Splitting Forward-Backward Penalty Scheme for Constrained Variational Problems

We study a forward backward splitting algorithm that solves the variational inequality

where

\mathcal{H} H

is a real Hilbert space,

A: \mathcal{H}\rightrightarrows \mathcal{H} A : H ⇉ H

is a maximal monotone operator,

\Phi: \mathcal{H}\to\mathbf{R} Φ : H → R

is a smooth convex function, and

N_C N C

is the outward normal cone to a closed convex set

C\subset\mathcal{H} C ⊂ H

. The constraint set

C C

is represented as the intersection of the sets of minima of two convex penalization function

\Psi_1:\mathcal{H}\to\mathbf{R} Ψ 1 : H → R

and

\Psi_2:\mathcal{H}\to\mathbf{R}\cup \{+\infty\} Ψ 2 : H → R ∪ { + ∞ }

. The function

\Psi_1 Ψ 1

is smooth, the function

\Psi_2 Ψ 2

is proper and lower semicontinuous. Given a sequence

(\beta_n) ( β n )

of penalization parameters which tends to infinity, and a sequence of positive time steps

(\lambda_n) ( λ n )

, the algorithm (SFBP),

n\geq 1 n ≥ 1

,

\ \left\{\begin{array}{rcl} x_1 & \in & \mathcal{H},\\ x_{n+1} & = & (I+\lambda_n A+\lambda_n\beta_n\partial\Psi_2)^{-1} (x_n-\lambda_n\nabla\Phi(x_n)-\lambda_n\beta_n\nabla\Psi_1(x_n)), \end{array}\right.   { x 1 ∈ H , x n + 1 = ( I + λ n A + λ n β n ∂ Ψ 2 ) − 1 ( x n − λ n ∇ Φ ( x n ) − λ n β n ∇ Ψ 1 ( x n ) ) ,

performs forward steps on the smooth parts and backward steps on the other parts. Under suitable assumptions, we obtain weak ergodic convergence of the sequence

(x_n) ( x n )

to a solution of the variational inequality. Convergence is strong when either

A A

is strongly monotone or

\Phi Φ

is strongly convex. We also obtain weak convergence of the whole sequence

(x_n) ( x n )

when

A A

is the subdifferential of a proper lower semicontinuous convex function. This provides a unified setting for several classical and more recent results, in the line of historical research on continuous and discrete gradient-like systems.