Complexity of integer quasiconvex polynomial optimization

We study a particular case of integer polynomial optimization: Minimize a polynomial F̂ on the set of integer points described by an inequality system F1 ≤ 0, . . . , Fs ≤ 0, where F̂ , F1, . . . , Fs are quasiconvex polynomials in n variables with integer coefficients. We design an algorithm solving this problem that belongs to the time-complexity class O(s) · lO(1) · dO(n) · 2O(n 3), where d ≥ 2 is an upper bound for the total degree of the polynomials involved and l denotes the maximum binary length of all coefficients. The algorithm is polynomial for a fixed number n of variables and represents a direct generalization of Lenstra’s algorithm in integer linear optimization. In the considered case, our complexity-result improves the algorithm given by Khachiyan and Porkolab for integer optimization on convex semialgebraic sets.