An affine covariant composite step method for optimization with PDEs as equality constraints

Abstract

AMS MSC 2000: 49M37, 90C55, 90C06We propose a composite step method, designed for equality constrained optimization with partial differential equations. Focus is laid on the construction of a globalization scheme, which is based on cubic regularization of the objective and an affine covariant damped Newton method for feasibility. We show finite termination of the inner loop and fast local convergence of the algorithm. We discuss preconditioning strategies for the iterative solution of the arising linear systems with projected conjugate gradient. Numerical results are shown for optimal control problems subject to a nonlinear heat equation and subject to nonlinear elastic equations arising from an implant design problem in craniofacial surgery.

A sequential homotopy method for mathematical programming problems

“…We apply Algorithm 1 to problem (27) on Ω = (0, 1) 2 with the target state u d (ξ) = 12(1 − ξ 1 )ξ 1 (1 − ξ 2 )ξ 2 from [41] and control bounds q l (ξ) = −50, q u (ξ) = min 50, 800 max ξ 1 − 1 2 2 , ξ 2 − 1 2 2 for the parameters a = 10 −p , b = 10 p for p = 0, . .…”

“…The field of globalized Newton methods based on differential equation methods applied to the Newton flow started in the early 1950s with Davidenko [19] and continues to raise scientific interest over the decades [20,8,33,23,14,50,41,51,22], predominantly due to the affine invariance properties of the Newton flow [21]. By trading the affine invariance of the Newton flow for the stability properties of the gradient flow, we obtain from a dynamical systems point of view the advantage of being repelled from maxima or saddle points when solving nonlinear optimization problems.…”

“…We apply the proposed method to the following benchmark problem adapted from [41]: Let Ω ⊂ R 2 be a bounded domain with Lipschitz boundary and let constants a, b, γ > 0, control bounds q l , q u ∈ L r (Ω), r ∈ (2, ∞], and a target function u d ∈ L 2 (Ω) be given. We solve the control-constrained quasilinear elliptic optimal control problem min 1 2…”

Optimal control of static contact in finite strain elasticity

“…Although it is possible to show the existence of optimal solutions, the numerical computation of such solutions poses significant challenges due to the contact constraints and the resulting non-smoothness. In order to apply the specialized algorithm developed in [21], we deploy the normal compliance approach to relax the constraints. Consequently, we obtain the regularized problem:…”

“…Thereafter, we combine a path-following method with an affine covariant composite step method as the inner solver for the numerical solution. This method was developed in [18,21] and has been proven to be well suited for large-scale problems involving nonlinear elasticity. Finally, we will present some numerical examples to assess the viability of our approach.…”