BORWin: Exact algorithm based on a Bi-Objective Relaxation for Window-constrained problems

TL;DR

It is shown that complementary upper bounds could be derived to further improve performance in some special cases and BOR-Win appears to be very efficient compared to state-of-the-art approaches.

Abstract

A mixed integer maximization problem involving several additional constraints defined with both a lower and an upper bound is considered. It is assumed that one of such constraints is more restrictive than the others. As it can be seen as a resource window constraint, it defines the so-called window-constrained problem. From a bi-objective perspective, a 2-phase algorithm, called BORWin, is devised. It stands for Bi-Objective Relaxation for Window-constrained problems. The first phase is generic for any window-constrained problem and provides a family of upper bounds based on a bi-objective relaxation of the additional constraints. It is shown that the latter bounds strongly relate to the Lagrangian dual bounds. The second phase is derived for a variant involving a graph structure, namely the window-constrained longest-path problem on an acyclic graph. The aim is to take advantage of the upper bounds to devise an efficient label extension algorithm. It is shown that complementary upper bounds could be derived to further improve performance in some special cases. A typical example is when the additional constraints have special knapsack structures. This is the case for the Hydro-Unit Commitment problem with a single plant (1-HUC). From numerical experiments for the 1-HUC, BOR-Win appears to be very efficient compared to state-of-the-art approaches.