The Complexity of Continuous Optimization

Abstract

Given a polynomial objective function f(x1,…,xn), we consider the problem of finding the maximum of this polynomial inside some convex set D = {x : Ax <= B}. We show that, under a complexity assumption, this extremum cannot be approximated by any polynomial-time algorithm, even exceedingly poorly. This represents an unusual interplay of discrete and continuous mathematics: using a combinatorial argument to get a hardness result for a continuous optimization problem.