A unified approach for smoothing approximations to the exact $$\ell _1$$ ℓ 1 -penalty for inequality-constrained optimization

In penalty methods for inequality-constrained optimization problems, the nondifferentiability of the exact $$\ell _1$$ ℓ 1 -penalty function limits the use of efficient smooth algorithms for solving the subproblems. In light of this, a great number of smoothing techniques has been proposed in the literature. In this paper we present, in a unified manner, results and methods based on functions that smooth and approximate the exact penalty function. We show that these functions define a class of algorithms that converges to global and local minimizers. This unified approach allows us to derive sufficient conditions that guarantee the existence of local minimizers for the subproblems and to establish a linear convergence rate for this class of methods, using an error bound-type condition. Finally, numerical experiments with problems of the CUTEst collection are presented to illustrate the computational performance of some methods from the literature which can be recovered as particular cases of our unified approach.