A classical approach for handling non-convex optimization problems involving bilinear terms is to approximate the bilinear expressions using McCormick relaxations. Such relaxations are central to global optimization methods, with the spatial branch-and-bound (sBB) algorithm being one of the most widely used approaches. In the spatial branching step, the algorithm partitions the feasible region using axis-aligned (parallel-axis) cuts, generating hyper-rectangular sub-region.
In this research, we investigate the impact of introducing 45\textdegree-oblique cuts instead of traditional axis-aligned cuts. We show that oblique cuts can provide improved directional bounds—either upper or lower—depending on the orientation of the cut. More precisely, we demonstrate that the upper (respectively, lower) bounds derived from the proposed oblique-envelope formulation are always tighter than the corresponding McCormick upper (respectively, lower) bounds over the associated feasible region.
Furthermore, we derive conditions under which the range of the true bilinear function over the feasible region induced by axis-aligned cuts exceeds the corresponding range over the feasible region induced by oblique cuts. These results provide theoretical justification for the potential tightening effect of oblique partitioning in spatial branch-and-bound algorithms.