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Journal of Convex Analysis 23 (2016), No. 2, 531--565
Copyright Heldermann Verlag 2016

Splitting Forward-Backward Penalty Scheme for Constrained Variational Problems

Marc-Olivier Czarnecki
Institut de Mathématiques et Modélisation, Université Montpellier 2, Place Eugène Bataillon, 34095 Montpellier cedex 5, France

Nahla Noun
Département de Mathématiques, Faculté des Sciences 1, Université Libanaise, Hadath, Beyrouth, Lebanon

Juan Peypouquet
Departamento de Matemática, Universidad Técnica Federico Santa María, Avenida España 1680, Valparaíso, Chile


\newcommand{\R}{\mathbf{R}} \renewcommand{\H}{\mathcal{H}} We study a forward backward splitting algorithm that solves the variational inequality \begin{equation*} A x +\nabla \Phi(x)+ N_C (x) \ni 0 \end{equation*} where $\H$ is a real Hilbert space, $A: \H\rightrightarrows \H$ is a maximal monotone operator, $\Phi: \H\to\R$ is a smooth convex function, and $N_C$ is the outward normal cone to a closed convex set $C\subset\H$. The constraint set $C$ is represented as the intersection of the sets of minima of two convex penalization function $\Psi_1:\H\to\R$ and $\Psi_2:\H\to\R\cup \{+\infty\}$. The function $\Psi_1$ is smooth, the function $\Psi_2$ is proper and lower semicontinuous. Given a sequence $(\beta_n)$ of penalization parameters which tends to infinity, and a sequence of positive time steps $(\lambda_n)$, the algorithm (SFBP), $n\geq 1$, \begin{equation*} \ \left\{\begin{array}{rcl} x_1 & \in & \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. \end{equation*} 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 $(x_n)$ to a solution of the variational inequality. Convergence is strong when either $A$ is strongly monotone or $\Phi$ is strongly convex. We also obtain weak convergence of the whole sequence $(x_n)$ when $A$ 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.

Keywords: Constrained convex optimization, forward-backward algorithms, hierarchical optimization, maximal monotone operators, penalization methods, variational inequalities.

MSC: 37N40, 46N10, 49M30, 65K05, 65K10, 90B50, 90C25

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