Lifted stationary points of sparse optimization with complementarity constraints

We aim to compute lifted stationary points of a sparse optimization problem ( $$P_{0}$$ P 0 ) with complementarity constraints. We define a continuous relaxation problem ( $$R_{\nu }$$ R ν ) that has the same global minimizers and optimal value with problem ( $$P_{0}$$ P 0 ). Problem ( $$R_{\nu }$$ R ν ) is a mathematical program with complementarity constraints (MPCC) and a difference-of-convex objective function. We define MPCC lifted-stationarity of ( $$R_{\nu }$$ R ν ) and show that it is weaker than directional stationarity, but stronger than Clarke stationarity for local optimality. Moreover, we propose an approximation method to solve ( $$R_{\nu }$$ R ν ) and an augmented Lagrangian method to solve its subproblem ( $$R_{\nu ,\sigma }$$ R ν , σ ), which relaxes the equality constraint in ( $$R_{\nu }$$ R ν ) with a tolerance $$\sigma $$ σ . We prove the convergence of our algorithm to an MPCC lifted-stationary point of problem ( $$R_{\nu }$$ R ν ) and use a sparse optimization problem with vertical linear complementarity constraints to demonstrate the efficiency of our algorithm on finding sparse solutions in practice.