A novel dual-decomposition method based on $p$-Lagrangian relaxation

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.