In this paper we introduce the concept of solving strategy for a linear sem
i-infinite programming problem, whose index set is arbitrary and whose coef
ficient functions have no special property at all. In particular, we consid
er two strategies which either approximately solve or exactly solve the app
roximating problems, respectively. Our principal aim is to establish a glob
al framework to cope with different concepts of well-posedness spread out i
n the literature. Any concept of well-posedness should entail different pro
perties of these strategies, even in the case that we are not assuming the
boundedness of the optimal set. In the paper we consider three desirable pr
operties, leading to an exhaustive study of them in relation to both strate
gies. The more significant results are summarized in a table, which allows
us to show the double goal of the paper. On the one hand, we characterize t
he main features of each strategy, in terms of certain stability properties
(lower and upper semicontinuity) of the feasible set mapping, optimal valu
e function and optimal set mapping. On the other hand, and associated with
some cells of the table, we recognize different notions of Hadamard well-po
sedness. We also provide an application to the analysis of the Hadamard wel
l-posedness for a linear semi-infinite formulation of the Lagrangian dual o
f a nonlinear programming problem.