Primal-Dual Extragradient Methods for Nonlinear Nonsmooth PDE-Constrained Optimization

Abstract

Abstract. We study the extension of the Chambolle-Pock primal-dual algorithm to nonsmooth optimization problems involving nonlinear operators between function spaces. Local convergence is shown under technical conditions including metric regularity of the corresponding primal-dual optimality conditions. We also show convergence for a Nesterov-type accelerated variant, provided one part of the functional is strongly convex. We show the applicability of the accelerated algorithm to examples of inverse problems with L 1 and L ∞ fitting terms as well as of state-constrained optimal control problems, where convergence can be guaranteed after introducing an (arbitrarily small, still nonsmooth) Moreau-Yosida regularization. This is verified in numerical examples.

Optimal Control of the Principal Coefficient in a Scalar Wave Equation

“…We replace u in (2) by u n . Due to the regularity assumptions (17) and the assumption that y 0 is constant on ω c , we have…”

“…Using weak * semi-continuity of norms (cf., e.g., [38, p. 63]), we can now pass to the limit in (19) to obtain (16), for those (y 0 , y 1 , f ) which satisfy the additional regularity assumption (17).…”

“…In this section, we address the numerical solution of (20) using a stabilized spacetime finite element discretization for second-order hyperbolic equations [15] and a nonlinear primal-dual proximal splitting algorithm [16][17][18]. Since we now consider a finite-dimensional optimization problem, we can include the constraint u ∈ U directly via the multi-bang penalty instead of enforcing it inside the state equation.…”

“…To solve (35), we extend the approach in [31] by applying the nonlinear primal-dual proximal splitting method from [16][17][18] together with a lifting trick. For this purpose, we write (35) (omitting the bold notation and the subscripts denoting vectors and discretizations from now on and assuming that…”

“…In the current work, we extend this approach to optimal control and identification of discontinuous coefficients in scalar wave equations by deriving under additional (natural) assumptions on the data the adapted higher regularity results for the wave equation based on elliptic maximal regularity theory [14]; see Assumption 3.1 and Proposition 3.6 below. We also address a suitable discretization of the problem using a stabilized finite element method [15] and its solution by a nonlinear primal-dual proximal splitting method [16][17][18].…”