This talk is concerned with optimizing nonlinear functions over the lattice points in a polyhedral set. We present three families of polynomial time algorithms for special cases of the general problem. Each such algorithm makes use of combinatorial, geometric or algebraic properties of the underlying problem. The first problem deals with optimizing nonlinear functions over a matroid. (Joint work with Jon Lee and Shmuel Onn). The second class of problems concerns convex n-fold integer minimization problems. (Joint work with Raymond Hemmecke and Shmuel Onn). Our last family of problems is to maximize polynomials over the integer points in a polytope when the dimension is fixed. Under mild assumptions we present an FPTAS for performing this task. (Joint work with Jesus De Loera, Matthias Koeppe and Raymond Hemmecke).