We develop an algorithmic theory of nonlinear optimization over sets
of integer points presented by inequalities or by oracles. Using a
combination of geometric and algebraic methods, involving zonotopes,
Graver bases, multivariate polynomials and Frobenius numbers, we provide
polynomial-time algorithms for broad classes of nonlinear combinatorial
optimization problems and integer programs in variable dimension.
I will overview this work, joint with many colleagues over the last few