However, in many applications, the max norm of tracking error is critical, which is a nonsmooth function and thus gradient-based ILC methods cannot be directly used.
It is well known that (f(x)=sum^{n}_{i=1} vert f_{i}(x) vert ) is a nonconvex nonsmooth function, and (h(x)=frac{1}{2} Vert x Vert ^{2}) is a simple convex function.
where F B ( B ) = 1 2 ∥ A BC t-1 ) − Y ∥ F 2 + 1 2 ∥ C t-1 ∥ F 2. This is a composite convex optimization problem involving the sum of a smooth function (F B (B)) and a nonsmooth function (λ∥B∥2,1).
Typically, the objective function involves a smooth function with Lipschitz continuous gradient, a linear composite nonsmooth function, and a nonsmooth function.
But it is a nonsmooth equation so we cannot use this method.
The result is a nonsmooth version of result due to Brézis and Nirenberg [27].
Since problem (2.1) is a nonsmooth equation, the classical Newton methods cannot be used to solve it.
Discretization of the underlying optimization model, which is a nonsmooth convex programming problem, leads to an overdetermined linear system that can be handled by interior point methods.
In this paper, we consider the structured nonconvex minimization problem min_{xinmathbb{R}^{n}} bigl{ F x):=f(x)+h(x bigr}, (1) where (f: mathbb{R}^{n}rightarrowmathbb{R}) is possibly a nonconvex nonsmooth function and (h: mathbb{R}^{n}rightarrow -infty,infty]) is a closed proper convex function.
In this paper, we study the minimization problem of the type (L x,y)=f(x)+R x,y)+g y)), where f and g are both nonconvex nonsmooth functions, and R is a smooth function we can choose.
