A Polynomial Chaos Approach to Robust Multi-Objective Optimization
Almost all real world optimization problems across a wide range of disciplines contain uncertainty. This is why the engineering community at large is increasingly focusing on robust optimization and the quantification of uncertainty. Uncertainty can derive from a variety of sources such as errors in measuring, difficulties in sampling, lack of knowledge, or future events that are not completely known at the time of sampling.
A Deterministic Approach to Multi-Objective Optimization
In conventional sampling, design parameters can often only be determined up to some tolerance level or they may vary according to a probability distribution. Traditional simulation tools produce results for fixed parameters which makes it hard to determine the level of confidence in the simulation results.
Deterministic approaches to multi-objective optimization do not consider the impact of variations due to uncertainty in the output. Consequently the selected optimal solution may be very unstable, sensitive to variations and can be the optimal solution only for a “lucky” configuration and/or only for a short instant.
Finding a Robust Solution to Multi-Objective Optimization
Robust optimization aims at finding a “robust solution” for which performance remains stable when exposed to small perturbations. In “A Polynomial Chaos Approach to Robust Multiobjective Optimization”, Silvia Poles of EnginSoft and Alberto Lovison of the Department of Pure and Applied Mathematics at the University of Padua, focus on the source of uncertainty coming from the stochasticity of the input parameters during the optimization process. They propose an approach based on Polynomial Chaos Expansions (PCE) to quantify and control uncertainty during the optimization process. They apply their method of solving multi-objective optimization problems where the objective functions are not analytically known, and for which evaluations are time consuming.
They concluded that an important advantage of this approach is that when Polynomial Chaos Expansions (PCE) is applied instead of other classical approaches such as Monte Carlo or Latin Hypercube samplings, there is a considerable reduction in the total number of evaluations carried out. This becomes increasingly important when the multiobjective optimization problem requires the evaluation of time consuming functions. In fact polynomial chaos may well be the only possibility that engineers have available to them to obtain robust solutions in the case of time consuming functions.