Minimizing Lipschitz-continuous strongly convex functions over integer points in polytopes

Michel Baes, Alberto Del Pia, Yurii E. Nesterov, Shmuel Onn, Robert Weismantel

Abstract: This paper is about the minimization of Lipschitz-continuous and strongly convex functions over integer points in polytopes. Our results are related to the rate of convergence of a black-box algorithm that iteratively solves special quadratic integer problems with a constant approximation factor. Despite the generality of the underlying problem, we prove that we can find efficiently, with respect to our assumptions regarding the encoding of the problem, a feasible solution whose objective function value is close to the optimal value. We also show that this proximity result is the best possible up to a factor polynomial in the encoding length of the problem.