Abstract
In this paper, we consider a multi-fidelity optimization under two types of constraints: prior constraints and posterior constraints. The prior constraints are prerequisite to execution of the simulation that computes the objective function value and the posterior constraint violation values, and are evaluated independently from the simulation with significantly lower computational time than the simulation. We have several simulators that approximately simulate the objective and constraint violation values with different trade-offs between accuracy and computational time. We propose an approach to solve the described constrained optimization problem with as little computational time as possible by utilizing multiple simulators. Based on a covariance matrix adaptation evolution strategy, we combines three algorithmic components: prior constraint handling technique, posterior constraint handling technique, and adaptive simulator selection technique for multi-fidelity optimization. We apply the proposed approach to a compliance minimization problem and show a promising convergence behavior.
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- 1.
We remark that if a simulator \(\mathcal {S}_k\) is computationally very cheap to evaluate, the computation time for \(\textsc {evaluate}_{k}\) may be dominated by the time for \(\textsc {repair}_\text {pri}\), which is computationally demanding compared to the other parts as it internally solves an minimization problem. In such a case, it may be more reasonable to record the computational time for \(\textsc {evaluate}_{k}\) to \(t_k\), rather than the total time of \(\mathcal {S}_{k}\) in \(\textsc {evaluate}_{k}\). This is also expected to affect the performance of the proposed strategy when it is implemented on a parallel machine, especially when the computational time for Line 2 to 6 in Algorithm 6 differs substantially among \(\ell = 1, \dots , \lambda \). It is highly involved with parallel computation and is out of scope of this paper. Further studies on this line is left for future work.
- 2.
Both the original MCR and MCR-mod rank the candidate solutions based only on the constraint violations when the candidates are all infeasible. On one hand it is nice as the search distribution is forced to sample feasible solutions with high probability. On the other hand, we empirically observe that it is trapped by a sub-optimal feasible point since it tends to satisfy the constraints at the beginning of the search without considering the objective function landscape. A possible solution is to include the objective function as a posterior constraint as \(h(\mathcal {S}(x)) = f(\mathcal {S}(x)) - f_\mathrm {thre}\), where \(f_\mathrm {thre}\) is the worst allowable objective value. In engineering optimization, it is often the case that the objective is to find a solution that satisfies all demands. In such a situation, \(f_\mathrm {thre}\) is available.
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Acknowledgement
This work is partially supported by JSPS KAKENHI Grant Number 19H04179 and 19J21892. We thank anonymous reviewers for their valuable comments.
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Akimoto, Y., Sakamoto, N., Ohtani, M. (2020). Multi-fidelity Optimization Approach Under Prior and Posterior Constraints and Its Application to Compliance Minimization. In: Bäck, T., et al. Parallel Problem Solving from Nature – PPSN XVI. PPSN 2020. Lecture Notes in Computer Science(), vol 12269. Springer, Cham. https://doi.org/10.1007/978-3-030-58112-1_6
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