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.

We consider two partitions over the space of linear semi-infinite programming parameters with a fixed index set and bounded coefficients (the functions of the constraints are bounded). The first one is the primal-dual partition inspired by consistency and boundedness of the optimal value of the linear semiinfinite optimization problems. The second one is a refinement 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 infinite compact Hausdorff topological space and the functions defining 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 refinements. In addition, a sufficient condition for the boundedness of the optimal set of the dual problem has been presented..

Optimization problems containing a finite number of variables and an infinite number of constraints are called semi-infinite programming problems. Under certain conditions, a class of these problems can be represented as bi-level programming problems. Bi-level problems are a particular class of optimization problems, in which there is another optimization problem embedded in one of the constraints. We reformulate a semi-infinite problem into a bi-level problem and then into a single-level nonlinear one by using the Kuhn–Tucker optimality conditions. The resulting reformulation allows us to employ a branch and bound scheme to optimally solve the problem. Computational experimentation over well-known instances shows the effectiveness of the developed method concluding that it is able to effectively solve linear semi-infinite programming problems. Additionally, some linear bi-level problems from literature are used to validate the robustness of the proposed algorithm.