New Necessary Optimality Conditions for Bilevel Programs by Combining the MPEC and Value Function Approaches

Abstract

The bilevel program is a sequence of two optimization problems where the constraint region of the upper level problem is determined implicitly by the solution set to the lower level problem. The classical approach to solving such a problem is to replace the lower level problem by its Karush-Kuhn-Tucker (KKT) condition and solve the resulting mathematical programming problem with equilibrium constraints (MPEC). In general the classical approach is not valid for nonconvex bilevel programming problems. The value function approach uses the value function of the lower level problem to define an equivalent single level problem. But the resulting problem requires a strong assumption, such as the partial calmness condition, for the KKT condition to hold. In this paper we combine the classical and the value function approaches to derive new necessary optimality conditions under rather weak conditions. The required conditions are even weaker in the case where the classical approach or the value function approach alone is applicable.

Solving Ill-posed Bilevel Programs

“…Note that special cases of this result on (P ) are derived in [10,21]. For further clarity on the fact that the result in Theorem 2.3 will effectively lead to standard results in the literature on KKT and value function reformulations of (P o ) and (P ), the interested reader is referred to [16][17][18] and to [45,47,48] for other approaches to derive optimality conditions for (P ). Finally, considering (P i ) where the lower-level solution function y(.)…”

Smoothing augmented Lagrangian method for nonsmooth constrained optimization problems

“…For a general bilevel program, the lower level problem may have equality and/or inequality constraints and the first order condition is the KKT condition. If the lower level has inequality constraints, then the resulting combined program is a nonsmooth mathematical program with complementarity constraints and necessary conditions of Clarke, Mordukhovich and Strong (C, M, S) type have been studied in Ye and Zhu [55] and Ye [52]. For the simple bilevel program we consider in this paper the constraint of the lower level problem is a fixed set Y independent of x.…”

“…However, Ye and Zhu [55] illustrated that the partial calmness condition is still too strong to hold for many bilevel problems (for example, the Mirrlees' problem) and proposed a new first order necessary optimality condition by considering the combined program with both the first order condition of the lower level problem and the value function constraint. For a general bilevel program, the lower level problem may have equality and/or inequality constraints and the first order condition is the KKT condition.…”

Variational Analysis of Marginal Functions with Applications to Bilevel Programming

“…An interesting relaxation of partial calmness was developed in [13] with applications to a discretized obstacle control problem governed by partial differential equations of the elliptic type. Paper [28] proposed another approach to optimistic bilevel programs combining the aforementioned MPEC and value function ones and taking advantages of both for problems with smooth data. We also mention related developments in [11] for semi-infinite and infinite bilevel programs with DC (difference of convex) data and in [1,26] for multiobjective bilevel programs.…”

“…The major goal of this paper is to derive new necessary optimality conditions for a class of bilevel programs the importance of which has been well recognized in optimization theory and applications; see, e.g., the book by Dempe [6], the more recent publications [7,8,10,26,28], and the bibliographies therein among other references that mostly concern bilevel programming in finite dimensions. Here we consider bilevel programs in infinite dimensions while all the results obtained are new in the standard framework of finite-dimensional spaces.…”