Integer Optimization on Convex Semialgebraic Sets

Abstract

Let Y be a convex set in \Re k 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 $\{ y_k \mid y=(y_1, . . . ,y_k) \in Y \cap \Ze^k \}$ is not empty, then the problem has an optimal solution $y^* \in Y \cap \Ze^k$ of binary length ld O(k4) . 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 polynomial-time solvability of linear integer programming in fixed dimension to semidefinite integer programming.