Optimality conditions for non-finite valued convex composite functions

Burke has recently developed second-order necessary and sufficient conditions for convex composite optimization in the case where the convex function is finite valued. In this note the authors present a technique for reducing the infinite valued case to the finite valued one. They then use this technique to extend the results in Burke to the case in which the convex function may take infinite values. The authors conclude by comparing these results with those established by Rockafellar for the piecewise linear-quadratic case.