MINLP Problems and Explanation-based Constraint Programming Guillaume Rochart 1,2 , Eric Monfroy 1 , and Narendra Jussien 2 1 LINA, FRE CNRS 2729 2, rue de la Houssini` ere – B.P. 92208 – F-44322 Nantes Cedex 3 2 D´ epartement Informatique, ´ Ecole des Mines de Nantes 4, rue Alfred Kastler – B.P. 20722 – F-44307 Nantes Cedex 3 Abstract. Numerous industrial problems can be modelled as MINLP problems combining both numeric and integer variables. Several meth- ods were proposed to solve these problems. But industrial applications need more than solving problems: dynamic problems, over-constrained problems, or explaining solver behaviour are features required by in- dustrial applications. Explanation-based constraint programming offers such tools. In this paper, we show how to apply explanation-based mech- anisms for mixed problems thanks to a generic framework. Last, some first experimental results are exposed: the overhead due to explanation managing is acceptable and can even speed up some resolutions. Introduction Numerous industrial problems can be modelled as MINLP (Mixed Integer Non- Linear Programming ) problems combining both numeric and integer variables: design of water or gas networks, automobile, aircraft, etc. [4]. These problems are really hard to solve: they combine the combinatorial nature of mixed integer programming and the intrinsic difficulty of nonlinear programs. Several methods were proposed to solve such problems [4]: branch-and-bound, extended cutting plane methods, generalised Bender’s decomposition, etc. But industrial applications need more than solving problems. Problems can be dynamic, this implies that constraints may be added or removed dynamically. Moreover, if no solution is found, the user often needs to know why the problem is over-constrained, or why the expected solution is inconsistent. Constraint programming offers generic models and tools to solve combinato- rial problems. Furthermore, explanation-based constraint programming provides tools to solve dynamically such problems and maintain explanations about the resolution: why a problem has no solution, why the optimum bound is reached, or how to improve a solution are informations that explanation-based constraint programming can provide. Such features are now well known for constraint pro- gramming over integer variables [6]. However only few works proposed solutions to extend it to real variables. [8] proposed to extend mac-dbt to solve numeric problems thanks to a dynamic domain splitting mechanism. But these works solve separately integer problems