  In this paper, we introduce a new optimization algorithm that is well suited
for solving parameter estimation problems. We call our new method cubic
regularized Newton with affine scaling (CRNAS). In contrast to so-called
first-order methods which rely solely on the gradient of the objective
function, our method utilizes the Hessian of the objective. As a result it is
able to focus on points satisfying the second-order optimality conditions, as
opposed to first-order methods that simply converge to critical points. This is
an important feature in parameter estimation problems where the objective
function is often non-convex and as a result there can be many critical points
making it is near impossible to identify the global minimum. An important
feature of parameter estimation in mathematical models of biological systems is
that the parameters are constrained by either physical constraints or prior
knowledge. We use an affine scaling approach to handle a wide class of
constraints. We establish that CRNAS identifies a point satisfying
$\epsilon$-approximate second-order optimality conditions within
$O(\epsilon^{-3/2})$ iterations. Finally, we compare CRNAS with MATLAB's
optimization solver fmincon on three different test problems. These test
problems all feature mixtures of heterogeneous populations, a problem setting
that CRNAS is particularly well-suited for. Our numerical simulations show
CRNAS has favorable performance, performing comparable if not better than
fmincon in accuracy and computational cost for most of our examples.
