We introduce the concept of disjunctive sum of squares for certifying nonnegativity of polynomials. Unlike the popular sum of squares approach where nonnegativity is certified by a single algebraic identity, the disjunctive sum of squares approach certifies nonnegativity with multiple algebraic identities which can be found in parallel. Our main result is a disjunctive Positivstellensatz proving that we can keep the degree of each algebraic identity as low as the degree of the polynomial whose nonnegativity is in question. Based on this result, we construct a semidefinite programming based converging hierarchy of lower bounds for the problem of minimizing a polynomial over a compact basic semialgebraic set, where the size of the largest semidefinite constraint is fixed throughout the hierarchy. We further prove a second disjunctive Positivstellensatz which leads to an optimization-free hierarchy for polynomial optimization. We specialize this result to the problem of proving copositivity of matrices. Finally, we describe how the disjunctive sum of squares approach can be combined with a branch-and-bound algorithm and we present numerical experiments on polynomial, copositive, and combinatorial optimization problems.

翻译：我们引入了析取平方和的概念，用于证明多项式的非负性。与流行的平方和方法（通过单个代数恒等式证明非负性）不同，析取平方和方法通过多个可并行求解的代数恒等式来证明非负性。我们的主要结果是一个析取正位置定理（Positivstellensatz），证明我们可以将每个代数恒等式的次数保持在与待证非负性多项式相同的次数上。基于此结果，我们构建了一个基于半定规划的收敛下界层次结构，用于求解紧致基本半代数集上的多项式极小化问题，其中该层次结构中最大的半定约束规模是固定的。我们进一步证明了第二个析取正位置定理，该定理为多项式优化提供了一种无优化层次结构。我们将此结果专门应用于矩阵共正性的证明问题。最后，我们描述了如何将析取平方和法与分支定界算法相结合，并给出了多项式优化、共正优化和组合优化问题的数值实验。