Integer Optimization on Convex Semialgebraic Sets

Let Y be a convex set inRk defined by polynomial inequalities and equations of degree at most d ≥ 2 with integer coefficients of binary length at most l. We show that if the set of optimal solutions of the integer programming problem min{yk | y = (y1, . . . , yk) ∈ Y ∩Zk} is not empty, then the problem has an optimal solution y∗ ∈ Y ∩Zk of binary length ld O(k ). For fixed k, our bound implies a polynomial-time algorithm for computing an optimal integral solution y∗. In particular, we extend Lenstra’s theorem on the polynomialtime solvability of linear integer programming in fixed dimension to semidefinite integer programming.