  This work describes PUSH, a primal heuristic combining Feasibility Pump and
Shifting. The main idea is to replace the rounding phase of the Feasibility
Pump with a suitable adaptation of the Shifting and other rounding heuristics.
The algorithm presents different strategies, depending on the nature of the
partial rounding obtained. In particular, we distinguish when the partial
solution is feasible, infeasible with potential candidates, and infeasible
without candidates. We used a threshold to indicate the percentage of variables
to round with our algorithm and which other to round to the nearest integer.
Most importantly, our algorithm tackles directly equality constraints without
duplicating rows. We select the parameters of our algorithm on the 19 instances
provided for the Mip Competition 2022. Finally, we compared our approach to
other start heuristics, like Simple Rounding, Rounding, Shifting, and
Feasibility Pump on the first 800 MIPLIB2017 instances ordered by the number of
non-zeros.
