## 49-XX CALCULUS OF VARIATIONS AND OPTIMAL CONTROL; OPTIMIZATION [See also 34H05, 34K35, 65Kxx, 90Cxx, 93-XX]

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#### Keywords

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- Test rig optimization (2014)
- Designing good test rigs for fatigue life tests is a common task in the auto- motive industry. The problem to find an optimal test rig configuration and actuator load signals can be formulated as a mathematical program. We in- troduce a new optimization model that includes multi-criteria, discrete and continuous aspects. At the same time we manage to avoid the necessity to deal with the rainflow-counting (RFC) method. RFC is an algorithm, which extracts load cycles from an irregular time signal. As a mathematical func- tion it is non-convex and non-differentiable and, hence, makes optimization of the test rig intractable. The block structure of the load signals is assumed from the beginning. It highly reduces complexity of the problem without decreasing the feasible set. Also, we optimize with respect to the actuators’ positions, which makes it possible to take torques into account and thus extend the feasible set. As a result, the new model gives significantly better results, compared with the other approaches in the test rig optimization. Under certain conditions, the non-convex test rig problem is a union of convex problems on cones. Numerical methods for optimization usually need constraints and a starting point. We describe an algorithm that detects each cone and its interior point in a polynomial time. The test rig problem belongs to the class of bilevel programs. For every instance of the state vector, the sum of functions has to be maximized. We propose a new branch and bound technique that uses local maxima of every summand.

- Minimization and Parameter Estimation for Seminorm Regularization Models with I-Divergence Constraints (2012)
- This papers deals with the minimization of seminorms \(\|L\cdot\|\) on \(\mathbb R^n\) under the constraint of a bounded I-divergence \(D(b,H\cdot)\). The I-divergence is also known as Kullback-Leibler divergence and appears in many models in imaging science, in particular when dealing with Poisson data. Typically, \(H\) represents here, e.g., a linear blur operator and \(L\) is some discrete derivative operator. Our preference for the constrained approach over the corresponding penalized version is based on the fact that the I-divergence of data corrupted, e.g., by Poisson noise or multiplicative Gamma noise can be estimated by statistical methods. Our minimization technique rests upon relations between constrained and penalized convex problems and resembles the idea of Morozov's discrepancy principle. More precisely, we propose first-order primal-dual algorithms which reduce the problem to the solution of certain proximal minimization problems in each iteration step. The most interesting of these proximal minimization problems is an I-divergence constrained least squares problem. We solve this problem by connecting it to the corresponding I-divergence penalized least squares problem with an appropriately chosen regularization parameter. Therefore, our algorithm produces not only a sequence of vectors which converges to a minimizer of the constrained problem but also a sequence of parameters which convergences to a regularization parameter so that the penalized problem has the same solution as our constrained one. In other words, the solution of this penalized problem fulfills the I-divergence constraint. We provide the proofs which are necessary to understand our approach and demonstrate the performance of our algorithms for different image restoration examples.

- On optimal control simulations for mechanical systems (2011)
- The primary objective of this work is the development of robust, accurate and efficient simulation methods for the optimal control of mechanical systems, in particular of constrained mechanical systems as they appear in the context of multibody dynamics. The focus is on the development of new numerical methods that meet the demand of structure preservation, i.e. the approximate numerical solution inherits certain characteristic properties from the real dynamical process. This task includes three main challenges. First of all, a kinematic description of multibody systems is required that treats rigid bodies and spatially discretised elastic structures in a uniform way and takes their interconnection by joints into account. This kinematic description must not be subject to singularities when the system performs large nonlinear dynamics. Here, a holonomically constrained formulation that completely circumvents the use of rotational parameters has proved to perform very well. The arising constrained equations of motion are suitable for an easy temporal discretisation in a structure preserving way. In the temporal discrete setting, the equations can be reduced to minimal dimension by elimination of the constraint forces. Structure preserving integration is the second important ingredient. Computational methods that are designed to inherit system specific characteristics – like consistency in energy, momentum maps or symplecticity – often show superior numerical performance regarding stability and accuracy compared to standard methods. In addition to that, they provide a more meaningful picture of the behaviour of the systems they approximate. The third step is to take the previ- ously addressed points into the context of optimal control, where differential equation and inequality constrained optimisation problems with boundary values arise. To obtain meaningful results from optimal control simulations, wherein energy expenditure or the control effort of a motion are often part of the optimisation goal, it is crucial to approxi- mate the underlying dynamics in a structure preserving way, i.e. in a way that does not numerically, thus artificially, dissipate energy and in which momentum maps change only and exactly according to the applied loads. The excellent numerical performance of the newly developed simulation method for optimal control problems is demonstrated by various examples dealing with robotic systems and a biomotion problem. Furthermore, the method is extended to uncertain systems where the goal is to minimise a probability of failure upper bound and to problems with contacts arising for example in bipedal walking.