Convex Inequalities Without Constraint Qualification nor Closedness Condition, and Their Applications in Optimization

AbstractGiven two convex lower semicontinuous extended real valued functions F and h defined on locally convex spaces, we provide a dual transcription of the relation ⋆$$ F\left( 0,\cdot \right) \geq h\left( \cdot \right) . $$Some results in this direction are obtained in the first part of the paper (Lemma 2, Theorem 1). These results then are applied to the case when the left-hand-side in (⋆) is the sum of two convex functions with a convex composite one (Theorem 2). In the spirit of previous works (Hiriart-Urruty and Phelps, J Funct Anal 118:154–166, 1993; Penot, J Convex Anal 3:207–219, 1996, 2005; Thibault, 1995, SIAM J Control Optim 35:1434–1444, 1997, etc.) we give in Theorem 3 a formula for the subdifferential of such a function without any qualification condition. As a consequence of that, we extend to the nonreflexive setting a recent result (Jeyakumar et al., J Glob Optim 36:127–137 2006, Theorem 3.2) about subgradient optimality conditions without constraint qualifications. Finally, we apply Theorem 2 to obtain Farkas-type lemmas and new results on DC, convex, semi-definite, and linear optimization problems.