Bilevel programs (BL) form a special class of optimization problems. They appear in many models in economics, game theory and mathematical physics. BL programs show a more complicated structure than standard finite problems. We study the so-called KKT-approach for solving bilevel problems, where the lower level minimality condition is replaced by the KKT-or the FJ-condition. This leads to a special structured mathematical program with complementarity constraints. We analyze the KKT-approach from a generic viewpoint and reveal the advantages and possible drawbacks of this approach for solving BL problems numerically.

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a b s t r a c tIn this paper we consider the variable inequalities problem, that is, to find a solution of the inclusion given by the sum of a function and a point-to-cone application. This problem can be seen as a generalization of the classical inequalities problem taking a variable order structure. Exploiting this relation, we propose two variants of the subgradient algorithm for solving the variable inequalities model. The convergence analysis is given under convex-like conditions, which, when the point-to-cone application is constant, contains the old subgradient schemes.

We consider a generalization of standard vector optimization which is called vector optimization with variable ordering structures. The problem class under consideration is characterized by a point-dependent proper cone-valued mapping: here, the concept of K-convexity of the incorporated mapping plays an important role. We present and discuss several properties of this class such as the cone of separations and the minimal variable K-convexification. The latter one refers to a general approach for generating a variable ordering mapping for which a given mapping is K-convex. Finally, this approach is applied to a particular case.