Subderivative

The subdifferential is a generalization of the gradient is not differentiable convex function. The subdifferential plays an important role in the analysis as well as the convex the convex optimization.

Definition

Let be a convex function. A vector is called subgradient of at the site, if for all

Where the standard scalar called. Intuitively, this is defined in one dimension (ie ) is the graph of the function on all of the linearization point on, the linearization is provided on the straight line with the slope. The subdifferential is the set of all subgradients of the point. According to the "supporting hyperplanes " theorem, the subdifferential of a convex function is continuous everywhere not empty.

Example

Subdifferential the function is given by:

Narrowness

Be steady and is limited. Then, the amount is limited.

Evidence

Be steady and is limited. Set with. Suppose is not limited, then there are for one and one with. Be. Thus are. We obtain the estimate

Is therefore not a subgradient. This is a contradiction.