Recently, it has been observed that several nondifferentiable minimization problems share the property that the question of whether a given point is optimal can be answered by solving a certain bounded least squares problem. If the resulting residual vector, r, vanishes then the current point is optimal. Otherwise, r is a descent direction. In fact, as will be seen, r points at the steepest descent direction. On the other hand, it is customary to characterize the optimality conditions (and the steepest descent vector) of a convex nondifferentiable function via its subdifferential. Also, it is well known that optimality conditions are usually related to theorems of the alternative. One aim of the present survey is to clarify the relations between these subjects. Another aim is to introduce a new type of theorems of the alternative. The new theorems characterize the optimality conditions of discrete l1 approximation problems and multifacility location problems, and provide a simple way to obtain the subdifferential and the steepest descent direction in such problems. A further objective of the review is to demonstrate that the ability to compute the steepest descent direction at degenerate dead points opens a new way for handling degeneracy in active set methods.
