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 $ (\mathbb {R}, +, \cdot ) $ (R,+,& sdot;). Namely, let \[ f, g_1, \ldots, g_l, h_1, \ldots, h_m \colon \mathbb{R}<^>n o \mathbb{R} \] f,g1,& mldr;,gl,h1,& mldr;,hm:Rn -> R be definable $ C<^>p $ Cp functions ( $ p \ge 2 $ p >= 2), and assume that f is nonnegative on the set \[ S := \{x \in \mathbb{R}<^>n \ | \ g_1(x) \ge 0, \ldots, g_l(x) \ge 0, h_1(x) = 0, \ldots, h_m(x) = 0 \}. \] S:={x is an element of Rn | g1(x)>= 0,& mldr;,gl(x)>= 0,h1(x)=0,& mldr;,hm(x)=0}. Under some natural hypotheses on zeros of f in S, we show that f is expressible in the form \[ f = \phi_0 + \sum_{i = 1}<^>l \phi_i g_i + \sum_{j =1}<^>m \psi_j h_j, \] f=phi 0+& sum;i=1l phi igi+& sum;j=1m psi jhj, where $ \phi _i, \psi _j \colon \mathbb {R}<^>n ightarrow \mathbb {R} $ phi i,psi j:Rn -> R are definable $ C<^>{p - 2} $ Cp-2-functions and each $ \phi _i $ phi i is a sum of squares of definable $ C<^>{p - 2} $ Cp-2-functions. As a consequence, we derive global optimality conditions which generalize the Karush-Kuhn-Tucker optimality conditions for nonlinear convex optimization.