The p-Lagrangian relaxation for separable nonconvex MIQCQP problems

Abstract

This paper presents a novel technique to compute Lagrangian bounds for nonconvex mixed-integer quadratically constrained quadratic programming problems presenting a separable structure (i.e., a separable problems) such as those arising in deterministic equivalent representations of two-stage stochastic programming problems. In general, the nonconvex nature of these models still poses a challenge to the available solvers, which do not consistently perform well for larger-scale instances. Therefore, we propose an appealing alternative algorithm that allows for overcoming computational performance issues. Our novel technique, named the p-Lagrangian decomposition, is a decomposition method that combines Lagrangian decomposition with mixed-integer programming-based relaxations. These relaxations are obtained using the reformulated normalised multiparametric disaggregation technique and can be made arbitrarily precise by means of a precision parameter p. We provide a technical analysis showing the convergent behaviour of the approach as the approximation is made increasingly precise. We observe that the proposed method presents significant reductions in computational time when compared with a previously proposed techniques in the literature and the direct employment of a commercial solver. Moreover, our computational experiments show that the employment of a simple heuristic can recover solutions with small duality gaps.

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

“…Employing either Algorithm 1 or 2 ensures the convergence to the optimal solution of the p-Lagrangian dual sub-problem (9). For the convergence of Algorithms 1 and 2 to the optimal solution of problem (6), please refer to the [27] and [10], respectively.…”

“…. µ |S| ), the p-Lagrangian dual function ( 7) provides a lower bound for the primal RNMDT p problem (5) [6]. Our objective is to find the tightest (i.e., the uppermost) lower bound.…”

“…Starting dual multipliers µ 0 for Algorithms 1 and 2 were set µ 0 = 0. To set the first-stage variables (x s 0 ) s∈S for Algorithm 2, we considered the solution of the p-Lagrangian dual function (6) for a fixed value of the dual variable µ = µ 0 . Following [10], to initialise (V s 0 ) s∈S , we took one arbitrary scenario (in our case the first scenario in S,i.e., s = 1) and set V 1 0 = {(x 1 0 , y 1 0 , w 1 0 )}.…”

“…To tackle problem (4), [6] developed an algorithm named p-Lagrangian decomposition. The p-Lagrangian decomposition method involves exploiting Lagrangian relaxation for decomposing the primal problem (4) into |S| independent sub-problems and employing the reformulated normalised multiparametric disaggregation technique (RNMDT) [5] to construct mixedinteger-based relaxations.…”

“…As suggested by the numerical results in [6], the p-Lagrangian decomposition demonstrated superior performance compared to commercial solver Gurobi [24]. Nevertheless, the p-Lagrangian decomposition algorithm has an important shortcoming related to the duality gap arising from the mixedinteger nature of the primal problem combined with the imprecision of the RNMDT relaxation.…”