An adaptive strategy on the error of the objective functions for uncertainty-based derivative-free optimization

Abstract

In this work, a strategy is developed to deal with the error affecting the objective functions in uncertainty-based optimization. We refer to the problems where the objective functions are the statistics of a quantity of interest computed by an uncertainty quantification technique that propagates some uncertainties of the input variables through the system under consideration. In real problems, the statistics are computed by a numerical method and therefore they are affected by a certain level of error, depending on the chosen accuracy. The errors on the objective function can be interpreted with the abstraction of a bounding box around the nominal estimation in the objective functions space. In addition, in some cases the uncertainty quantification methods providing the objective functions also supply the possibility of adaptive refinement to reduce the error bounding box. The novel method relies on the exchange of information between the outer loop based on the optimization algorithm and the inner uncertainty quantification loop. In particular, in the inner uncertainty quantification loop, a control is performed to decide whether a refinement of the bounding box for the current design is appropriate or not. In single-objective problems, the current bounding box is compared to the current optimal design. In multi-objective problems, the decision is based on the comparison of the error bounding box of the current design and the current Pareto front. With this strategy, fewer computations are made for clearly dominated solutions and an accurate estimate of the objective function is provided for the interesting, non-dominated solutions. The results presented in this work prove that the proposed method improves the efficiency of the global loop, while preserving the accuracy of the final Pareto front

Surrogate-assisted Bounding-Box approach for optimization problems with tunable objectives fidelity

“…The Bounding-Box approach proposed in [22] is a strategy permitting to choose for each design a fidelity level l to compute the fitness function. It aims at optimizing the trade-off between accuracy and computational cost and relies on the Boxed Pareto dominance rule defined above.…”

“…In [22], the recursive Bounding-Box approach shows very good parsimony only during the first optimization iterations. Afterwards, most visited designs are located close to the Pareto front and require high-fidelity computations.…”

“…To this extent, the classical and recursive approaches are presented. The later has been developed in [22]. Note that the following development is written in the design space, but can easily be transposed to the objective space.…”

“…They also introduce an extended concept of the Pareto dominance rule. In Fusi et al [22], these concepts have been used for online resource allocation in the context of robust and reliability-based optimization, where the objective functions are statistics of a quantity of interest. The Bounding-Box approach allows to tune the fidelity of the statistics computation depending on the closeness to the Pareto front.…”

“…The first contribution of this paper relies on providing some mathematical evidence about the convergence of the Bounding-Box approach for multi-objective optimization problem with adaptively tuned objective values. The work [22] revealed that while substantial gains can be obtained in the early stages of the optimization, the convergence of the optimizer toward the Pareto front lessen the efficiency of the Bounding-Box approach. This behavior is due to the increasing number of efficient designs when the optimizer finds the optimal area.…”