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：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表现为非凸混合整数二次约束二次规划模型的两阶段随机规划问题。通过调整精度因子p的取值，p-分支定界法所得解的精度可进行任意调节。该方法融合了两项关键技术：其一是名为p-拉格朗日分解的技术，可为原始非凸MIQCQP问题生成具有可分离结构的对偶问题混合整数松弛；其二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，并确保在获得最优解时满足整数性与非预期性条件。本文还通过将两种对偶问题求解方法作为子程序，对p-分支定界法的计算效率进行了对比分析：分别是近端束方法和Frank-Wolfe渐进对冲算法。后者通过在经典渐进对冲算法的内循环中，引入类似Frank-Wolfe方法的线性化步长插值机制。在随机生成的算例上测试表明，p-分支定界法的计算效率优于商业求解器Gurobi。