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We introduce Scylla, a primal heuristic for mixed-integer optimizationproblems. It exploits approximate solves of the Linear Programming relaxationsthrough the matrix-free Primal-Dual Hybrid Gradient algorithm with specializedtermination criteria, and derives integer-feasible solutions viafix-and-propagate procedures and feasibility-pump-like updates to the objectivefunction. Computational experiments show that the method is particularly suitedto instances with hard linear relaxations.

Scylla: a matrix-free fix-propagate-and-project heuristic for mixed-integer optimization