On the complexity of optimization over the standard simplex

We review complexity results for minimizing polynomials over the standard simplex and unit hypercube. In addition, we derive new results on the computational complexity of approximating the minimum of some classes of functions (including Lipschitz continuous functions) on the standard simplex. The main tools used in the analysis are Bernstein approximation and Lagrange interpolation on the simplex combined with an earlier result by de Klerk et al. [A PTAS for the minimization of polynomials of fixed degree over the simplex, Theoretical Computer Science 361 (2–3) (2006) 210–225].