Mixed-integer nonlinear optimization encompasses a broad class of problems that present both theoretical and computational challenges. We propose a new type of method to solve these problems based on a branch-and-bound algorithm with convex node relaxations. These relaxations are solved with a Frank-Wolfe algorithm over the convex hull of mixed-integer feasible points instead of the continuous relaxation via calls to a mixed-integer linear solver as the linear oracle. The proposed method computes feasible solutions while working on a single representation of the polyhedral constraints, leveraging the full extent of mixed-integer linear solvers without an outer approximation scheme and can exploit inexact solutions of node subproblems.

翻译：混合整数非线性优化涵盖了一类同时具有理论挑战和计算挑战的广泛问题。我们提出了一种基于分支定界算法与凸节点松弛求解此类问题的新型方法。这些松弛问题通过Frank-Wolfe算法在混合整数可行点的凸包上进行求解，而非通过调用混合整数线性求解器作为线性预言机进行连续松弛。所提出的方法在单一的约束多面体表示上进行操作，无需外逼近方案即可充分利用混合整数线性求解器的全部功能，同时能够兼容节点子问题非精确解的使用，并在此过程中计算出可行解。