266,822 research outputs found
Learning to Branch in Combinatorial Optimization with Graph Pointer Networks
Branch-and-bound is a typical way to solve combinatorial optimization
problems. This paper proposes a graph pointer network model for learning the
variable selection policy in the branch-and-bound. We extract the graph
features, global features and historical features to represent the solver
state. The proposed model, which combines the graph neural network and the
pointer mechanism, can effectively map from the solver state to the branching
variable decisions. The model is trained to imitate the classic strong
branching expert rule by a designed top-k Kullback-Leibler divergence loss
function. Experiments on a series of benchmark problems demonstrate that the
proposed approach significantly outperforms the widely used expert-designed
branching rules. Our approach also outperforms the state-of-the-art
machine-learning-based branch-and-bound methods in terms of solving speed and
search tree size on all the test instances. In addition, the model can
generalize to unseen instances and scale to larger instances
An overview of Stackelberg pricing in networks
The Stackelberg pricing problem has two levels of decision making: tariff setting by an operator, and then selection of the cheapest alternative by customers. In the network version, an operator determines tariffs on a subset of the arcs that he owns. Customers, who wish to connect two vertices with a path of a certain capacity, select the cheapest path. The revenue for the operator is determined by the tariff and the amount of usage of his arcs. The most natural model for the problem is a (bilinear) bilevel program, where the upper level problem is the pricing problem of the operator, and the lower level problem is a shortest path problem for each of the customers. This paper contains a compilation of theoretical and algorithmic results on the network Stackelberg pricing problem. The description of the theory and algorithms is generally informal and intuitive. We redefine the underlying network of the problem, to obtain a compact representation. Then we describe a basic branch-and-bound enumeration procedure. Both concepts are used for complexity issues and for the development of algorithms: establishing NP-hardness, approximability, special cases solvable in polynomial time, and an efficient exact branch-and-bound algorithm.Economics ;
An overview of Stackelberg pricing in networks
The Stackelberg pricing problem has two levels of decision making: tariff setting by an operator, and then selection of the cheapest alternative by customers. In the network version, an operator determines tariffs on a subset of the arcs that he owns. Customers, who wish to connect two vertices with a path of a certain capacity, select the cheapest path. The revenue for the operator is determined by the tariff and the amount of usage of his arcs. The most natural model for the problem is a (bi-linear) bilevel program, where the upper level problem is the pricing problem of the operator, and the lower level problem is a shortest path problem for each of the customers. This manuscript contains a compilation of theoretical and algorithmic results on the Stackelberg pricing problem. The description of the theory and algorithms is generally informal and intuitive. We redefine the underlying network of the problem, to obtain a compact representation. Then, we describe a basic branch-and-bound enumeration procedure. Both concepts are used for complexity issues and the development of algorithms: establishing NP-hardness, approximability, and polynomially solvable cases, and an efficient exact branch-and-bound algorithm.mathematical applications;
Exact Combinatorial Optimization with Graph Convolutional Neural Networks
Combinatorial optimization problems are typically tackled by the
branch-and-bound paradigm. We propose a new graph convolutional neural network
model for learning branch-and-bound variable selection policies, which
leverages the natural variable-constraint bipartite graph representation of
mixed-integer linear programs. We train our model via imitation learning from
the strong branching expert rule, and demonstrate on a series of hard problems
that our approach produces policies that improve upon state-of-the-art
machine-learning methods for branching and generalize to instances
significantly larger than seen during training. Moreover, we improve for the
first time over expert-designed branching rules implemented in a
state-of-the-art solver on large problems. Code for reproducing all the
experiments can be found at https://github.com/ds4dm/learn2branch.Comment: Accepted paper at the NeurIPS 2019 conferenc
Project scheduling with modular project completion on a bottleneck resource.
In this paper, we model a research-and-development project as consisting of several modules, with each module containing one or more activities. We examine how to schedule the activities of such a project in order to maximize the expected profit when the activities have a probability of failure and when an activity’s failure can cause its module and thereby the overall project to fail. A module succeeds when at least one of its constituent activities is successfully executed. All activities are scheduled on a scarce resource that is modeled as a single machine. We describe various policy classes, establish the relationship between the classes, develop exact algorithms to optimize over two different classes (one dynamic program and one branch-and-bound algorithm), and examine the computational performance of the algorithms on two randomly generated instance sets.Scheduling; Uncertainty; Research and development; Activity failures; Modular precedence network;
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