In this paper, we propose a novel $p$-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 $p$-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 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 $p$-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 $p$-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 linearization steps similar to those taken in the Frank-Wolfe method as an inner loop in the classic progressive heading.

翻译：本文提出了一种新型$p$-分支定界方法，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$p$的数值，可任意控制$p$-分支定界法所生成解的精度。该方法融合了两项关键技术：第一项技术称为$p$-拉格朗日分解，它为原始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改进版本，用于求解拉格朗日对偶问题，并确保最优解满足整数性与非预期性条件。通过在随机生成实例上的测试，该$p$-分支定界法的效率已被验证优于商业求解器Gurobi。本文还针对以两种替代解法（近端束方法与Frank-Wolfe渐进对冲方法）作为子程序的对偶问题求解过程，开展了$p$-分支定界法效率的对比分析。其中，后一种算法依赖于线性化步骤的插值操作（类似于Frank-Wolfe方法中作为经典渐进对冲内循环的处理方式）。