Optimization of a convex program with a polynomial perturbation

Abstract

We consider the problem of minimizing a convex function plus a polynomial p over a convex body K . We give an algorithm that outputs a solution x whose value is within ϵ range K ( p ) of the optimum value, where range K ( p ) = sup x ∈ K p ( x ) − inf x ∈ K p ( x ) . When p depends only on a constant number of variables, the algorithm runs in time polynomial in 1 / ϵ , the degree of p , the time to round K and the time to solve the convex program that results by setting p = 0 .