A Note on Nonconvex Minimax Theorem with Separable Homogeneous Polynomials

The minimax theorem for a convex-concave bifunction is a fundamental theorem in optimization and convex analysis, and has a lot of applications in economics. In the last two decades, a nonconvex extension of this minimax theorem has been well studied under various generalized convexity assumptions. In this note, by exploiting the hidden convexity (joint range convexity) of separable homogeneous polynomials, we establish a nonconvex minimax theorem involving separable homogeneous polynomials. Our result complements the existing study of nonconvex minimax theorem by obtaining easily verifiable conditions for the nonconvex minimax theorem to hold.