A Polyhedral Frobenius Theorem with Applications to Integer Optimization

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We prove a representation theorem of projections of sets of integer points by an integer matrix $W$. Our result can be seen as a polyhedral analogue of several classical and recent results related to the Frobenius problem. Our result is motivated by a large class of nonlinear integer optimization problems in variable dimension. Concretely, we aim to optimize $f(Wx)$ over a set $\mathcal{F} = P\cap \mathbb{Z}^n$, where $f$ is a nonlinear function, $P\subset \mathbb{R}^n$ is a polyhedron, and $W\in \mathbb{Z}^{d\times n}$. As a consequence of our representation theorem, we obtain a general efficient transformation from the latter class of problems to integer linear programming. Our bounds depend polynomially on various important parameters of the input data leading, among others, to first polynomial time algorithms for several classes of nonlinear optimization problems.