In this paper, 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 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 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.

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非凸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-分支定界方法生成解的精度。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，它为原始非凸MIQCQP问题生成具有可分离结构的对偶问题的混合整数松弛；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，确保最优解满足整数性约束和非预期性条件。通过随机生成实例测试，p-分支定界方法的效率优于商业求解器Gurobi。本文还比较了两种备选子问题求解方法对p-分支定界方法效率的影响，分别是近端束方法和Frank-Wolfe渐进对冲算法。后一种算法在经典渐进对冲框架的内循环中，采用类似于Frank-Wolfe方法的线性化步骤插值技术。