Chapter 13
Risk Measures in the Context of Robust
and Reliability Based Optimization
Elisa Morales Tirado and Domenico Quagliarella
Abstract The base concepts of robust optimization are described and detailed in
this chapter. In particular, the approach based on risk measurements is introduced,
after a first quick review of the classical deterministic approach. In this context,
the use of some special classes of risk measures, used in financial engineering, is
reported along with the main advantages and drawbacks related to their mathematical features. The usage of these risk measures will be then illustrated in an example
problem of robust aerodynamic design optimization. The focus is also given to
advanced techniques for error and confidence interval estimations and how they
can be used in the context of robust optimization to improve the overall efficiency
and effectiveness of the process.
Keywords Risk measure · Robust optimization · Aerodynamic shape design ·
Coherent risk measure
13.1 Introduction
Solving optimization problems in the presence of uncertainty is of fundamental
importance in many research and application fields ranging from economics [20]
to engineering [14] to decision theory. Uncertainty can be present at various
levels in an optimization problem and influence its solution in different ways. An
important and significant source of uncertainty can be, to begin with, intrinsic to the
mathematical model adopted to describe the physical [7], biological or economic
system on whose parameters one wants to intervene to alter its behaviour by
adapting it to their needs. This one is a case of epistemic uncertainty, which has
distinct characters respect to stochastic uncertainty. The latter originates in the
E. M. Tirado () · D. Quagliarella
Italian Aerospace Research Centre, Capua, Italy
e-mail: e.moralestirado@cira.it; d.quagliarella@cira.it
© Springer Nature Switzerland AG 2021
M. Vasile (ed.), Optimization Under Uncertainty with Applications to Aerospace
Engineering, https://doi.org/10.1007/978-3-030-60166-9_13
411
Risk Measures in the Context of Robust
and Reliability Based Optimization
Elisa Morales Tirado and Domenico Quagliarella
Abstract The base concepts of robust optimization are described and detailed in
this chapter. In particular, the approach based on risk measurements is introduced,
after a first quick review of the classical deterministic approach. In this context,
the use of some special classes of risk measures, used in financial engineering, is
reported along with the main advantages and drawbacks related to their mathematical features. The usage of these risk measures will be then illustrated in an example
problem of robust aerodynamic design optimization. The focus is also given to
advanced techniques for error and confidence interval estimations and how they
can be used in the context of robust optimization to improve the overall efficiency
and effectiveness of the process.
Keywords Risk measure · Robust optimization · Aerodynamic shape design ·
Coherent risk measure
13.1 Introduction
Solving optimization problems in the presence of uncertainty is of fundamental
importance in many research and application fields ranging from economics [20]
to engineering [14] to decision theory. Uncertainty can be present at various
levels in an optimization problem and influence its solution in different ways. An
important and significant source of uncertainty can be, to begin with, intrinsic to the
mathematical model adopted to describe the physical [7], biological or economic
system on whose parameters one wants to intervene to alter its behaviour by
adapting it to their needs. This one is a case of epistemic uncertainty, which has
distinct characters respect to stochastic uncertainty. The latter originates in the
E. M. Tirado () · D. Quagliarella
Italian Aerospace Research Centre, Capua, Italy
e-mail: e.moralestirado@cira.it; d.quagliarella@cira.it
© Springer Nature Switzerland AG 2021
M. Vasile (ed.), Optimization Under Uncertainty with Applications to Aerospace
Engineering, https://doi.org/10.1007/978-3-030-60166-9_13
411