New regularity conditions for strong and total Fenchel–Lagrange duality in infinite dimensional spaces

We give new regularity conditions for convex optimization problems in separated locally convex spaces. We completely characterize the stable strong and strong Fenchel–Lagrange duality. Then we give similar statements for the case when a solution of the primal problem is assumed as known, obtaining complete characterizations for the so-called total and stable totalFenchel–Lagrange duality  , respectively. For particular settings the conditions that we consider turn into some constraint qualifications already used by different authors, like Farkas–Minkowski CQ, locally Farkas–Minkowski CQ and basic CQ, and we rediscover and improve some recent results from the literature.