Can dual values smoothing lead to generating duplicate columns?

Question

When using smoothing for dual values stabilization in column generation, the duals used in the subproblem lie on the segment joining the stability center (inside the dual feasible region) and the current dual values yielded by the last master problem solution (outside the dual feasible region unless the problem is solved to optimality).

Does this imply that it is possible to generate a column which is already in the column pool? I am thinking that, with respect to the duals used in the subproblem, an existing column can still have strictly negative reduced cost. Therefore, the subproblem could generate it again. Is this correct?

I am referring to the smoothing method proposed, e.g., in this paper:

A. Pessoa, R. Sadykov, E. Uchoa, F. Vanderbeck (2018) Automation and Combination of Linear-Programming Based Stabilization Techniques in Column Generation. INFORMS Journal on Computing 30(2):339-360. https://doi.org/10.1287/ijoc.2017.0784

Answer 1

You solve the pricing problem using the smoothed dual values, but then you analyse the reduced cost of obtained columns using original dual values (i.e., those coming from the master LP). If the column is already in the pool, it means that the original reduced cost is non-negative, and one does not usually add this column to the master LP. If all obtained columns have non-negative original reduced cost, then the so-called misprice happens, and one should solve the pricing problem again with different dual values (either original dual values or dual values smoothed with a different coefficient).