2014/08/05 by Marc-Olivier Czarnecki, Czarnecki, Marc-Olivier, Nahla Noun +3 · 1 citation
Computer Science · Engineering · Medicine · #Optimization and Variational Analysis #Sparse and Compressive Sensing Techniques #Aortic aneurysm repair treatments
paper · pdf · doi:10.48550/arxiv.1408.0974
We study a forward backward splitting algorithm that solves the variational inequality A x +∇ Φ(x)+ NC (x) \ni 0 where H is a real Hilbert space, A: H\rightrightarrows H is a maximal monotone operator, Φ: H→ℝ is a smooth convex function, and NC is the outward normal cone to a closed convex set C⊂ H. The constraint set C is represented as the intersection of the sets of minima of two convex penalization function Ψ1:H→ℝ and Ψ2: H→ℝ∪ \+∞\. The function Ψ1 is smooth, the function Ψ2 is proper and lower semicontinuous. Given a sequence (βn) of penalization parameters which tends to infinity, and a sequence of positive time steps (λn), the algorithm \x1 · amp; ∈ · amp; H,
xn+1 · amp; = · amp; (I+λn A+λnβn∂Ψ2)-1(xn-λn∇Φ(xn)-λnβn∇Ψ1(xn)), n≥ 1.. 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 (xn) to a solution of the variational inequality. Convergence is strong when either A is strongly monotone or Φ is strongly convex. We also obtain weak convergence of the whole sequence (xn) 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.