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978-3-8439-1644-8, Reihe Mathematik
Robust Nonlinear Programming with Discretized PDE Constraints using Second-order Approximations
183 Seiten, Dissertation Technische Universität Darmstadt (2013), Softcover, B5
In most cases, real-world applications are subject to uncertainty in one way or another. Typically, parameters that describe, for example, a physical system can only be measured to a specific precision. Some parameters may even be impossible to measure and only estimates based on expert opinion are available. In particular, the actual values of these parameters are uncertain.
It is indisputable that the mentioned uncertainties may cause serious economic as well as safety consequences. Thus, it is an important task to develop and apply techniques that help to reduce and control the influence of uncertainty. Nowadays, mathematical optimization methods are a standard tool in various areas of application. Therefore, the applied optimization techniques clearly should be capable of taking into account the presence of uncertainty.
Essentially, there exist two main approaches to explicitly consider uncertainty within the framework of mathematical optimization. Stochastic optimization and robust optimization. This thesis is concerned with robust optimization, particularly, in the context of shape optimization under uncertainty, where the design optimization of load-carrying mechanical systems is the guiding practical application. This requires to examine uncertain optimization problems with constraints that are given by partial differential equations (PDEs). The discretization of such problems leads to general (nonconvex) nonlinear programs with discretized PDE constraints under uncertainty, which is the topic of this work.
We present a novel second-order approximation approach for the robust counterpart of general nonlinear optimization problems with uncertain parameters that are restricted to an ellipsoidal uncertainty set. The proposed method models the effects of uncertainty more accurately than the linearization technique developed by Diehl et al. in 2006 and Zhang in 2007. The need for better approximations as well as the applicability of the newly developed second-order approach is shown by considering discretized geometry optimization problems from structural mechanics, where uncertainty in the acting forces is examined. The applicability and the efficiency of the presented robust optimization methods are demonstrated by giving numerical results for the shape optimization of two generic load-carrying mechanical components. We compare non-robust and robust solutions by considering the associated nominal and worst-case objective function values.