Solvability And Primal-dual Partitions Of The Space Of Continuous Linear Semi-infinite Optimization Problems

Abstract

Different partitions of the parameter space of all linear semi-infinite programming problems with a fixed compact set of indices and continuous right and left hand side coefficients have been considered in this paper. The optimization problems are classified in a different manner, e.g., consistent and inconsistent, solvable (with bounded optimal value and nonempty optimal set), unsolvable (with bounded optimal value and empty optimal set) and unbounded (with infinite optimal value). The classification we propose generates a partition of the parameter space, called second general primal-dual partition. We characterize each cell of the partition by means of necessary and sufficient, and in some cases only necessary or sufficient conditions, assuring that the pair of problems (primal and dual), belongs to that cell. In addition, we show non emptiness of each cell of the partition and with plenty of examples we demonstrate that some of the conditions are only necessary or sufficient. Finally, we investigate various questions of stability of the presented partition.

Recent contributions to linear semi-infinite optimization: an update

“…As a …nal observation in this section, we point out (following [68]) that weak genericity is related with the so-called primal-dual partitions of (see, e.g., [15], [116], [173] and references therein).…”

Primal-dual partitions in linear semi-infinite programming with bounded coefficients

“…Therefore, π 2 / ∈ Π 1 1 in the case of bounded coefficients. The following example shows that the facts that c ∈ int M and that π satisfies the strong Slater condition are not necessary conditions for π to belong to Π 1 1 . Example 4.3.…”

“…The following statements are true: 1 1 if and only if c ∈ int M and π satisfies the Slater condition; ii) π ∈ Π 2 1 if and only if 0 n 1 / ∈ cl K, c ∈ int M and π does not satisfy the Slater condition; iii) π ∈ Π 3 1 if and only if c ∈ M \ int M and π satisfies the Slater condition; iv) π ∈ Π 4 1 if and only if 0 n 1 / ∈ cl K, c ∈ M \ int M and π does not satisfy the Slater condition.…”

“…We consider again the parameter of Example 4.1 and we show that the facts that c ∈ int M and that π satisfes the strong Slater condition are not sufficient for ensuring that π ∈ Π 1 1 . Example 4.2.…”

“…The partitions where T is a compact topological space and the functions a and b are continuous (i.e., the continuous case), have been analyzed in [1,5,6]. In particular, the latter reference deals with the consistence and boundedness of the optimal value of the problems and these properties define the primal-dual partition.…”