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Research paper thumbnail of Convex Inequalities Without Constraint Qualification nor Closedness Condition, and Their Applications in Optimization

Given two convex lower semicontinuous extended real valued functions F and h de…ned on locally co... more Given two convex lower semicontinuous extended real valued functions F and h de…ned on locally convex spaces, we provide a dual transcription of the relation (?) F (0;) h () : Some results in this direction are obtained in the …rst part of the paper (Lemma 2, Proposition 1), and they are applied to the case when the left-hand-side in (?) is the sum of two convex functions with a convex composite one (Theorem 1). In the spirit of previous works ([16], [23], [24], [26], [27], etc.) we give in Theorem 2 a formula for the subdifferential of such a function without any quali…cation condition. As a consequence of that, we extend to the nonre ‡exive setting a recent result ([22, Theorem 3.2]) about subgradient optimality conditions without constraint quali…cations. Finally, we apply Theorem 1 to obtain Farkas-type lemmas and new results on DC, convex, semi-de…nite, and linear optimization problems.

Research paper thumbnail of Convex Inequalities Without Constraint Qualification nor Closedness Condition, and Their Applications in Optimization

Given two convex lower semicontinuous extended real valued functions F and h de…ned on locally co... more Given two convex lower semicontinuous extended real valued functions F and h de…ned on locally convex spaces, we provide a dual transcription of the relation (?) F (0;) h () : Some results in this direction are obtained in the …rst part of the paper (Lemma 2, Proposition 1), and they are applied to the case when the left-hand-side in (?) is the sum of two convex functions with a convex composite one (Theorem 1). In the spirit of previous works ([16], [23], [24], [26], [27], etc.) we give in Theorem 2 a formula for the subdifferential of such a function without any quali…cation condition. As a consequence of that, we extend to the nonre ‡exive setting a recent result ([22, Theorem 3.2]) about subgradient optimality conditions without constraint quali…cations. Finally, we apply Theorem 1 to obtain Farkas-type lemmas and new results on DC, convex, semi-de…nite, and linear optimization problems.

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