Optimality Conditions for Bilevel Programming: An Approach Through Variational Analysis

Abstract

In this article, we focus on the study of bilevel programming problems where the feasible set in the lower-level problem is an abstract closed convex set and not described by any equalities and inequalities. In such a situation, we can view them as MPEC problems and develop necessary optimality conditions. We also relate various solution concepts in bilevel programming and establish some new connections. We study in considerable detail the notion of partial calmness and its application to derive necessary optimality conditions and also give some illustrative examples.

Appendix

We are now going to demonstrate how to obtain the estimation of $\partial \varphi ( \bar{x})$ as given in (3.5). We shall begin with a general setting of a parametric optimization problem given as

$$\begin{aligned} \min _{y} f ( x, y ) \quad \text{ subject } \text{ to }\quad y \in K (x). \end{aligned}$$

where f(x, y) is say an extended-valued proper lower-semicontinuous map function and further K is set-valued map with closed convex and non-empty values for each x. Further, assume that S is an inner-semicompact map. Our question is that can we estimate $\partial \varphi (\bar{x})$ where $\varphi$ is given as

$$\begin{aligned} \varphi ( x) = \inf _y \{ f(x, y) : y \in K(x) \}. \end{aligned}$$

In order to do so, we need to consider the notion of the asymptotic subdifferential or singular subdifferential of a lower-semicontinuous function. Note that if h is a proper lower-semicontinuous function then we define the asymptotic subdifferential at $\bar{x}$ as

$$\begin{aligned} \partial ^\infty _L h ( \bar{x} ) = \{ v \in \mathbb {R}^n : ( v, 0 ) \in N_{epi f} ( \bar{x}, h ( \bar{x}) ) \}. \end{aligned}$$

In the context of the given parametric optimization problem in this section, let us assume that the following qualification condition holds at $( \bar{x}, \bar{y}) \in gph S$. The condition says that

$$\begin{aligned} \partial ^\infty _L f ( \bar{x}, \bar{y}) \cap ( -N_{gph K} ( \bar{x}, \bar{y}) )= \{ (0,0) \}. \end{aligned}$$

(3.6)

If this qualification condition holds, then Mordukhovich [17] shows us that

$$\begin{aligned} \partial _L \varphi ( \bar{x}) \subseteq \bigcup _{ \bar{y} \in S ( \bar{x})} \left\{ x^* + D^* K( \bar{x} | \bar{y})(y^*) : ( x^*, y^*) \in \partial _L f ( \bar{x}, \bar{y}) \right\} . \end{aligned}$$

(3.7)

Note that since we consider f to be finite-valued and jointly convex we conclude that f is locally Lipschitz and hence we know from Mordukhovich that

$$\begin{aligned} \partial ^\infty _L f ( \bar{x}, \bar{y}) = \{ (0,0) \}. \end{aligned}$$

Hence, the qualification condition given in (3.6) is automatically holding. Further note that as $K (x) = K$ we have for any $y^* \in \mathbb {R}^m$.

$$\begin{aligned} D^*K( \bar{x} | \bar{y}) ( y^*) = \{ 0 \}. \end{aligned}$$

This shows that in our setting of a fully convex problem with a compact and convex K we have

$$\begin{aligned} \partial \varphi (\bar{x} ) \subseteq \bigcup _{ \bar{y} \in S ( \bar{x})} \{ x^* \in \mathbb {R}^n : x^* \in \partial _x f ( \bar{x}, \bar{y}) \}. \end{aligned}$$

Exercises

• 1. Write down the proof of Theorem 3.1 in detail.

• 2. Let us now assume that in the problem (OBP) the feasible set of the lower-level problem is independent of x, i.e. $K(x) = K$. Further, let us assume that $K = \{ x \in \mathbb {R}^n : g_i(x) \le 0, i = 1, \ldots p \}$, where each $g_i$ is a convex function. With this additional information how can you write down the problem (OBP) in terms of the optimality conditions of the lower-level problem. Note that the constraints functions $g_i$, $i = 1, \ldots , p$ must now appear in the model. Assuming that each local minimum of the problem (OBP) is also a local minimum of the problem where the lower-level is represented in terms of optimality conditions we urge the reader to now write down the optimality condition for the bilevel problem. Choose appropriate assumptions.

• 3. This is possibly an open question still to best of our knowledge. Can we replace the notion of partial calmness by some more tractable condition. One can of course see [12] but again calmness conditions are not so easy to check. It would be indeed important to have a better alternative to partial calmness.

• 4. Write down the proof of Theorem 3.2 in detail.