We present a novel relaxation framework for general mixed-integer nonlinear programming (MINLP) grounded in computational geometry. Our approach constructs polyhedral relaxations by convexifying finite sets of strategically chosen points, iteratively refining the approximation to converge toward the simultaneous convex hull of factorable function graphs. The framework is underpinned by three key contributions: (i) a new class of explicit inequalities for products of functions that strictly improve upon standard factorable and composite relaxation schemes; (ii) a proof establishing that the simultaneous convex hull of multilinear functions over axis-aligned regions is fully determined by their values at corner points, thereby generalizing existing results from hypercubes to arbitrary axis-aligned domains; and (iii) the integration of computational geometry tools, specifically voxelization and QuickHull, to efficiently approximate feasible regions and function graphs. We implement this framework and evaluate it on randomly generated polynomial optimization problems and a suite of 619 instances from \texttt{MINLPLib}. Numerical results demonstrate significant improvements over state-of-the-art benchmarks: on polynomial instances, our relaxation closes an additional 20--25\% of the optimality gap relative to standard methods on half the instances. Furthermore, compared against an enhanced factorable programming baseline and Gurobi's root-node bounds, our approach yields superior dual bounds on approximately 30\% of \texttt{MINLPLib} instances, with roughly 10\% of cases exhibiting a gap reduction exceeding 50\%.

翻译：本文提出一种基于计算几何的新型松弛框架，用于求解一般混合整数非线性规划问题。该方法通过凸化策略选取有限点集构建多面体松弛，并迭代精化近似过程，最终收敛至可分解函数图的同步凸包。该框架包含三个关键创新：(i) 提出一类新的函数乘积显式不等式，其约束严格优于标准可分解松弛与复合松弛方案；(ii) 证明轴对齐区域上多线性函数的同步凸包完全由角点函数值确定，从而将现有超立方体结论推广至任意轴对齐定义域；(iii) 集成计算几何工具（体素化与QuickHull算法）高效逼近可行域与函数图。我们在随机生成的多项式优化问题及\texttt{MINLPLib}库的619个实例上实现并评估该框架。数值结果表明相较于当前最优基准有显著提升：在半数多项式实例中，本松弛方法额外缩小20-25%的优化间隙。此外，相较增强可分解规划基线与Gurobi根节点界值，该方法在约30%的\texttt{MINLPLib}实例中取得更优对偶界，其中约10%的案例间隙缩减幅度超过50%。