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The strengthening of linear relaxations and bounds of mixed integer linearprograms has been an active research topic for decades. Enumeration-basedmethods for integer programming like linear programming-based branch-and-boundexploit strong dual bounds to fathom unpromising regions of the feasible space.In this paper, we consider the strengthening of linear programs via a compositeof Dantzig-Wolfe and Fenchel decompositions. We provide geometricinterpretations of these two classical methods. Motivated by these geometricinterpretations, we introduce a novel approach for solving Fenchel sub-problemsand introduce a novel decomposition combining Dantzig-Wolfe and Fencheldecompositions in an original manner. We carry out an extensive computationalcampaign assessing the performance of the novel decomposition on theunsplittable flow problem. Very promising results are obtained when the newapproach is compared to classical decomposition methods.

On the integration of Dantzig-Wolfe and Fenchel decompositions via directional normalizations