I was going through this nice paper ” A Simple and General Duality Proof for Wasserstein Distributionally Robust Optimization”, and one quick qu on applying Theorem 1 to my poject: What if in my problem, I can show as an additional lemma, that the necessary condition for optimality is that the primal problem’s inequality constraint has to actually be a strict equality(if not, we can always slightly modify our distribution to yield a better value, contradicting optimality).
what kind of additional benefits do we get from that additional strong condition? If we have that additional equation, I'm hoping that we can use that to directly get the value of the lagrange multiplier $\lambda$ by plugging in the equation I get from the first order condition from the inner maximization of the dual problem. This allows me to bypass solving the outer minimization problem with respect to $\lambda$; I just plugin the $\lambda$ I got from solving the equation. This would be very beneficial, because in my case, the outer minimization is pretty dirty to characterize analytically(it would be immediate going for numerical solutions) because the objective func becomes a fractional function(numerator is cubic in $\lambda$, denominator is quadratic in $\lambda$ , with a lot of floating parameters…) Any suggestions, pointing out to literature, general proof on these minimax convex duality results, would help.