  We develop a new approximation theory for linear and quadratic interpolation
models, suitable for use in convex-constrained derivative-free optimization
(DFO). Most existing model-based DFO methods for constrained problems assume
the ability to construct sufficiently accurate approximations via
interpolation, but the standard notions of accuracy (designed for unconstrained
problems) may not be achievable by only sampling feasible points, and so may
not give practical algorithms. This work extends the theory of
convex-constrained linear interpolation developed in [Hough & Roberts, SIAM J.
Optim, 32:4 (2022), pp. 2552-2579] to the case of linear regression models and
underdetermined quadratic interpolation models.
