Surrogate-model-based optimization
Published:
As part of my personal goal to improve my writing skill, I will be posting several notes (or documentation) I used in learning some tools or concepts I utilize in my current research. These topics will be mostly about structural reliability, optimization and disaster risk. Along the way, I will be able to deepen my understanding on the fundamental principles and at the same time it will be good exercise to articulate ideas in a clear and concise manner.
For this post, my topic will be on surrogate-model-based optimization or Bayesian optimization.
Optimization as an engineering tool
I have learned about optimization during my master’s degree studies. I was extremely fascinated by how mimicking the behavior of flocking animals can be used to solve engineering problems. It was very rewarding that we were tasked to code manually our own particle swarm optimizer.
I see optimization techniques as a tool that engineers can (or should) use in design tasks or decision making. One of the main task of a structural engineer is decide the building specifications, that is, materials to be used, dimension and cross-sectional areas of structure and many more. Due to many design choices to make: what materials to use? what should the dimension of the columns, beams, etc.? Structural design is best framed as an optimization problem (OP). Mathematically, it is stated as
\[\begin{aligned} \min_{x} \quad & f(x) \\ \text{subject to} \quad & g_i(x) \le 0, \quad i = 1,\dots,m \\ & h_j(x) = 0, \quad j = 1,\dots,p \end{aligned}\]The goal is to find the best design or optimal design $x^{\ast}$ that will minimized the objective function $f(\cdot)$ and satisfy the constraint functions, $g_i(\cdot)$ and $h_j(\cdot)$. This is a standard constrained optimization problem. In structural optimization, the objective function to minimize is often the construction or material costs of the building while the constraints are related to deformation or stress thresholds. The requirements (geometric, deformation and strength) set by building codes are usually the constraints used in structural optimization.
Solution to the OP stated above are extensively described in [1] and [2]. The primary hurdle in using optimization in engineering design would be the computational cost, that is, either evaluating the objective function or the constraints function. For example, if the design you are doing requires performing a nonlinear analysis then
Reference
- Nocedal, J., & Wright, S. J. (2006). Numerical optimization (2nd ed). Springer.
- Martins, J. R. R. A., & Ning, A. (2022). Engineering design optimization. Cambridge University Press. https://doi.org/10.1017/9781108980647