Necessary optimality conditions for optimistic bilevel programming problems using set-valued programming

On the solution of convex bilevel optimization problems

“…Dempe et al [11]), it is closely related to (generalized) semi-infinite programming ((G)SIP) (see Günzel et al. [12], Stein [13], Stein and Still [14], Weber [15]), mathematical programs with equilibrium (complementarity) constraints (Outrata et al [16]), nonsmooth (Dempe et al [3]), set-valued (Dempe and Pilecka [17]) and (mixed-)integer optimization (Audet et al [18]); ideas of semidefinite optimization can be used to solve it (Jeyakumar et al [19]). …”

Optimality conditions for MPECs in terms of directional upper convexifactors

“…For a locally Lipschitz function, most known subdi¤erentials are convexi…cators and these subdi¤erentials may contain the convex hull of a convexi…cator [16]. Noting that convexi…cators admitted by discontinuous functions may be unbounded and because the boundedness of convexi…cators is of crucial importance in many wellknown results, Dempe and Pilecka [3] developed the concept of directional convexi…cators. They were able to create a convexi…cator for a given lower semicontinuous function using directional convexi…cators, presuming convexity and closedness of the set of continuity directions (see [3, Corollary 2 and Proposition 1]).…”

“…Motivated by the above work of Dempe and Pilecka [3], we investigate necessary and su¢ cient optimality conditions for (MPEC) where data functions are not necessarily smooth/locally Lipschitz/convex/continuous.…”

“…To achieve our goal, we introduce an alternative stationarity concept and a generalized Abadie-type regularity condition using directional upper (semi-regular) convexi…cator; and, assuming the feasible set is locally star-shaped, we show that alternative stationarity is in fact a …rst-order necessary optimality condition for MPECs. Unlike Dempe and Pilecka [3] and Gadhi et al [14], we do not assume that the sets of all continuity directions are convex or compact. Under some directional generalized convexities, we establish su¢ cient optimality conditions for (M P EC) : Notice that directional upper semi-regular convexi…cators are not necessarily upper semi-regular convexi…cators; moreover, they may not even be directional upper regular convexi…cators (see Example 11).…”

An Investigation of the Optimistic Solution to the Linear Trilevel Programming Problem

“…In bilevel programming problems, optimistic and pessimistic approaches are proposed to resolve such ambiguities. In the optimistic approach [16], the decision maker of the lower level (follower) is motivated to choose an optimal solution that has the best result for the upper-level decision maker (leader) among one's own alternative optimal solutions. In contrast, in the pessimistic approach [17], the second-level decision maker is required to choose an optimal solution that has the worst result for the first level decision maker.…”