On set containment characterization and constraint qualification for quasiconvex programming (Q637546)

Let \(X\) be a locally convex Hausdorff topologycal vector space. Set containment problems consist of characterizing the inclusion \(A \subset B\), where \(A=\{ x\in X |\;\forall i \in I, f_i(x) \leq 0 \}\), \(B=\{ x\in X |\;\forall j \in J, h_j(x) \leq 0 \}\), and \(f_i, h_j: X \to R\cup \{\pm \infty \}\). Recently, the set containment characterization for convex programming, under the convexity of \(f_i\), \(i \in I\), and the linearity or the concavity of \(h_j\), \(j \in J\), was established. In this paper, dual characterizations of the containment of a convex set with quasiconvex inequality constraints are investigated. A Lagrange-type duality and a closed cone constraint qualification are described, and it is shown that this constraint qualification is the weakest constraint qualification for the duality.