  Constrained second-order convex optimization algorithms are the method of
choice when a high accuracy solution to a problem is needed, due to their local
quadratic convergence. These algorithms require the solution of a constrained
quadratic subproblem at every iteration. We present the \emph{Second-Order
Conditional Gradient Sliding} (SOCGS) algorithm, which uses a projection-free
algorithm to solve the constrained quadratic subproblems inexactly. When the
feasible region is a polytope the algorithm converges quadratically in primal
gap after a finite number of linearly convergent iterations. Once in the
quadratic regime the SOCGS algorithm requires $\mathcal{O}(\log(\log
1/\varepsilon))$ first-order and Hessian oracle calls and $\mathcal{O}(\log
(1/\varepsilon) \log(\log1/\varepsilon))$ linear minimization oracle calls to
achieve an $\varepsilon$-optimal solution. This algorithm is useful when the
feasible region can only be accessed efficiently through a linear optimization
oracle, and computing first-order information of the function, although
possible, is costly.
