The alignment principle

The notion of "alignment" between points in a primal space and points in a dual space can be used to characterize optimality conditions for many common optimization problems. This gives a nice geometric interpretation of what it means for points to be optimal, in settings including 1) regularization in machine learning, 2) gauge duality, and 3) linear conic optimization. I will give an overview of what the alignment principle is, relate it to a generalized notion of sparsity, and discuss its application in the three settings above.