A novel approach for enclosing all homogeneous azeotropes in multicomponent mixtures is presented. The thermodynamic criteria for azeotropy are outlined, and mathematical equations for each criterion are developed. The global optimization approach is based on developing convex underestimators which are coupled with a branch and bound framework in which upper and lower bounds on the solution are refined by successively partitioning the target region into small disjoint rectangles. The objective of such an approach is to enclose all global minima since each global minimum corresponds to a homogeneous azeotrope. Because of the nature of the thermodynamic equations which describe the behavior of the liquid phase, the constraint equations are highly nonlinear and nonconvex. The success of this approach depends upon constructing valid convex lower bounds for each nonconvex function in the constraints. Four different thermodynamic models are studied, the Wilson, NRTL, UNIQUAC, and UNIFAC equations. Tight convex lower bounding functions are found for the nonconvex terms in each model. The unique element of the proposed approach is that it offers a theoretical guarantee of enclosing all homogeneous azeotropes. The effectiveness of the proposed approach is illustrated in several example problems.