In many mathematical optimization applications dual variables are an important output of the solving process, due to their role as price signals. When dual solutions are not unique, different solvers or different computers, even different runs in the same computer if the problem is stochastic, often end up with different optimal multi-pliers that also depend on the discretization of the data.
From the perspective of a decision maker, this variability makes the price signals less reliable and, hence, less useful. We address this issue for a particular family of linear and quadratic programs by proposing a solution procedure that, among all possible optimal multipliers, systematically yields the one with the smallest norm.
The approach, based on penalization techniques of nonlinear programming, amounts to a regularization in the dual of the original problem. As the penalty parameter tends to zero, convergence of the primal sequence and, more critically, of the dual is shown under natural assumptions. The methodology is illustrated on a battery of two-stage stochastic linear programs.
The variance of the Lagrange Multiplier regarding different discretizations is investigated in theoretical and numerical aspects We show the positive impact of the regularization in the price distribution of the Northern Europe hydro-generation system. This real-life example, set in a two stage perspective, helps us to better understand price signals in regularized and non-regularized settings.