswick May 19, 2006 Abstract We propose a novel solution approach for polynomial programming problems with equality constraints. By means of a generic transformation, we show that solution schemes for the, typically simpler, problem without equalities can be used to address the problem with equalities. In particular, we propose new solution schemes for mixed binary programs, pure 0–1 quadratic programs, and the stable set problem. Keywords: Polynomial programming, 0–1 programming, stable set problem, LMI approximations. 1 Introduction A polynomial program is an optimization problem whose objective and constraints are given by multivariate polynomials. Formally, a polynomial program is a problem of the form: ρ = inf x f (x) s.t. g i (x) ≥ 0, i =1,...,d h j (x)=0, j =1,...,v (1) where f (x), g i (x), i =1,...,d and h j (x), j =1,...,v are given polynomials in n variables. An increasingly active research trend is to use a combination of real algebraic geometry and numer- ical optimization tools to derive new solution schemes for polynomial programs. Some instances of the growing literature in this area are the articles by de Klerk and Pasechnik [6], Lasserre [10], Laurent [13], Parrilo [17, 18], Parrilo and Sturmfels [19], and Sturmfels, Demmel, and Nie [24]; along with previous work by Nesterov [15], and Shor [21]. As a result of this research trend, linear and semidefinite programming algorithms are now successfully used to address polynomial programs that arise in areas such as combinatorial optimization, signal processing, mathematical finance, quadratic programming, physics, and probability (see, e.g., [2, 3, 6, 7, 8, 11]). * Supported by NSF grant CCF-0092655. † jfp@andrew.cmu.edu ‡ jvera@andrew.cmu.edu § Partially supported by NSERC grant 31814-05, and NSF grant DMI-0098427. lzuluaga@unb.ca 1