An augmented Lagrangian multiplier method based on a CHKS smoothing function for solving nonlinear bilevel programming problems

Bilevel programming techniques deal with decision processes involving two decision makers with a hierarchical structure. In this paper, an augmented Lagrangian multiplier method is proposed to solve nonlinear bilevel programming (NBLP) problems. An NBLP problem is first transformed into a single level problem with complementary constraints by replacing the lower level problem with its Karush–Kuhn–Tucker optimality condition, which is sequentially smoothed by a Chen–Harker–Kanzow–Smale (CHKS) smoothing function. An augmented Lagrangian multiplier method is then applied to solve the smoothed nonlinear program to obtain an approximate optimal solution of the NBLP problem. The asymptotic properties of the augmented Lagrangian multiplier method are analyzed and the condition for solution optimality is derived. Numerical results showing viability of the approach are reported.