On Computing the Mordukhovich Subdifferential Using Directed Sets in Two Dimensions

Abstract

The Mordukhovich subdifferential is highly important in the variational and non-smooth analysis and optimization, but it may often be hard to calculate it. Here we propose a method of computing the Mordukhovich subdifferential of differences of sublinear (DS) functions applying the directed subdifferential of differences of convex (DC) functions. We restrict ourselves to the two-dimensional case mainly for simplicity of the proofs and for the visualizations. The equivalence of the Mordukhovich symmetric subdifferential (the union of the corresponding subdifferential and superdifferential), to the Rubinov subdifferential (the visualization of the directed subdifferential), is established for DS functions in two dimensions. The Mordukhovich subdifferential and superdifferential are identified as parts of the Rubinov subdifferential. In addition it is possible to construct the directed subdifferential in a way similar to the Mordukhovich one by considering outer limits of Frechet subdifferentials. The results are extended to the case of DC functions. Examples illustrating the obtained results are presented.