Abstract
We consider two partitions over the space of linear semi-infi nite programming parameters with a fi xed index set and bounded coefficients (the functions of the constraints are bounded). The fi rst one is the primal-dual partition inspired by consistency and boundedness of the optimal value of the linear semi-infi nite optimization problems. The second one is a re finement of the primal-dual partition that arises considering the boundedness of the optimal set. These two partitions have been studied in the continuous case, this is, the set of indices is a compact infi nite compact Hausdorff topological space and the functions de fining the constraints are continuous. In this work, we present an extension of this case. We study same topological properties of the cells generated by the primal-dual partitions and characterize their interior. Through examples, we show that the results characterizing the sets of the partitions in the continuous case are neither necessary nor sufficient in both refi nements. In addition, a sufficient condition for the boundedness of the optimal set of the dual problem has been presented.
Highlights
We associate with each triplet π ∈ Π = Bn × B × R a primal problem P: inf c′x s. t. atx ≥ bt, t ∈ T
In R+(T) we consider the norms l∞ and l1. As both primal and dual problems are defined with the same data a, b and c, these are represented by the triplet π ∶= (a, b, c)
The following theorem shows that the characterization of the interior of the sets that are generated with the primal-dual partition, in the case of bounded coefficients, is like the continuous case
Summary
In R+(T) we consider the norms l∞ and l1 As both primal and dual problems are defined with the same data a, b and c, these are represented by the triplet π ∶= (a, b, c). The interior of the sets generated through the partition is studied. In the present paper a refinement of the primal-dual partition is presented. We present a condition that implies the boundedness of the optimal set of the dual problem
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