A hybrid semismooth quasi-Newton method for nonsmooth optimal control with PDEs

Abstract

We propose a semismooth Newton-type method for nonsmooth optimal control problems. Its particular feature is the combination of a quasi-Newton method with a semismooth Newton method. This reduces the computational costs in comparison to semismooth Newton methods while maintaining local superlinear convergence. The method applies to Hilbert space problems whose objective is the sum of a smooth function, a regularization term, and a nonsmooth convex function. In the theoretical part of this work we establish the local superlinear convergence of the method in an infinite-dimensional setting and discuss its application to sparse optimal control of the heat equation subject to box constraints. We verify that the assumptions for local superlinear convergence are satisfied in this application and we prove that convergence can take place in stronger norms than that of the Hilbert space if initial error and problem data permit. In the numerical part we provide a thorough study of the hybrid approach on two optimal control problems, including an engineering problem from magnetic resonance imaging that involves bilinear control of the Bloch equations. We use this problem to demonstrate that the new method is capable of solving nonconvex, nonsmooth large-scale real-world problems. Among others, the study addresses mesh independence, globalization techniques, and limited-memory methods. We observe throughout that algorithms based on the hybrid methodology are several times faster in runtime than their semismooth Newton counterparts. Keywords Semismooth Newton methods • Quasi-Newton methods • Superlinear convergence • Nonsmooth optimal control • Bloch equations Mathematics Subject Classification 49M05 • 49M15 • 49K20 • 49-04 • 90-04 • 90C06 • 90C30 • 90C53 • 90C56 • 90C90 B Armin Rund

An interior point-proximal method of multipliers for convex quadratic programming

“…Indeed, there have been developed approaches based on trust-region strategies (e.g. see [1,16,23,44,56]), as well as line-search strategies based on smooth penalty functions (e.g. see the developments on the forward-backward envelope (FBE) [49,50,65,66] or developments based on the proximal point method [24]).…”

“…see [24,33,40,46,50]). Nonetheless, more recently there have been developed certain globalized Newtonlike schemes, specialized to the case of L 1 -regularized PDE-constrained optimization (see [43,52]), in which the ℓ 1 term as well as the box constraints in (P) are both explicitly handled. We should notice, however, that globalized Newton-like schemes applied to (P) need additional assumptions on the smooth Hessian matrix Q, as well as the constraint matrix A, since otherwise, the stability of the related Newton linear systems, arising as subproblems, might be compromised.…”

“…Those include globalized (smooth, semi-smooth, quasi or proximal) Newton methods (e.g. see [13,32,43,45,46,50,52,56,65]), variants of the proximal point method (e.g. see [20,24,25,38,39,41,49,59]), or interior point methods (IPMs) applied to a reformulation of (P) (e.g.…”

OPTE special issue on PDE-constrained optimization

“…The paper (Mannel and Rund 2021) by Florian Mannel and Armin Rund combines a quasi-Newton approach with a semismooth Newton method to solve nonsmooth optimal control problems. The cost function is the sum of a smooth part and a closed convex part.…”

On the order of convergence of Broyden’s method

“…For the case of smooth systems of equations, the surveys [2,8,16,25] cover many aspects and provide further references, so we restrict ourselves to mentioning [7,15,31] that develop the local convergence theory of Broyden's method. Works that are too recent to be included in the surveys are, for instance, concerned with the extension of Broyden's method to constrained nonlinear systems of equations [24], to set-valued mappings [3], and to single-and set-valued nonsmooth problems in infinite-dimensional spaces [1,23,27]. Broyden's method is also used in implicit deep learning, cf.…”