Non-smooth optimisation applied to improving steam turbine designs

Thesis

This project is motivated by Siemens’ desire to improve the efficiency of their steam turbines. Siemens has developed a software called 3dv1d which tests a theoretical steam turbine design both for efficiency and practicality. By using 3dv1d to define black box objective and constraint functions, we can find optimal steam turbine designs by applying mathematical optimisation software.

We begin by investigating the optimisation algorithm currently used by Siemens, which is known to yield good steam turbine designs for Siemens even though it does not converge to a point which satisfies optimality conditions. This investigation reveals that this algorithm fails because the efficiency function is not continuously differentiable. Since the algorithm currently used is a smooth optimisation algorithm, this means its failure follows from the fact that it is not suitable to the problem.

Next we explore the literature for existing non-smooth optimisation (NSO) algorithms which may be suitable for solving the problem. We identify one of these as being most applicable, and test its effectiveness only to find that it runs too slowly to be useful in practice.

Our continued exploration of the topic of non-smooth optimisation led us to the particular topic of exact line search for non-smooth optimisation, and more specifically non-smooth 1D optimisation. Here we have noted that many of the objections to using ELS do not apply in the context of NSO. Moreover, while the topic of 1D optimisation has been well developed, there is a surprising absence of any algorithm capable of robust super-linear convergence when applied to 1D NS functions. Our main contribution to mathematics through this project has been to construct a new algorithm to fill this void.

In addition to this, we have also addressed the larger problem motivated by Siemens: to robustly locate local optima for functions like that implied by 3dv1d. While not being as rigorous as in 1D optimisation, we have constructed a heuristic for the purpose of solving problems like Siemens’ which we demonstrate both to perform robustly and often find the best known solution.

Authors: Grant-Peters, J

• DOI: 10.5287/ora-zvx5vpqrp
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: exact linesearch; non-smooth optimisation
• Subjects: Mathematical optimization; Applied mathematics