Surrogate-based algorithm for computing an upper bound set for the 0/1 bi-objective bi-dimensional knapsack problem
Résumé
In this work, we consider the binary knapsack problem with two objectives and two dimensions. Our purpose is to compute a tight upper bound set for the set of nondominated points of this problem. To do this we consider the surrogate relaxation. Its principle is to aggregate the constraints of the problem according to a multiplier u. Thus we obtain a single dimensional problem. In the single-objective case, the solution of this relaxed problem allows to obtain an upper bound whose quality relies on the choice of the multiplier. The problem consisting of finding the tightest possible bound is called dual surrogate problem.
With two objectives, we propose to solve the convex relaxation of the surrogate relaxation. It gives a upper bound set for the 2O2DKP. The upper bound sets obtained with different multipliers are not necessarily comparable. However the intersection of the search spaces induced by those bound sets defines a tighter upper bound set.
We propose a generalization of the dual surrogate problem for the bi-objective case. We call it the optimal convex surrogate (OCS) upper bound set. This upper bound set can be obtained using all the possible multipliers. Thus this is the tighest possible upper bound set based on the convex relaxation of the surrogate relaxation for the 2O2DKP. Even if the number of possible multipliers is infinite, the number of different upper bound sets is finite. Properties of the different multipliers u and dominance relation between upper bound sets are studied. An exact and an approximative method to compute the OCS upper bound set are derived.
We tested numerically those two methods on a 2O2DKP benchmark and we compare them a state-of-the-art approximative method. The results obtained are encouraging for the use of this bound set computation in an exact solving method for the 2O2DKP.