An exact approach for the bilevel knapsack problem with interdiction constraints and extensions

We consider the bilevel knapsack problem with interdiction constraints, an extension of the classic 0–1 knapsack problem formulated as a Stackelberg game with two agents, a leader and a follower, that choose items from a common set and hold their own private knapsacks. First, the leader selects some items to be interdicted for the follower while satisfying a capacity constraint. Then the follower packs a set of the remaining items according to his knapsack constraint in order to maximize the profits. The goal of the leader is to minimize the follower’s total profit. We derive effective lower bounds for the bilevel knapsack problem and present an exact method that exploits the structure of the induced follower’s problem. The approach strongly outperforms the current state-of-the-art algorithms designed for the problem. We extend the same algorithmic framework to the interval min–max regret knapsack problem after providing a novel bilevel programming reformulation. Also for this problem, the proposed approach outperforms the exact algorithms available in the literature

Solving Bilevel Knapsack Problem using Graph Neural Networks

The Bilevel Optimization Problem is a hierarchical optimization problem with two agents, a leader and a follower. The leader make their own decisions first, and the followers make the best choices accordingly. The leader knows the information of the followers, and the goal of the problem is to find the optimal solution by considering the reactions of the followers from the leader's point of view. For the Bilevel Optimization Problem, there are no general and efficient algorithms or commercial solvers to get an optimal solution, and it is very difficult to get a good solution even for a simple problem. In this paper, we propose a deep learning approach using Graph Neural Networks to solve the bilevel knapsack problem. We train the model to predict the leader's solution and use it to transform the hierarchical optimization problem into a single-level optimization problem to get the solution. Our model found the feasible solution that was about 500 times faster than the exact algorithm with $1.7\%$ optimal gap. Also, our model performed well on problems of different size from the size it was trained on.Comment: 27 pages, 2 figure

A New General-Purpose Algorithm for Mixed-Integer Bilevel Linear Programs

International audienceBilevel optimization problems are very challenging optimization models arising in many important practical contexts, including pricing mechanisms in the energy sector, airline and telecommunication industry, transportation networks, critical infrastructure defense, and machine learning. In this paper, we consider bilevel programs with continuous and discrete variables at both levels, with linear objectives and constraints (continuous upper level variables, if any, must not appear in the lower level problem). We propose a general-purpose branch-and-cut exact solution method based on several new classes of valid inequalities, which also exploits a very effective bilevel-specific preprocessing procedure. An extensive computational study is presented to evaluate the performance of various solution methods on a common testbed of more than 800 instances from the literature and 60 randomly generated instances. Our new algorithm consistently outperforms (often by a large margin) alternative state-of-the-art methods from the literature, including methods exploiting problem-specific information for special instance classes. In particular, it solves to optimality more than 300 previously unsolved instances from the literature. To foster research on this challenging topic, our solver is made publicly available online

A New General-Purpose Algorithm for Mixed-Integer Bilevel Linear Programs

Bilevel optimization problems are very challenging optimization models arising in many important practical contexts, including pricing mechanisms in the energy sector, airline and telecommunication industry, transportation networks, critical infrastructure defense, and machine learning. In this paper, we consider bilevel programs with continuous and discrete variables at both levels, with linear objectives and constraints (continuous upper level variables, if any, must not appear in the lower level problem). We propose a general-purpose branch-and-cut exact solution method based on several new classes of valid inequalities, which also exploits a very effective bilevel-specific preprocessing procedure. An extensive computational study is presented to evaluate the performance of various solution methods on a common testbed of more than 800 instances from the literature and 60 randomly generated instances. Our new algorithm consistently outperforms (often by a large margin) alternative state-of-the-art methods from the literature, including methods exploiting problem-specific information for special instance classes. In particular, it solves to optimality more than 300 previously unsolved instances from the literature. To foster research on this challenging topic, our solver is made publicly available online

A branch-and-cut algorithm for the Edge Interdiction Clique Problem

Given a graph G and an interdiction budget k∈N, the Edge Interdiction Clique Problem (EICP) asks to find a subset of at most k edges to remove from G so that the size of the maximum clique, in the interdicted graph, is minimized. The EICP belongs to the family of interdiction problems with the aim of reducing the clique number of the graph. The EICP optimal solutions, called optimal interdiction policies, determine the subset of most vital edges of a graph which are crucial for preserving its clique number. We propose a new set-covering-based Integer Linear Programming (ILP) formulation for the EICP with an exponential number of constraints, called the clique-covering inequalities. We design a new branch-and-cut algorithm which is enhanced by a tailored separation procedure and by an effective heuristic initialization phase. Thanks to the new exact algorithm, we manage to solve the EICP in several sets of instances from the literature. Extensive tests show that the new exact algorithm greatly outperforms the state-of-the-art approaches for the EICP