Scaling, Proximity, and Optimization of Integrally Convex Functions

In discrete convex analysis, the scaling and proximity properties for the class of L$^\natural$-convex functions were established more than a decade ago and have been used to design efficient minimization algorithms. For the larger class of integrally convex functions of $n$ variables, we show here that the scaling property only holds when $n \leq 2$, while a proximity theorem can be established for any $n$, but only with a superexponential bound. This is, however, sufficient to extend the classical logarithmic complexity result for minimizing a discrete convex function of one variable to the case of integrally convex functions of any fixed number of variables.