We study the unequal circle-circle non-overlapping constraints, a form of reverse convex constraints that often arise in optimization models for cutting and packing applications. The feasible region induced by the intersection of circle-circle non-overlapping constraints is highly non-convex, and standard approaches to construct convex relaxations for spatial branch-and-bound global optimization of such models typically yield unsatisfactory loose relaxations. Consequently, solving such non-convex models to guaranteed optimality remains extremely challenging even for the state-of-the-art codes. In this paper, we apply a purpose-built branching scheme on non-overlapping constraints and utilize strengthened intersection cuts and various feasibility-based tightening techniques to further tighten the model relaxation. We embed these techniques into a branch-and-bound code and test them on two variants of circle packing problems. Our computational studies on a suite of 75 benchmark instances yielded, for the first time in the open literature, a total of 54 provably optimal solutions, and it was demonstrated to be competitive over the use of the state-of-the-art general-purpose global optimization solvers.

我们研究不相等的圆-圆非重叠约束，这是反向凸约束的一种形式，通常在切割和包装应用的优化模型中出现。由圆-圆不重叠约束的交集引起的可行区域是高度非凸的，构造此类模型的空间分支和边界全局优化的凸松弛的标准方法通常会产生不令人满意的松弛。因此，即使对于最先进的代码，求解此类非凸模型以确保最优性仍然极具挑战性。在本文中，我们对非重叠约束应用了专门构建的分支方案，并利用增强的相交切口和各种基于可行性的拧紧技术来进一步拧紧模型松弛。我们将这些技术嵌入到分支定界代码中，并在两种圆形填充问题变体上进行了测试。我们在一组75个基准实例上进行的计算研究在公开文献中首次得出了总共54个可证明的最佳解决方案，并且事实证明，与使用最新的通用解决方案相比，该解决方案具有竞争优势用途的全局优化求解器。