The generalized gradient of at, denoted by, is defined to be the subdifferential of the convex function at, that is, (2.3).
Then the subdifferential of is defined as follows: (4.1).
That is, F hat{x})+G(hat{x})=min_{xin H}F x)+G x quad Leftrightarrowquad 0in nabla F hat{x})+partial G(hat{x}), where ∇F is the gradient of F and ∂G is the subdifferential of G defined by partial G x)=bigl{ zin H: langle y-x,zrangle+G x leq G(y-x,zrangle+G x leqG y.
The set of minimizers of f is defined to be arg min y ∈ H f ( y ) = { z ∈ H : f ( x ) ≤ f ( y ) for all y ∈ H }, and the subdifferential of f is defined as ∂ f ( x ) = { z ∈ H : 〈 y − x, z 〉 ≤ f ( y ) − f ( x ), ∀ y ∈ H }. for all x ∈ H.
Let be the subdifferential of, where is as defined in (3.3).
The proximal point algorithm of Martinet [35] and Rockafellar [36] was introduced to approximate a solution of (0in Au) where A is the subdifferential of some convex functional defined on a real Hilbert space.
Let T : H → 2 H be the multi-valued map defined by T x = ∂ f ( x ) ∀ x ∈ H, where ∂ f ( x ) is the subdifferential of f at x and is defined by ∂ f ( x ) = { z ∈ H : 〈 z, y − x 〉 ≤ f ( y ) − f ( x ) ∀ y ∈ H }. It is well known that for every x ∈ H, ∂ f ( x ) is nonempty, weakly closed and convex.
The subdifferential of f at x is defined as partial f(x)=bigl{ xiinRe^{N}mid f y geq f(x)+ langlexi, y-x rangle, forall yinRe^{N}bigr}.
If is -subdifferentiable at, then we define the -subdifferential of at as follows: (3.6).
The subdifferential of f at x is the convex set defined by ∂ f ( x ) = { x ∗ ∈ X ∗ : f ( x ) + 〈 x ∗, y − x 〉 ≤ f ( y ), ∀ y ∈ X }. (2.1).
