Cardinality objective nonlinear programs for facility capacity expansion

In this paper, we consider a class of mathematical programs with inequality and equality constraints where the objective involves a cardinality penalty. The introduction of cardinality penalty can make the model automatically generate sparse solutions, but solving the cardinality objective nonlinear program is highly challenging since the objective function is discontinuous. We first give a continuous approximation and discuss its relationship with the original problem. Second, we propose a proximal augmented Lagrangian method for finding a weak directional(d)-stationary point of the continuous approximation. The proposed algorithm is a novel combination of the classical augmented Lagrangian method and proximal gradient algorithm. We prove that the proposed method globally converges to a weak d-stationary point of the continuous approximation, which is stronger than Clarke stationary point. Third, we demonstrate that the cardinality objective nonlinear program is a better model for the facility capacity expansion problem, which can generate key capacity expansion locations to avoid the high operating costs caused by expanding a large number of facilities. Finally, a systematic computational study on two capacity expansion problems is presented. The numerical results demonstrate the benefit of the cardinality objective nonlinear program and the effectiveness of the proposed algorithm.