A Novel l1 Exact Penalty Function Method and Its Application
TL;DR
A continuously differentiable smoothing approximation that preserves the exactness of the original penalty while enabling efficient numerical solution and faster convergence and improved objective values compared with the traditional non-smooth penalty method is proposed.
Abstract
Penalty function methods are fundamental for solving constrained optimization problems, yet the widely used l1 exact penalty function is non-differentiable and therefore incompatible with gradient-based algorithms. This paper proposes a continuously differentiable smoothing approximation that preserves the exactness of the original penalty while enabling efficient numerical solution. The proposed smoothing function is shown to be C1 on R, with explicit error bounds and a computable lower bound for the penalty parameter that guarantees exactness under standard constraint qualifications. An iterative algorithm is developed and its convergence is proved. Numerical experiments on benchmark problems validate the effectiveness of the approach. The method is then applied to a policy-driven university financial risk management model with twelve decision variables, three conflicting objectives, and multiple regulatory constraints. Results demonstrate faster convergence and improved objective values compared with the traditional non-smooth penalty method, confirming the practical utility of the proposed smoothing technique.