Minimizing Polynomials Over Semialgebraic Sets

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This paper concerns a method for finding the minimum of a polynomial on a semialgebraic set, i.e., a set in ᵐ defined by finitely many polynomial equations and inequalities, using the Karush-Kuhn-Tucker (KKT) system and sum of squares (SOS) relaxations. This generalizes results in the recent paper njwgrad, which considers minimizing polynomials on algebraic sets, i.e., sets in ᵐ defined by finitely many polynomial equations. Most of the theorems and conclusions in njwgrad generalize to semialgebraic sets, even in the case where the semialgebraic set is not compact. We discuss the method in some special cases, namely, when the semialgebraic set is contained in the nonnegative orthant ⁿ+ or in box constraints [a,b]ₙ. These constraints make the computations more efficient.