Mixed integer lifted cover inequalities for knapsack problems with a single continuous variable

We present a separation procedure for a class of valid inequalities (mixed lifted cover inequalities) for the mixed binary knapsack set with a single continuous variable. It consists of the following steps: (i) the continuous variable is fixed to a given value, (ii) a valid inequality for the corresponding binary knapsack set is computed, (iii) the inequality is lifted using an iterative procedure to become valid for the mixed integer knapsack set. Computational experience on a wide set of mixed integer programming instances from MIPLIB and Mittleman sets is carried out, showing that our cutting plane algorithm leads to significant improvement in lower bounds, with negligible additional computational effort.