Design optimization using multiple dominance relations

Abstract

A challenge in engineering design is to choose suitable objectives and constraints from many quantities of interest, while ensuring an optimization is both meaningful and computationally tractable. We propose an optimization formulation that can take account of more quantities of interest than existing formulations, without reducing the tractability of the problem. This formulation searches for designs that are optimal with respect to a binary relation within the set of designs that are optimal with respect to another binary relation. We then propose a method of finding such designs in a single optimization by defining an overall ranking function to use in optimizers, reducing the cost required to solve this formulation. In a design under uncertainty problem, our method obtains the most robust design that is not stochastically dominated faster than a multiobjective optimization. In a car suspension design problem, our method obtains superior designs according to a k-optimality condition than previously suggested multiobjective approaches to this problem. In an airfoil design problem, our method obtains designs closer to the true lift/drag Pareto front using the same computational budget as a multiobjective optimization. K E Y W O R D Saerodynamics, engineering design, optimization, probabilistic methods, shape design, suspension design, transonic *A binary relation is a comparison of two designs that determines whether one is better than the other, and for an optimal design no better design exists with respect to the relation. 3 Int J Numer Methods Eng. 2020;121:2481-2502.wileyonlinelibrary.com/journal/nme

Configuration Optimization of a Hydrogen-Kerosene Hybrid Combustion Aircraft Retrofit

“…For this, a binary relation (denoted by ⪯) can be defined such that a design 𝑥 ∈ X dominates another design 𝑦 ∈ X if 𝑥 ⪯ 𝑦. With this, the set of optimal designs in X with respect to a binary relation can be defined, and is denoted by min(X, ⪯) [25]:…”

“…In Multi-objective Optimization (MO), a candidate design 𝑥 ∈ X dominates another design 𝑦 ∈ X if all 𝑚 objectives in 𝑥 are strictly smaller than the corresponding 𝑚 objectives in 𝑦. This is known as Pareto dominance, and is denoted by 𝑥 ⪯ Pareto 𝑦 [25,29].…”

“…Similarly, a pair of designs is Pareto equivalent if neither design Pareto dominates each other (𝑥 ⪯̸ Pareto 𝑦 and 𝑦 ⪯̸ Pareto 𝑥). The Pareto front consists of the performance vectors of the optimal designs according to ⪯ Pareto , and is denoted PF(X) [25]:…”

“…The alternative proposed in [25] is known as Multiple Dominance Relations (MDR). The key concept behind MDR is to use 𝑛 nested binary relations instead, finding sets of optimized designs within sets of optimized designs [25]:…”

“…Further, the relative importance of the objectives might not necessarily be constant or quantifiable. To address these issues, this investigation will use Multiple Dominance Relations (MDR) [25] within the optimization routine. With this, the goals of the study can now be summarised:…”