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.

翻译：约束二阶凸优化算法因具有局部二次收敛性，在高精度求解问题时成为首选方法。这类算法每步迭代需求解一个带约束的二次子问题。我们提出二阶条件梯度滑动（SOCGS）算法，该算法使用无投影方法不精确求解带约束的二次子问题。当可行域为多面体时，算法在经过有限步线性收敛迭代后，原始间隙呈二次收敛。进入二次收敛阶段后，SOCGS算法需调用$\mathcal{O}(\log(\log 1/\varepsilon))$次一阶和海瑟矩阵预言，以及$\mathcal{O}(\log (1/\varepsilon) \log(\log1/\varepsilon))$次线性最小化预言，即可得到$\varepsilon$-最优解。当可行域仅能通过线性优化预言高效访问，且函数一阶信息虽可计算但代价高昂时，该算法具有实用价值。