Abstract
In this paper, we present some strong and total Fenchel dualities for convex programming problems with data uncertainty within the framework of robust optimization in locally convex Hausdorff vector spaces. By using the properties of the epigraph of the conjugate functions, we give some new constraint qualifications, which characterizes completely the strong duality and the stable strong duality. Moreover, some sufficient and/or necessary conditions for the total duality and converse duality are also obtained.
Highlights
Let X and Y be real locally convex Hausdorff topological vector spaces, whose dual spaces, X∗ and Y ∗, are endowed with the weak∗-topologies w∗(X∗, X) and w∗(Y ∗, Y ), respectively
It is well known that the optimal values of these problems, v(P) and v(D), respectively, satisfy the so-called weak duality (i.e., v(P) ≥ v(D)), but a duality gap may occur (i.e., we may have v(P) > v(D))
A challenge in convex analysis has been to give sufficient conditions which guarantee the strong duality, that is, v(P) = v(D) and (D) has at least an optimal solution, and guarantee the converse strong duality, which corresponds to the situation in which v(P) = v(D) and (P) has at least an optimal solution
Summary
Let X and Y be real locally convex Hausdorff topological vector spaces, whose dual spaces, X∗ and Y ∗, are endowed with the weak∗-topologies w∗(X∗, X) and w∗(Y ∗, Y ), respectively. It is well known that the optimal values of these problems, v(P) and v(D), respectively, satisfy the so-called weak duality (i.e., v(P) ≥ v(D)), but a duality gap may occur (i.e., we may have v(P) > v(D)). Li et al considered the following uncertain convex programming problem (cf [ ]):. Our main aim in the present paper is to give some new regularity conditions which completely characterize the strong dualities and the stable strong dualities between (P) and (D) and between (P ) and (D), and provide sufficient and/or necessary conditions for the total duality, the converse dualities between (P) and (D). Some sufficient and/or necessary conditions for the total duality and the converse duality between (P) and (D) are given in Sections and , respectively
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