Convex Quadratic Approximation

For some applications it is desired to approximate a set of m data points in IR with a convex quadratic function. Furthermore, it is required that the convex quadratic approximation underestimate all m of the data points. It is shown here how to formulate and solve this problem using a convex quadratic function with s = (n+ 1)(n+ 2)/2 parameters, s ≤ m, so as to minimize the approximation error in the L norm. The approximating function is q(p, x), where p ∈ IR is the vector of parameters, and x ∈ IR. The Hessian of q(p, x) with respect to x (for fixed p) is positive semi-definite, and its Hessian with respect to p (for fixed x) is shown to be positive semi-definite and of rank ≤ n. An algorithm is described for computing an optimal p∗ for any specified set of m data points, and computational results (for n = 4, 6, 10, 15) are presented showing that the optimal q(p∗, x) can be obtained efficiently. It is shown that the approximation will usually interpolate s of the m data points.