Separation of convex sets by Clarke subdifferential

Abstract

In this article we consider a separation technique proposed in J. Grzybowski, D. Pallaschke, and R. Urbański (A pre-classification and the separation law for closed bounded convex sets, Optim. Method Softw. 20(2005), pp. 219–229) for separating two convex sets A and B with another convex set C. We prove that in a finite dimension C can be chosen as the Clarke subdifferential at the origin of , where p_A , p_B denotes the support functions of A and B respectively.

Keywords:

• data classification
• convex analysis
• non-smooth optimization

Notes

Note

1. Let (X,d) a metric space. The set is called the closed interval. A set A is said to be d-convex, if [x,y] ⊂ A for arbitrary x, y ∈ A 14,17.