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

Abstract

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.

Introduction

A bilevel programming (BLP) problem is characterized by a nested optimization problem with two levels, namely, an upper and a lower level, in a hierarchy where the constraint region of the upper level problem is implicitly determined by the lower level optimization problem. Such problems arise in game-theoretic and decomposition related applications in engineering design. Recently, these problems have received increasing attention and have been applied in various aspects of resource planning, management, and policy making, including water resource management, and financial, land-use, production, and transportation planning [1].

After a real-world problem has been modeled as a bilevel decision problem, how to find the optimal solution for this decision problem becomes very important. However, the nested optimization model described above is quite difficult to solve. It has been proved that solving a linear bilevel program is NP-hard [2], [3]. In fact, even searching for a locally optimal solution of a linear bilevel program is strongly NP-hard [4]. Herminia and Carmen [5] pointed out that the existence of multiple optima for the lower-level problem can result in an inadequate formulation of the bilevel problem. Moreover, the feasible region of the problem is usually non-convex, without necessarily being a connected set, and empty, making it difficult to solve BLP problems.

Existing methods for solving BLP problems can be classified into the following categories [6]: extreme-point search, transformation, descent and heuristic, interior point, and intelligent computation approaches. Recently, the latter approach has become an important computational method for solving BLP problems. Genetic algorithm [7], [8], [9], Neural networks [10], Tabu search [11], [12], [13], ant colony optimization [14], and particle swarm optimization [15], [16], [17] are typical intelligent algorithms for solving BLP problems. In all these algorithms, the BLP problem is first reduced to a one-level problem with complementary constraints by replacing the lower level problem with its Karush–Kuhn–Tucker (KKT) optimality condition. However, the one-level complementary slackness problem is non-convex and non-differential. The major difficulty in solving this type of problem is that its constraints fail to satisfy the standard Mangasarian–Fromovitz constraint qualification at any feasible point, and that general intelligent algorithms may fail to uncover such local minima or even mere stationary points.

To avoid the difficulty of the nonsmooth constraints, a sequence of smooth problems by using smoothing functions have been progressively approximated this nonsmooth problem. To date, there have been many smoothing functions, such as perturbed Fischer–Burmeister (FB), Chen–Harker–Kanzow–Smale (CHKS), Neural networks, uniform, and Picard smoothing functions [18], [19], [20]. Thereinto, perturbed FB and CHKS smoothing functions have been applied widely to solve BLP problems. Facchinei et al. [21] transformed mathematical program with equilibrium constraints (MPEC), including bilevel programming problems as a particular case, into an equivalent one-level nonsmooth optimization problem, which was then approximated progressively by applying CHKS smoothing function to a sequence of smooth, regular problems that approximate the nonsmooth problem. Finally, it was solved by standard available software for constrained optimization. Dempe [22] introduced a smoothing technique by applying perturbed FB function and Lagrangian function method to solve a BLP problem, approximated by a sequence of smooth optimization problems. Wang et al. [23] proposed an approximate programming method based on the simplex method to research the general bilevel nonlinear programming model by applying perturbed FB smoothing function. Etoa [24] presented a smoothing sequential quadratic programming using perturbed FB functional to compute a solution of a quadratic convex bilevel programming problem. In general, perturbed FB and CHSK smoothing functions are very closely related. These two functions are very similar except for the difference of a constant. In what follows, we choose to focus on the CHKS smoothing function to approximate nonsmooth problem.

The purpose of this work is to design an efficient algorithm for solving general nonlinear bilevel programming (NBLP) models. Different from the algorithms in [22], [23], where Lagrangian optimization and simplex method were used to solve a smooth problem approximated by perturbed FB smoothing function, we apply a CHKS smoothing function and an augmented Lagrangian multiplier method to solve NBLP problem, and focus on the detailed analysis on the condition for solution optimality. The remainder of this paper is organized as follows: Section 2 presents the formulation and basic definitions of bilevel programming, while Section 3 introduces the smoothing method for nonlinear complementarity problems. In Section 4 we explain the augmented Lagrangian multiplier method in detail and derive the condition for asymptotic stability, solution feasibility, and solution optimality. Numerical examples are reported in Section 5 and Section 6 concludes the paper.