Sensitivity Analysis for Two-Level Value Functions with Applications to Bilevel Programming

Abstract

Abstract. This paper contributes to a deeper understanding of the link between a now conventional framework in hierarchical optimization spread under the name of the optimistic bilevel problem and its initial more difficult formulation that we call here the original optimistic bilevel optimization problem. It follows from this research that, although the process of deriving necessary optimality conditions for the latter problem is more involved, the conditions themselves do notto a large extent-differ from those known for the conventional problem. It has been already well recognized in the literature that for optimality conditions of the usual optimistic bilevel program appropriate coderivative constructions for the set-valued solution map of the lower-level problem could be used, while it is shown in this paper that for the original optimistic formulation we have to go a step further to require and justify a certain Lipschitz-like property of this map. This occurs to be related to the local Lipschitz continuity of the optimal value function of an optimization problem constrained by solutions to another optimization problem; this function is labeled here as the twolevel value function. More generally, we conduct a detailed sensitivity analysis for value functions of mathematical programs with extended complementarity constraints. The results obtained in this vein are applied to the two-level value function and then to the original optimistic formulation of the bilevel optimization problem, for which we derive verifiable stationarity conditions of various types entirely in terms of the initial data.

Solving Ill-posed Bilevel Programs

“…At first, as already mentioned above, a local optimal solution of (P o ) generates a local optimal solution of (P ), but the converse in not true, cf. [13]. In the latter paper, it is also shown that the optimality conditions of (P o ) are equivalent to those of (P ) under a mild assumption.…”

“…has to be inner semicontinuous [13]. This assumption is strong and fails for very simple examples, as shown in the latter reference.…”

“…This picture highlights the typical behavior of the corresponding graphs for ill-posed bilevel programs. Going from a smooth function F in (P ), the best we can usually get for ϕ o and ϕ p is lower-semicontinuity and upper-semicontinuity, respectively, at points where S is multivalued [13]. As for F, it is set-valued at such points.…”

“…Using a KKT reformulation for S (1.4), a combination of solvers is used in [4] to construct a procedure to compute global optimal solutions for a special case of (P p ). For the first time, optimality conditions for problems (P o ) and (P p ) were derived in [13] and [14], respectively. Though a small number of publications have recently focused on the computation of stationary/local optimal solutions (without imposing lower-level uniqueness) of (P ), see [11,27,44], we are not aware of any method that locally solves (P o ).…”

Variational Analysis of Marginal Functions with Applications to Bilevel Programming

“…A detailed study of the original optimistic bilevel program (1.4) and the corresponding two-level value function is given [9]. The challenging issue in the value function approach is the nonsmoothness of problem (1.8) due to the intrinsic nondifferentiability of the value function (1.7), even when all the original data are smooth.…”