On convex quadratic approximation

Let n greater than or equal to 1 and f : R-n --> R a convex function. Given distinct points z(1), z(2).... z(N) in R-n we consider the problem of finding a quadratic function g : Rn --> R such that parallel to[f (z(1)) - g(z(1))..... f(z(N)) - g(z(N))]parallel to is minimal for a given norm parallel to (.) parallel to. For the Euclidean norm this is the well-known quadratic least squares problem. (If the norm is not specified we will simply refer to g as the quadratic approximation.) In this paper we prove the result that the quadratic approximation is not necessarily convex for n greater than or equal to 2, even though it is convex if n = 1. This result has many consequences both for the field of statistics and optimization. We show that the best convex quadratic approximation can be obtained in the multivariate case by using semidefinite programming techniques.