Optimization is Not Enough: Why Problem Formulation Deserves Equal Attention

TL;DR

The paper investigates how problem formulation influences black-box optimization outcomes in the topology and fiber-path design of laminated composites. By coupling Moving Morphable Components (MMC) with Lamination Parameters (LP) and using the Lamination Parameter Interpolation Method (LPIM), it enables a compact, physically grounded design space where topology and material steering can be optimized concurrently or sequentially under a volume constraint and connectivity penalties. Empirical results show that context-agnostic, concurrent optimization often yields infeasible or suboptimal designs, while a sequential, domain-informed approach yields better performance and interpretability, especially within limited evaluation budgets. The work advocates for embedding domain knowledge in optimization pipelines and calls for new benchmarks that reward physically informed, context-aware strategies in black-box optimization for engineering design.

Abstract

Black-box optimization is increasingly used in engineering design problems where simulation-based evaluations are costly and gradients are unavailable. In this context, the optimization community has largely analyzed algorithm performance in context-free setups, while not enough attention has been devoted to how problem formulation and domain knowledge may affect the optimization outcomes. We address this gap through a case study in the topology optimization of laminated composite structures, formulated as a black-box optimization problem. Specifically, we consider the design of a cantilever beam under a volume constraint, intending to minimize compliance while optimizing both the structural topology and fiber orientations. To assess the impact of problem formulation, we explicitly separate topology and material design variables and compare two strategies: a concurrent approach that optimizes all variables simultaneously without leveraging physical insight, and a sequential approach that optimizes variables of the same nature in stages. Our results show that context-agnostic strategies consistently lead to suboptimal or non-physical designs. In contrast, the sequential strategy yields better-performing and more interpretable solutions. These findings underscore the value of incorporating, when available, domain knowledge into the optimization process and motivate the development of new black-box benchmarks that reward physically informed and context-aware optimization strategies.

Figures (10)

• Figure 1: Schematics of the cantilever beam test case. The depicted elements have a length of 1 in both $x$ and $y$ directions.
• Figure 2: Parametrization and LSF of a single MMC. Adapted from raponi_kriging-assisted_2019.
• Figure 3: Examples of Interpolation Curves by using the Lamination Parameter Interpolation Method. Adapted from serhat_lamination_2019.
• Figure 4: Master nodes used for the interpolation of LPs.
• Figure 6: Convergence plot of the algorithms under concurrent (solid lines) and sequential (dashed lines) approaches, averaged over 20 runs. The black dotted line marks the transition between the two stages of the sequential strategy.