In this paper we present different regularity conditions that equivalently characterize various ε-duality gap statements (with ε ≥ 0) for constrained optimization problems and their Lagrange and Fenchel-Lagrange duals in separated locally convex spaces, respectively. These regularity conditions are formulated by using epigraphs and ε-subdifferentials. When ε = 0 we rediscover recent results on stable strong and total duality and zero duality gap from the literature. MSC: PreliminariesMotivated by recent results on stable strong and total duality for constrained convex optimization problems in [2,6,7,9,13,17] and the ones on zero duality gap in [15,16] we introduce in this paper several regularity conditions which characterize ε-duality gap statements (with ε ≥ 0) for a constrained optimization problem and its Lagrange and FenchelLagrange dual problems, respectively. The regularity conditions we provide in Section 2 are based on epigraphs, while the ones in Section 3 on ε-subdifferentials. In this way we extend many of the results in the mentioned papers, which are recovered as special cases when ε = 0, delivering thus generalizations of the classical Farkas-Minkowski and basic constraint qualifications. Moreover some statements in [6-8, 15, 16], which arise from our results in the special *

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