Remarks on infinite dimensional duality
The convex like optimization problem \(f(x)\rightarrow \min , g(x)\in -C\) with \(x\) in a TVS and \(g(x)\) in a normed space, \(C\) a convex order cone is considered. A strong (Lagrange) duality result derived in [\textit{P. Daniele, S. Giuffré, G. Idone} and \textit{A. Maugeri}, Math. Ann. 339, No. 1, 221--239 (2007; Zbl 1356.49064)] uses the empty intersection of the tangent cone at the optimal point w.r.t. \(\tilde M\) with \((0,\infty)\times \{0\}\) as constrained qualification. Hereby is \(\tilde M\) the Minkowski sum of the used order cone \(\mathbb{R}_+ \times C\) and the image of the non feasible points by \((f,g)\). Let now \(M\supset \{(0,0)\}\) be the Minkowski sum of the order cone \(\mathbb{R}_+ \times C\) and the whole image of \((f-f(x_0),g)\). In the paper is shown, that the same duality result holds when the above constrained qualification is replaced by the following condition: There is an element \(\xi\) in \(M\) and an outer normal \(\eta^*\) from the dual space at zero of \(M\) such that \(\eta^* \xi <0\). Hence ``the interior point free'' primal constrained qualification can be replaced by some strong inequality with respect to the outer normal cone. The given approach works for spaces where the order cone of the constraint \(g\) does not contain interior points e.g. for \(l^2\) with the order cone of nonnegative sequences.
scientific article; zbMATH DE number 5702524
scientific article