A novel dual-decomposition method for non-convex two-stage stochastic mixed-integer quadratically constrained quadratic problems

TL;DR

This paper introduces a flexible p-branch-and-bound method for solving complex non-convex two-stage stochastic MIQCQP problems, demonstrating superior efficiency over existing solvers through comparative analysis.

Contribution

The paper presents a novel p-branch-and-bound approach combining p-Lagrangian decomposition and dual decomposition, with adjustable solution precision for non-convex stochastic programming.

Findings

The method's solution precision can be arbitrarily adjusted.

The p-branch-and-bound outperforms Gurobi on test instances.

Comparative analysis shows superior efficiency over alternative dual solution methods.

Abstract

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.