## 1 Introduction  
Hybridizing heuristics with mathematical programming and Machine Learning (ML) techniques is a prominent research field to design optimization algorithms, these techniques having complementary advantages and drawbacks [1]. Matheuristics, hybridizing mathematical programming and heuristics, use the modeling facilities of mathematical approaches for industrial optimization problems, possibly highly-constrained ones which are usually difficult for meta-heuristics, with a good scalability to face large size instances, which is a characteristic of meta-heuristics and the bottleneck for exponential methods like Branch&Bound (B&B) tree search [2].  
Usually, hybrid algorithms compute primal solutions of optimization problems as in [3], rare works use such hybridization to improve dual bounds. Using dual heuristics for Mixed Integer Linear Programming (MILP) is mentioned in [4]: \"In linear programming based branch-and-bound algorithms, many heuristics have been developed to improve primal solutions, but on the dual side we rely solely on cutting planes to improve dual bounds\". Until the 2000s, many efforts on improving the quality of dual bounds relied on improving cut generation to speed up B&B approaches [5] or using Lagrangian bounds like the Danzig-Wolfe reformulation [6]. In the 2000s, primal heuristics inside B&B search improved exact MILP solving, having earlier good primal solutions in the B&B search allows for pruning earlier nodes and to converge faster [7]. However, heuristics might be considered as the \"dark side\" of MILP solving [7].  
In practice, heuristics and matheuristics are useful to design heuristic separation or pricing in Branch &Cut (B&C) or Branch& Price (B&P) methods: finding quickly and heuristically a violated constraint or a negative reduced cost column speeds up exact algorithms [8]. In such cases, heuristics are useful for a quicker computation of the bounds given by exact approaches. Dual heuristics may furnish quicker dual bounds with an additional gap to the optimal values or mathematical relaxations. Surrogate relaxations guide the search of primal solution with dual heuristics [9; 10]. Aggregation and disaggregation techniques provide also dual and primal bounds, used in network flow models [11] or scheduling problems [12]. We note that surrogate optimization and aggregation techniques have a recent interest in nonlinear optimization [13].  
This paper investigates the interest of hybridizing ML and matheuristics to provide dual heuristics for a highly constrained optimization problems from a real life industrial problem. The case study is the EURO/ROADEF Challenge 2010, to optimize the refueling and maintenance planning of nuclear power plants [14]. The best primal solutions were mainly obtained with a local search approach [15]. It was an open question after the Challenge to have dual bounds for this large scale problem. The first dual bounds used optimal computations of very relaxed problems [16]. Our preliminary works improved these dual bounds with aggregation and reduction techniques to have tractable MILP computations of dual bounds [17]. New dual bounds are provided in this paper, illustrating the interest of hybridizing exact methods, heuristics, and ML techniques.  
This paper is organized as follows. Section 2 gives an overview of the problem constraints. Section 3 presents the state-of-the art of the solving approaches. Section 4 presents MILP relaxations and pre-processing reduction to compute dual bounds with smaller MILP. Section 5 analyzes how dual bounds are provided using different scenario decomposition and how ML techniques guide the choice of appropriate dual bounds. The computational results are analyzed in Section 6. Conclusions and perspectives are drawn in Section 7.