Looking for sparsity is nowadays crucial to speed up the training of large-scale neural networks. Projections onto the$\ell_{1}$and$\ell_{1,\propto}$are among the most efficient techniques to sparsify and reduce the overall cost of neural networks. In this paper, we introduce a new projection algorithm for the$\ell_{1,\infty}$norm ball. Its worst-case time complexity is$\mathcal{O}(nm+J\log(nm))$for a matrix in$\mathbb{R}^{n\times m}. J$is a term that tends to 0 when the sparsity is high, and to$n\times m$in the worst case. The algorithm is easy to implement and it is guaranteed to converge to the exact solution in finite time. Moreover, we propose to incorporate the$\ell_{1,\infty}$ball projection while training an auto encoder to enforce feature selection and sparsity of the weights. Sparsification appears in the encoder to primarily do feature selection due to our application in biology, where only a very small part ($<$2%) of the data is relevant. We show that in both the biological and general cases of sparsity, our method is the fastest.

In robust optimization, finding a solution that solely respects the constraints is not enough. Usually, the uncertainty and unknown parameters of the model are represented by random variables. In such conditions, a good solution is a solution robust to most-likely assignments of these random variables. Recently, the Confidence constraint has been introduced by Mercier-Aubin et al. in order to enforce this type of robustness in constraint programming. Unfortunately, it is restricted to a conjunction of binary inequalities In this paper, we generalize the Confidence constraint to any constraint and propose an implementation based on Multi-valued Decision Diagrams (MDDs). The Confidence constraint is defined over a vector of random variables. For a given constraint C, and given a threshold, the Confidence constraint ensures that the probability for C to be satisfied by a sample of the random variables is greater than the threshold. We propose to use MDDs to represent the constraints on the random variables. MDDs are an efficient tool for representing combinatorial constraints, thanks to their exponential compression power. Here, both random and decision variables are stored in the MDD, and propagation rules are proposed for removing values of decision variables that cannot lead to robust solutions. Furthermore, for several constraints, we show that decision variables can be omitted from the MDD because lighter filtering algorithms are sufficient. This leads to gain an exponential factor in the MDD size. The experimental results obtained on a chemical deliveries problem in factories – where the chemicals consumption are uncertain – shows the efficiency of the proposed approach.

In this paper, we propose a model to schedule the next 24 hours of operations in a bulk cargo port to unload bulk cargo trains onto stockpiles. It is a problem that includes multiple parts such as splitting long trains into shorter ones and the routing of bulk material through a configurable network of conveyors to the stockpiles. Managing such trains (up to three kilometers long) also requires specialized equipment. The real world nature of the problem specification implies the necessity to manage heterogeneous data. Indeed, when new equipment is added (e.g. dumpers) or a new type of wagon comes in use, older or different equipment will still be in use as well. In this paper, we provide a detailed presentation of this real world problem and its associated data. This allows us to propose an effective constraint programming model to solve this problem. We also discuss the model design and the different implementations of the propagators that we used in practice. Finally, we show how this model, coupled with a large neighborhood search, was able to find 24 hour schedules efficiently

Heuristics are one of the most important tools to guide search to solve combinatorial problems. They are often specifically designed for one single problem and require both expertise and implementation work. Generic frameworks like SAT or CSP have developed heuristics that obey general principles like first fail or are able to learn and adapt from the exploration of the search tree like Dom/wDeg. In SAT, the classic VSIDS heuristic falls into both categories. The question of whether it is possible to learn from solving existing problems has been addressed for a long time by portfolio solvers where the best heuristic is chosen by Machine Learning from hand-crafted features, and more recently with Deep Learning by embedding this knowledge into a Graph Neural Network (GNN). In this paper, we build upon the latter category by proposing a new heuristic based on Deep Reinforcement Learning using two GNNs with adversarial rewards. We show that our method reduces the number of fails to get the first solution by more than 50% compared to MiniSat. This work shows the advantages of this type of techniques to extract structural and contextual knowledge from past solving experience.

Multi-valued Decision Diagrams (MDDs) have been extensively studied in the last ten years. Recently, efficient algorithms implementing operators such as reduction, union, intersection, difference, etc., have been designed. They directly deal with the graph structure of the MDD and a time reduction of several orders of magnitude in comparison to other existing algorithms have been observed. These operators have permitted a new look at MDDs, because extremely large MDDs can finally be manipulated as shown by the models used to solve complex application in music generation. However, MDDs become so large (50GB) that minutes are sometimes required to perform some operations. In order to accelerate the manipulation of MDDs, parallel algorithms are required. In this paper, we introduce such algorithms. We carefully design them in order to overcome inherent difficulties of the parallelization of sequential algorithms such as data dependencies, software lock-out, false sharing, or load balancing. As a result, we observe a speed-up , i.e. ratio between parallel and sequential runtimes, growing linearly with the number of cores.

Recent developments of efficient propagators, operations and creation methods for MDDs allow us to directly build efficient MDD-based models, without the need for intermediate data structures. In this paper, we take another step in this direction by improving the propagators of cost MDDs. In addition, we introduce a soft MDD propagator in order to deal with unsatisfiable problems. This directly offers cost and soft versions for table constraints and any constraints which can be represented by an MDD (regular, slide, knapsack...).