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Duality Theorems for Separable Convex Programming Without Qualifications

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Abstract

In the research of mathematical programming, duality theorems are essential and important elements. Recently, Lagrange duality theorems for separable convex programming have been studied. Tseng proves that there is no duality gap in Lagrange duality for separable convex programming without any qualifications. In other words, although the infimum value of the primal problem equals to the supremum value of the Lagrange dual problem, Lagrange multiplier does not always exist. Jeyakumar and Li prove that Lagrange multiplier always exists without any qualifications for separable sublinear programming. Furthermore, Jeyakumar and Li introduce a necessary and sufficient constraint qualification for Lagrange duality theorem for separable convex programming. However, separable convex constraints do not always satisfy the constraint qualification, that is, Lagrange duality does not always hold for separable convex programming. In this paper, we study duality theorems for separable convex programming without any qualifications. We show that a separable convex inequality system always satisfies the closed cone constraint qualification for quasiconvex programming and investigate a Lagrange-type duality theorem for separable convex programming. In addition, we introduce a duality theorem and a necessary and sufficient optimality condition for a separable convex programming problem, whose constraints do not satisfy the Slater condition.

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Acknowledgments

The authors are grateful to the editor and anonymous referees for many comments and suggestions improved the quality of the paper. This work was partially supported by JSPS KAKENHI Grant Numbers 15K17588, 25400205.

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  1. Department of Mathematics, Shimane University, Matsue, Shimane, Japan

    Satoshi Suzuki & Daishi Kuroiwa

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  1. Satoshi Suzuki
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  2. Daishi Kuroiwa
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Correspondence to Satoshi Suzuki.

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Suzuki, S., Kuroiwa, D. Duality Theorems for Separable Convex Programming Without Qualifications. J Optim Theory Appl 172, 669–683 (2017). https://doi.org/10.1007/s10957-016-1003-1

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