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.