Semidefinite programming for gradient and Hessian computation in maximum entropy estimation

We consider the classical problem of estimating a density on [0,1] via some maximum entropy criterion. For solving this convex optimization problem with algorithms using first-order or second-order methods, at each iteration one has to compute (or at least approximate) moments of some measure with a density on [0,1], to obtain gradient and Hessian data. We propose a numerical scheme based on semidefinite programming that avoids computing quadrature formula for this gradient and Hessian computation.