The authors present some necessary and sufficient conditions, which completely characterize the strong and total Lagrange duality, respectively, for convex optimization problems in separated locally convex spaces. The authors also prove similar statements for the problems obtained by perturbing the objective functions of the primal problems by arbitrary linear functionals. In the particular case, to deal with convex optimization problems having infinitely many convex inequalities as constraints, the conditions considered in this paper become the so-called Farkas-Minkowski and locally Farkas-Minkowski conditions for systems of convex inequalities used in the literature. In some situations, the locally Farkas-Minkowski condition turns out to be equivalent to the basic constraint qualification (BCQ) used in the literature. See [J.-B. Hiriart-Urruty
and C. Lemarechal
, Convex analysis and minimization algorithms. Part 1: Fundamentals. Grundlehren der Mathematischen Wissenschaften. 305. (Berlin): Springer- Verlag. (1993; Zbl 0795.49001
)] and H. Hu
[Math. Oper. Res., 30, No. 4, 956–965 (2005; Zbl 05279651
)]. Different results are also rediscovered as special cases and some of them are improved in their original context.