In this paper, we propose the novel $\textit{p}$-branch-and-bound method for solving two-stage stochastic programming problems whose deterministic equivalents are represented by mixed-integer quadratically constrained quadratic programming (MIQCQP) models. The precision of the solution generated by the $\textit{p}$-branch-and-bound method can be arbitrarily adjusted by altering the value of the precision factor $\textit{p}$. The proposed method combines two key techniques. The first one, named $\textit{p}$-Lagrangian decomposition, generates a mixed-integer relaxation of a dual problem with a separable structure for a primal 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 in the optimal solution. The $\textit{p}$-branch-and-bound method's efficiency has been tested on randomly generated instances and demonstrated superior performance over commercial solver Gurobi. This paper also presents a comparative analysis of the $\textit{p}$-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 heading.

翻译：本文提出了一种新颖的$p$-分支定界法，用于求解确定性等价形式为混合整数二次约束二次规划（MIQCQP）模型的两阶段随机规划问题。通过调整精度因子$p$的值，该方法可任意控制所生成解的精度。该方法融合了两项关键技术：第一项技术称为$p$-Lagrangian分解，能为原始MIQCQP问题的对偶问题生成一个具有可分离结构的混合整数松弛形式；第二项技术是经典对偶分解方法的变体，用于求解Lagrangian对偶问题，确保最优解同时满足整数性和非预测性条件。我们在随机生成的实例上测试了$p$-分支定界法的效率，结果表明其性能优于商业求解器Gurobi。本文还比较了以两种不同求解对偶问题的算法作为子程序时$p$-分支定界法的效率差异，这两种算法分别是近端束方法和Frank-Wolfe渐进对冲算法。后者算法依赖于线性化步骤的插值，其内循环与经典渐进对冲算法中的Frank-Wolfe方法步骤类似。