Concrete motivating cases
We appreciate the reviewer's feedback. Here we provide two concrete examples of robust optimization, hoping they can better clarify the motivation of our paper.
**[Case 1] Design & Manufacture of wireless antenna**
In wireless communication systems (e.g., 4G/WiFi), the antenna is an important device for sending & receiving electromagnetic waves. It typically consists of conductive elements, such as wires, rods, or metal surfaces. These elements are precisely shaped and positioned to determine the antenna's characteristics, *e.g.*, the radiation pattern, polarization, frequency range, and etc. However, during its manufacturing or fabrication process, machine errors occur inevitably because of various factors, such as inaccuracies in the machining equipment, variations in material properties, and imprecise assembly techniques. These errors deviate the design parameters from the desired specifications and can have significant effects on antenna performance [1].
For example, suppose we want to optimize the length of the antenna $x$ to maximize its gain $y$ at a frequency range of $[1.424, 2.75)$ GHz. Let the machine error be described by a probability distribution $P(x)$, the antenna gain of the final product is $f(x^* +\delta), \delta \sim P(x^*)$, where $x^*$ is the global optimum of the target function. For some antennas, the global optimum can be quite "unstable": a small perturbation in $x$ can degrade its performance dramatically (e.g., the RKHS function in Figure 4a of the paper). In these cases, instead of finding the global optimum, we aim to find a robust optimum whose average performance is best under the machine errors:
$$
x^r = \text{argmax}\_{x} \int_\delta f(x+\delta) d\_{P(x)}
$$
Moreover, as the machine errors derive from multiple random sources, its probability distribution $P(x)$ can be quite complex.
**[Case 2] Robust robot grasp**
Yet another motivating case comes from the robot grasp [2]. In this problem, we aim to control a robot hand to grasp an object. In particular, we need to decide the optimal translations and rotations of the robot hand: $\mathbf{x}=(\Delta_x, \Delta_y, \Delta_z, \theta_x, \theta_y, \theta_z)$. However, controller errors and execution noises happen during grasp and deviate hand configuration $x^{\prime} = \mathbf{x} + \mathbf{\delta}, \mathbf{\delta} \sim P(\mathbf{x}) $, preventing the precise positioning of the robot hand in the desired pose for a stable grasp. In addition, the grasp performance is often measured via a wrench space analysis [3] and its discontinuous nature sometimes renders it quite sensitive to a small configuration change, which further complicates this problem.
As such, when planning for a robotic grasp, we must consider the impact of the uncertainty in the input space (positioning errors) to ensure a robust grasp:
$$
\mathbf{x}^r = \text{argmax}\_{x} \int_{\delta} f(x+\delta) d\_{P(x)}
$$
### Summary
In general, both of these two cases involve the design parameters being perturbed before the evaluation by random errors and the targeting function can be quite unstable, i.e., a small change in $x$ may result in a large fluctuation in $y$. This kind of problem is quite common in the real world, say semiconductor design [4], and a more complete application list can be found in [5].
Locating the robust optimum in these tasks can be formed as an optimization problem in the expected form and robust optimization methods can be used. Compared with the other methods, our solution can model complex input distributions comprehensively and perform the optimization efficiently.
**Reference**
[1] Stutzman, Warren L., and Gary A. Thiele. Antenna theory and design. John Wiley & Sons, 2012.
[2] Nogueira, José, et al. "Unscented Bayesian optimization for safe robot grasping." *2016 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)*. IEEE, 2016.
[3] Weisz, Jonathan, and Peter K. Allen. "Pose error robust grasping from contact wrench space metrics." *2012 IEEE international conference on robotics and automation*. IEEE, 2012.
[4] Ng, Tsan Sheng, Yang Sun, and John Fowler. "Semiconductor lot allocation using robust optimization." *European Journal of Operational Research* 205.3 (2010): 557-570.
[5] Gabrel, Virginie, Cécile Murat, and et al. “Recent Advances in Robust Optimization: An Overview.” *European Journal of Operational Research* 235, no. 3 (June 2014): 471–83.