Torsten Bosse

One-Shot Approaches to Design Optimzation

The paper describes general methodologies for the solution of design optimization problems. In particular we outline the close relations between a fixed point solver based piggy back approach and a Reduced SQP method in Jacobi and Seidel variants. The convergence rate and general efficacy is shown to be strongly dependent on the characteristics of the state equation and the objective function. In the QP scenario where the state equation is linear and the objective quadratic, finite termination in two steps is obtained by the Seidel variant with Newton state solver and perfect design space preconditioning. More generally, it is shown that the retardation factor between simulation and optimization is bounded below by 2 with the difference depending on a cross-term representing the total sensitivity of the adjoint equation with respect to the design.

Optimal Design with Bounded Retardation for Problems with Non-separable Adjoints

In the natural and enginiering sciences numerous sophisticated simulation models involving PDEs have been developed. In our research we focus on the transition from such simulation codes to optimization, where the design parameters are chosen in such a way that the underlying model is optimal with respect to some performance measure. In contrast to general non-linear programming we assume that the models are too large for the direct evaluation and factorization of the constraint Jacobian but that only a slowly convergent fixed-point iteration is available to compute a solution of the model for fixed parameters. Therefore, we pursue the so-called One-shot approach, where the forward simulation is complemented with an adjoint iteration, which can be obtained by handcoding, the use of Automatic Differentiation techniques, or a combination thereof. The resulting adjoint solver is then coupled with the primal fixed-point iteration and an optimization step for the design parameters to obtain an optimal solution of the problem. To guarantee the convergence of the method an appropriate sequencing of these three steps, which can be applied either in a parallel (Jacobi) or in a sequential (Seidel) way, and a suitable choice of the preconditioner for the design step are necessary. We present theoretical and experimental results for two choices, one based on the reduced Hessian and one on the Hessian of an augmented Lagrangian. Furthermore, we consider the extension of the One-shot approach to the infinite dimensional case and problems with unsteady PDE constraints.

Cubic overestimation and secant updating for unconstrained optimization of C2, 1 functions

The discrepancy between an objective function f and its local quadratic model f(x)+∇ f(x)⊤ s+s⊤ H(x) s/2 ≈ f(x+s) at the current iterate x is estimated using a cubic term q |s|3/3. Potential steps are chosen such that they minimize (or at least significantly reduce) the overestimating function ∇ f(x)⊤ s+s⊤ B s/2+q |s|3/3 with B ≈ H(x). This ensures f(x+s)<f(x) unless the approximating Hessian B=B⊤ differs significantly from H(x) or the scalar q>0 is too small. Either one or both quantities may be updated after unsuccessful and successful steps alike. For an algorithm employing both the symmetric rank one update and a shifted version of the BFGS formula we show that either∈f |∇ f|=0 or sup |B|=∞, provided the Hessian H(x) is Lipschitz on some neighbourhood of a bounded level set. Superlinear convergence is theoretically expected and numerically observed but not yet proven.

On Hessian- and Jacobian-Free SQP Methods - a Total Quasi-Newton Scheme with Compact Storage

In this paper we describe several modifications to reduce the memory requirement of the total quasi-Newton method proposed by Andreas Griewank et al.

The Relative Cost of Function and Derivative Evaluations in theCUTEr Test Set

The CUTEr test set represents a testing environment for nonlinear optimization solvers containing more than 1,000 academic and applied nonlinear problems. It is often used to verify the robustness and performance of nonlinear optimization solvers. In this paper we perform a quantitative analysis of the CUTEr test set. As a result we see that some paradigms of nonlinear optimization and Automatic Differentiation can be verified whereas others need to be questioned. Furthermore, we will show that the CUTEr test set is probably biased, i.e., solvers that use exact derivatives and sparse linear algebra are likely to perform advantageously compared to solvers employing directional derivatives and low-rank updating.

Die magische Quadratur des Superhirns

Die von Robin Wersig in der ZDF-Sendung „Deutschlands Superhirn 2011“ am 28. Dezember 2011 [14] behandelte Aufgabe wird mathematisch formuliert und ihre – in gewissem Sinne – minimale Losung beschrieben. Diese beruht auf einer von Janisch 1859 veroffentlichten magischen Springertour uber das Schachbrett. Die Kenntnis dieses geometrisch einpragsamen Pfades erlaubt die Belegung des Quadrates durch einfaches Abzahlen, eventuell unter Auslassung einer einzigen Zahl. Die so gefundene Losung minimiert das Maximum und die Spreizung der 64 Belegungswerte. Eine Losung der Aufgabe mit geschlossenen Augen verlangt neben der genauen Kenntnis der Springertour nur die Durchfuhrung einer ganzzahligen Division durch 8 mit Rest. Nach unserer Erfahrung konnen gerade junge Menschen sich das Verfahren in einigen Tagen mental zu eigen machen.