Abstract
ABSTRACTIt is important to design engineering systems to be robust with respect to uncertainties in the design process. Often, this is done by considering statistical moments, but over-reliance on statistical moments when formulating a robust optimization can produce designs that are stochastically dominated by other feasible designs. This article instead proposes a formulation for optimization under uncertainty that minimizes the difference between a design's cumulative distribution function and a target. A standard target is proposed that produces stochastically non-dominated designs, but the formulation also offers enough flexibility to recover existing approaches for robust optimization. A numerical implementation is developed that employs kernels to give a differentiable objective function. The method is applied to algebraic test problems and a robust transonic airfoil design problem where it is compared to multi-objective, weighted-sum and density matching approaches to robust optimization; several advantages over these existing methods are demonstrated.
Highlights
A traditional optimization considers a quantity of interest of a system q as a function of controllable design variables x, to find a design that minimizes q
The method is applied to algebraic test problems and a robust transonic airfoil design problem where it is compared to multi-objective, weighted-sum and density matching approaches to robust optimization; several advantages over these existing methods are demonstrated
A pure MO approach can be used in order to find the robust Pareto front (Dodson and Parks 2009; Ghisu, Jarrett, and Parks 2011; Keane 2009; Lee, Periaux, Onate, Gonzalez, and Qin 2011), but this is computationally expensive, so often they are combined into a single objective using a weighted sum (WS) (Lee and Kwon 2006; Padulo, Campobasso, and Guenov 2011; Zhang and Hosder 2013)
Summary
A traditional optimization considers a quantity of interest of a system q as a function of controllable design variables x, to find a design that minimizes q. The importance of including uncertainties in the design process is becoming increasingly recognized: a designer instead defines a measure of the behaviour of the quantity of interest q under uncertainty as the objective function to optimize. Formulating this problem effectively for engineering design is the field of robust optimization (RO); a good overview of available RO methods is provided in Beyer and Sendhoff (2007). The qualitative purpose of doing a basic robust optimization is taken to be finding a design that maximizes the likelihood of achieving as good a performance as possible, in which case it makes more sense to deal with probability distributions in their entirety instead of considering μ and σ 2 as competing
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