Strong duality for infinite-dimensional vector-valued programming problems
Let X, Y, and Z be locally convex real topological vector spaces, \(A\subset X\) a convex subset, and let \(C\subset Y\), \(E\subset Z\) be cones. Let f: \(X\to Z\) be E-concave and g: \(X\to Y\) be C-concave functions. We consider a concave programming problem with respect to an abstract cone and its strong dual problem as follows: \[ (P)\quad \max imize\quad f(x),\quad subject\quad to\quad x\in A,\quad g(x)\in C, \] \[ (SD)\quad \min imize\quad \{\cup_{\phi \in C\quad +}\max \{(f+\phi \circ g)(A):\quad E\}\}, \] where C \(+\) denotes the set of all nonnegative continuous linear operators from Y to Z and (SD) is the strong dual problem to (P). The authors find a necessary condition of strong saddle point for Problem (P) and establish the strong duality relationships between Problems (P) and (SD).
locally convex real topological vector spaces
necessary condition of strong saddle point
scientific article; zbMATH DE number 4068652
scientific article