Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: 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. Hover, separable convex constraints do not always satisfy the constraint qualification, that is, Lagrange duality does not always hold for separable convex programming. ,  study duality theorems for separable convex programming without any qualifications.  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,  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.
['Satoshi Suzuki', 'Daishi Kuroiwa']
I am writing a research paper on a idea Duality Theorems for Separable Convex Programming Without Qualifications
[{"from": "human", "value": "I am writing a research paper on a idea Duality Theorems for Separable Convex Programming Without Qualifications"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: 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. Hover, separable convex constraints do not always satisfy the constraint qualification, that is, Lagrange duality does not always hold for separable convex programming. ,  study duality theorems for separable convex programming without any qualifications.  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,  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."}]