Understanding Badly and Well-Behaved Linear Matrix Inequalities Via Semi-infinite Optimization

In this paper, we use a linear semi-infinite optimization approach to study badly and well-behaved linear matrix inequalities. We utilize a result on uniform LP duality of linear semi-infinite optimization problems to prove recent results obtained by Pataki. Such an approach not only provides alternative proofs of known results, but also gives new insights about badly and well-behaved linear matrix inequalities in terms of a cone and a linear subspace associated with the corresponding linear semi-infinite systems. Furthermore, when the linear matrix inequality constraint of the primal semidefinite optimization problem is badly behaved, we give a characterization of objective functions for the primal linear semidefinite optimization problem with which strong duality holds.