A single-level reformulation of mixed integer bilevel programming problems

Abstract

This paper focuses on the single-level reformulation of mixed integer bilevel programming problems (MIBLPP). Due to the existence of lower-level integer variables, the popular approaches in the literature such as the first-order approach are not applicable to the MIBLPP. In this paper, we reformulate the MIBLPP as a mixed integer mathematical program with complementarity constraints (MIMPCC) by separating the lower-level continuous and integer variables. In particular, we show that global and local minimizers of the MIBLPP correspond to those of the MIMPCC respectively under suitable conditions.

Introduction

In this paper, we consider the mixed integer bilevel programming problem (MIBLPP) of the form

$$\begin{array}{cc}\underset{x,y,z}{min}\hfill & F(x,y,z)\hfill \\ \text{s.t.}\hfill & x\in X(y,z)\text{,}\hfill \\ \hfill & (y,z)\in S\left(x\right)\text{,}\hfill \end{array}$$

where some of the upper-level decision variables $x$ are integer-valued, $F:{\Re }^{n_1+n_2+n_3}\to \Re$, $X(y,z)$ is a nonempty closed subset in ${\Re }^{n_1}$ for any $(y,z)\in {\Re }^{n_2+n_3}$, and $S\left(x\right)$ is the set of global minimizers for the lower-level problem parameterized by $x$

$$\begin{array}{cc}\hfill \underset{y,z}{min}& f(x,y,z)\hfill \\ \hfill \text{s.t.}& (y,z)\in \Xi \left(x\right)\text{,}\hfill \end{array}$$

where $y$ represents the lower-level continuous variables, $z$ represents the lower-level integer variables, $f:{\Re }^{n_1+n_2+n_3}\to \Re$ is a continuously differentiable function with respect to $y$, and $\Xi \left(x\right)$ is defined as

$$\Xi \left(x\right)≔\{(y,z)\in {\Re }^{n_2}\times Z:g(x,y,z)\le 0,h(x,y,z)=0\}$$

with a set of integers $Z$ in ${\Re }^{n_3}$ and continuously differentiable functions $g:{\Re }^{n_1+n_2+n_3}\to {\Re }^p,h:{\Re }^{n_1+n_2+n_3}\to {\Re }^q$ with respect to $y$. Throughout this paper, we assume that the cardinality $\left|Z\right|$ of $Z$ is finite and $n_2>0$.

Bilevel programming problems have been intensively investigated; see, e.g., the review articles  [6], [7]. Almost all investigations in the literature are devoted to problems with only continuous variables in the lower-level problem for which the popular way is to reformulate it as a single-level problem, although there exist many practical problems with integer variables in the lower-level problem; see, e.g.,  [2], [4], [9], [18]. There have also been some progresses made on the numerical methods for solving linear MIBLPPs; see, e.g,  [2], [3], [17], [20], [23], [24].

It is desirable to know whether there exists a minimizer before solving the MIBLPP. Vicente et al.  [22] studied the existence of minimizers of linear MIBLPP for different cases corresponding to particularizations of the upper-level and lower-level variables. Unfortunately, for the case where there exist lower-level integer variables and upper-level joint constraints, it is a difficult task to give the existence of minimizers. In this paper, we assume that a minimizer exists for the MIBLPP and focus on deriving its single-level reformulation.

To the best of our knowledge, there are only a few publications on developing numerical methods for solving MIBLPPs with nonlinear functions  [10], [12], [16]. Our proposed single-level reformulation in this paper makes it possible to develop more approaches for solving MIBLPPs as what has been done for bilevel programming problems with only continuous variables in the lower-level problem.

When there is no lower-level integer variable ($n_3=0$), we may replace the lower-level problem of the MIBLPP with its Karush–Kuhn–Tucker (KKT) conditions, resulting in a mathematical program with complementarity constraints (MPCC). Their relations in the sense of global and local minimizers have been established in  [8] provided that the lower-level problem is convex and Slater’s constraint qualification is satisfied. When there are lower-level integer variables ($n_3>0$), the lower-level problem is clearly nonconvex and thus we cannot replace the lower-level problem with its KKT conditions directly. In this paper, we reformulate the MIBLPP as a mixed integer mathematical program with complementarity constraints (MIMPCC) by separating the lower-level continuous and integer variables. In order to investigate the relations between the MIBLPP and the MIMPCC, we assume that for any lower-level integer variables, the lower-level problem is solvable and convex with respect to the continuous variables. We show that the global minimizers of the MIBLPP correspond to those of the MIMPCC provided that for any lower-level integer variables, the lower-level problem satisfies the generalized Slater’s constraint qualification with respect to the continuous variables. Since bilevel programming is generally nonconvex, it is difficult to find a global minimizer and in some cases, one needs to be happy with obtaining a local minimizer. Thus, it is also necessary to investigate the relations between the local minimizers of the MIBLPP and the MIMPCC. The stronger Slater’s constraint qualification and the so-called restricted inf-compactness condition are required to ensure that the local minimizers of the MIBLPP correspond to those of the MIMPCC.

The rest of this paper is organized as follows. In Section  2 we give some background materials and preliminary results. In Section  3 we reformulate the MIBLPP as a single-level MIMPCC. In particular, we show that the global and local minimizers of the MIBLPP correspond to those of the MIMPCC respectively under some suitable conditions. Section  4 concludes the paper.

Notation: For any vectors $a,b\in {\Re }^n$, $a\perp b$ means that vector $a$ and vector $b$ are perpendicular. Given a point $x\in {\Re }^n$ and a function $\phi :{\Re }^n\to {\Re }^m$, $\nabla \phi \left(x\right)$ denotes the transposed Jacobian of $\phi$ at $x$.