Two novel deterministic global optimization algorithms for nonconvex mixed-integer problems (MINLPs) are proposed, using the advances of the alpha BB algorithm for nonconvex NLPs of Adjiman et al. the special structure mixed-integer alpha BB algorithm (SMIN-alpha BB) addresses problems with nonconvexities in the continuous variables and linear and mixed-bilinear participation of the binary variables. The general structure mixed-integer alpha BB algorithm (GMIN-alpha BB) is applicable to a very general class of problems for which the continuous relaxation is twice continuously differentiable. Both algorithms are developed using the concepts of branch-and-bound, but they differ in their approach to each of the required steps. The SMIN-alpha BB algorithm is based on the convex underestimation of the continous functions, while the GMIN-alpha BB algorithm is centered around the convex relaxation of the entire problem. Both algorithms rely on optimization or interval-based variable-bound updates to enhance efficiency. A series of medium-size engineering applications demonstrates the performance of the algorithms. Finally, a comparison of the two algorithms on the same problems highlights the value of algorithms that can handle binary or integer variables without reformulation.