ABSTRACT. Let {C_{0}, C_{1}, ..., C_{m}} be a collection of closed convex subsets with nonempty intersection C in a Hilbert space X. A geometric characterization is given of those collections for which the Karush-Kuhn-Tucker (or Lagrange multiplier) conditions are valid for characterizing optimal minimal solutions for any convex continuous function f : C → R over C. They are precisely those sets {C_{0}, C_{1}, ..., C_{m}} which have the “strong conical hull intersection property” that was introduced in [5].