We propose the novel p-branch-and-bound method for solving two-stage stochastic programming problems whose deterministic equivalents are represented by non-convex mixed-integer quadratically constrained quadratic programming (MIQCQP) models. The precision of the solution generated by the p-branch-and-bound method can be arbitrarily adjusted by altering the value of the precision factor p. The proposed method combines two key techniques. The first one, named p-Lagrangian decomposition, generates a mixed-integer relaxation of a dual problem with a separable structure for a primal non-convex MIQCQP problem. The second one is a version of the classical dual decomposition approach that is applied to solve the Lagrangian dual problem and ensures that integrality and non-anticipativity conditions are met once the optimal solution is obtained. This paper also presents a comparative analysis of the p-branch-and-bound method efficiency considering two alternative solution methods for the dual problems as a subroutine. These are the proximal bundle method and Frank-Wolfe progressive hedging. The latter algorithm relies on the interpolation of linearisation steps similar to those taken in the Frank-Wolfe method as an inner loop in the classic progressive hedging. The p-branch-and-bound method's efficiency was tested on randomly generated instances and demonstrated superior performance over commercial solver Gurobi.

翻译：我们提出了一种新颖的p分支定界方法，用于求解两阶段随机规划问题，其确定等价形式由非凸混合整数二次约束二次规划模型表示。通过调整精度因子p的取值，p分支定界方法所生成解的精度可任意调节。该方法融合了两项关键技术：第一项技术称为p-拉格朗日分解，用于生成原始非凸混合整数二次约束二次规划问题对偶问题的混合整数松弛形式，该对偶问题具有可分离结构；第二项技术是经典对偶分解方法的一个变体，用于求解拉格朗日对偶问题，并确保在获得最优解时满足整数性和非预期性条件。本文还通过将两种替代求解方法作为子程序应用于对偶问题，对p分支定界方法的效率进行了对比分析。这两种方法分别是近端束方法和弗兰克-沃尔夫渐进对冲算法。后者依赖于线性化步骤的插值技术，其内循环与经典渐进对冲算法中的弗兰克-沃尔夫方法步骤类似。通过在随机生成的实例上进行测试，p分支定界方法展现出优于商业求解器Gurobi的性能。