In this paper, we analyze conic minimax convex polynomial optimization problems. Under a suitable regularity condition, an exact conic programming relaxation is established based on a positivity characterization of a max function over a conic convex system. Further, we consider a general conic minimax \(\rho \) -convex polynomial optimization problem, which is defined by appropriately extending the notion of conic convexity of a vector-valued mapping. For this problem, it is shown that a Karush-Kuhn-Tucker condition at a global minimizer is necessary and sufficient for ensuring an exact relaxation with attainment of the conic programming relaxation. The exact conic programming relaxations are applied to SOS-convex polynomial programs, where appropriate choices of the data allow the associated conic programming relaxation to be reformulated as a semidefinite programming problem. In this way, we can further elaborate the obtained results for other special settings including conic robust SOS-convex polynomial problems and difference of SOS-convex polynomial programs.
