Abstract:In this paper, we study the stochastic knapsack problem with expectation constraint. We solve the relaxed version of this problem using a stochastic gradient algorithm in order to provide upper bounds for a branch-and-bound framework. Two approaches to estimate the needed gradients are studied, one based on Integration by Parts and one using Finite Differences. The Finite Differences method is a robust and simple approach with efficient results despite the fact that estimated gradients are biased, meanwhile In… Show more
“…These bounds are used in the branch-and-bound framework. Kosuch et al (2017) studied the stochastic knapsack problem with expectation constraints and proposed to solve the problem with the relaxed version of this problem using a stochastic gradient algorithm to provide upper bounds for a branch-and-bound framework. Blado & Toriello (2021) considered a two-stage stochastic multiple knapsack problem together with a set of possible disturbances with known probability of occurrence and proposed two branch-and-price approaches to solve it.…”
The knapsack problem is basic in combinatorial optimization and possesses several variants and expansions. In this paper, we focus on the multi-objective stochastic quadratic knapsack problem with random weights. We propose a Multi-Objective Memetic Algorithm With Selection Neighborhood Pareto Local Search (MASNPL). At each iteration of this algorithm, crossover, mutation, and local search are applied to a population of solutions to generate new solutions that would constitute an offspring population. Then, we use a selection operator for the best solutions to the combined parent and offspring populations. The principle of the selection operation relies on the termination of the non-domination rank and the crowding distance obtained respectively by the Non-dominated Sort Algorithm and the Crowding-Distance Computation Algorithm. To evaluate the performance of our algorithm, we compare it with both an exact algorithm and the NSGA-II algorithm. Our experimental results show that the MASNPL algorithm leads to significant efficiency.
“…These bounds are used in the branch-and-bound framework. Kosuch et al (2017) studied the stochastic knapsack problem with expectation constraints and proposed to solve the problem with the relaxed version of this problem using a stochastic gradient algorithm to provide upper bounds for a branch-and-bound framework. Blado & Toriello (2021) considered a two-stage stochastic multiple knapsack problem together with a set of possible disturbances with known probability of occurrence and proposed two branch-and-price approaches to solve it.…”
The knapsack problem is basic in combinatorial optimization and possesses several variants and expansions. In this paper, we focus on the multi-objective stochastic quadratic knapsack problem with random weights. We propose a Multi-Objective Memetic Algorithm With Selection Neighborhood Pareto Local Search (MASNPL). At each iteration of this algorithm, crossover, mutation, and local search are applied to a population of solutions to generate new solutions that would constitute an offspring population. Then, we use a selection operator for the best solutions to the combined parent and offspring populations. The principle of the selection operation relies on the termination of the non-domination rank and the crowding distance obtained respectively by the Non-dominated Sort Algorithm and the Crowding-Distance Computation Algorithm. To evaluate the performance of our algorithm, we compare it with both an exact algorithm and the NSGA-II algorithm. Our experimental results show that the MASNPL algorithm leads to significant efficiency.
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