Global Optimality Principles for Polynomial Optimization Problems over Box or Bivalent Constraints by Separable Polynomial Approximations∗

In this paper we present necessary conditions for global optimality for polynomial problems over box or bivalent constraints using separable polynomial relaxations. We achieve this by completely characterizing global optimality of separable polynomial problems with box as well as bivalent constraints. Then, by employing separable polynomial under-estimators, we establish sufficient conditions for global optimality for classes of polynomial optimization problems with box or bivalent constraints. The underestimators are constructed using the sum of squares convex polynomials. The significance of our optimality condition is that they can be numerically checked by solving semi-definite programming problems. We illustrate the versatility of our optimality conditions by simple numerical examples. ∗The authors are grateful to the referees and the editor for their constructive comments and helpful suggestions which have contributed to the final preparation of the paper. Research was partially supported by a grant from the Australian Research Council †Department of Applied Mathematics, University of New South Wales, Sydney 2052, Australia. Email:[email protected] ‡Department of Applied Mathematics, University of New South Wales, Sydney 2052, Australia. Email: [email protected] §Department of Mathematics and Statistics, University of Jaffna, Jaffna, Sri Lanka. Email: [email protected]. The work of this author was completed while he was visiting the University of New South Wales, Sydney, Australia.