  In this paper we introduce a new dual program, which is representable as a
semi-definite linear programming problem, for a primal convex minimax
programming model problem and show that there is no duality gap between the
primal and the dual whenever the functions involved are SOS-convex polynomials.
Under a suitable constraint qualification, we derive strong duality results for
this class of minimax problems. Consequently, we present applications of our
results to robust SOS-convex programming problems under data uncertainty and to
minimax fractional programming problems with SOS-convex polynomials. We obtain
these results by first establishing sum of squares polynomial representations
of non-negativity of a convex max function over a system of SOS-convex
constraints. The new class of SOS-convex polynomials is an important subclass
of convex polynomials and it includes convex quadratic functions and separable
convex polynomials. The SOS-convexity of polynomials can numerically be checked
by solving semi-definite programming problems whereas numerically verifying
convexity of polynomials is generally very hard.
