AbstractThis note presents a brute force approach tolinearly constrained programmingin non-convex optimization; our aim here is to illustrate a general methodology which can be applied to construct tailor-made algorithms in specific applications.In essence, the facial decomposition method constructs anon-redundant listof all faces of the polyhedral setP⊂Rn. Each face is characterized by a linear program in a given affine subspace ofRn. This list is conveniently displayed in a tree structure which represents the set of nodes to be searched (typically for optimality).