Representations of nonnegative functions and global optimality conditions

This paper addresses sum-of-squares representations of nonnegative functions that are definable in o-minimal structures on (R,+,⋅). Namely, let f,g1,…,gl,h1,…,hm:Rn→R be definable Cp functions (p≥2), and assume that f is nonnegative on the set S:={x∈Rn | g1(x)≥0,…,gl(x)≥0,h1(x)=0,…,hm(x)=0}.Under some natural hypotheses on zeros of f in S, we show that f is expressible in the form f=ϕ0+∑i=1lϕigi+∑j=1mψjhj,where ϕi,ψj:Rn→R are definable Cp−2-functions and each ϕi is a sum of squares of definable Cp−2-functions. As a consequence, we derive global optimality conditions which generalize the Karush–Kuhn–Tucker optimality conditions for nonlinear convex optimization.