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

TL;DR

This paper introduces BORWin, an exact algorithm leveraging bi-objective relaxation to efficiently solve window-constrained problems, particularly effective for hydro-unit commitment with complex constraints.

Contribution

It presents a novel two-phase algorithm combining bi-objective relaxation and graph-based methods for solving window-constrained problems more efficiently.

Findings

BORWin provides tight upper bounds related to Lagrangian dual bounds.

The algorithm is highly effective for hydro-unit commitment problems.

Numerical experiments show superior performance over existing methods.

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.