How to solve a design centering problem

Abstract

This work considers the problem of design centering. Geometrically, this can be thought of as inscribing one shape in another. Theoretical approaches and reformulations from the literature are reviewed; many of these are inspired by the literature on generalized semi-infinite programming, a generalization of design centering. However, the motivation for this work relates more to engineering applications of robust design. Consequently, the focus is on specific forms of design spaces (inscribed shapes) and the case when the constraints of the problem may be implicitly defined, such as by the solution of a system of differential equations. This causes issues for many existing approaches, and so this work proposes two restriction-based approaches for solving robust design problems that are applicable to engineering problems. Another feasible-point method from the literature is investigated as well. The details of the numerical implementations of all these methods are discussed. The discussion of these implementations in the particular setting of robust design in engineering problems is new.

Appendices

Appendix

Convergence of bounding method from Sect. 5.1.1

Let $[\mathbf {v}^U, \mathbf {w}^L] \subset [\mathbf {v}^L, \mathbf {w}^U] \subset [{\bar{\mathbf {y}}}^L, {\bar{\mathbf {y}}}^U] \subset \mathbb {R}^{n_y}$. Let

$$\begin{aligned} Y_j^{v}&= [v_1^L, w_1^U] \times \dots \times [v_{j-1}^L, w_{j-1}^U] \times [v_j^L, v_j^U] \times {[}v_{j+1}^L, w_{j+1}^U] \times \dots \times [v_{n_y}^L, w_{n_y}^U],\\ Y_j^{w}&= [v_1^L, w_1^U] \times \dots \times [v_{j-1}^L, w_{j-1}^U] \times [w_j^L, w_j^U] \times [v_{j+1}^L, w_{j+1}^U] \times \dots \times [v_{n_y}^L, w_{n_y}^U], \end{aligned}$$

for each $j \in \{1, \dots , n_y\}$. Then it is easy to see that the “outer” interval $[\mathbf {v}^L, \mathbf {w}^U]$ is a subset of $[\mathbf {v}^U, \mathbf {w}^L] \cup \big ( \bigcup _j (Y_j^v \cup Y_j^w) \big )$; for any $\mathbf {y}\in [\mathbf {v}^L, \mathbf {w}^U]$, each component $y_j$ lies in one of $[v_j^L, v_j^U]$, $[v_j^U, w_j^L]$, or $[w_j^L,w_j^U]$, which is included in the definition of one of $Y_j^v$, $[\mathbf {v}^U, \mathbf {w}^L]$, or $Y_j^w$.

Thus

$$\begin{aligned} \mathrm {vol}([\mathbf {v}^L, \mathbf {w}^U])&\le \mathrm {vol}([\mathbf {v}^U, \mathbf {w}^L]) + \sum _j \mathrm {vol}(Y_j^v) + \mathrm {vol}(Y_j^w) \\&= \mathrm {vol}([\mathbf {v}^U, \mathbf {w}^L]) + \sum _j ( (v_j^U - v_j^L) + (w_j^U - w_j^L) ) \prod _{k \ne j} (w_k^U - v_k^L). \end{aligned}$$

Let $\alpha = \,\mathrm {diam}([{\bar{\mathbf {y}}}^L, {\bar{\mathbf {y}}}^U])$. Then $(w_k^U - v_k^L) \le \alpha$ for each k. If $X' = [\mathbf {v}^L,\mathbf {v}^U] \times [\mathbf {w}^L, \mathbf {w}^U]$, then $(v_j^U - v_j^L) + (w_j^U - w_j^L) \le 2 \,\mathrm {diam}(X')$ for each j. Putting all these inequalities together one obtains

$$\begin{aligned} \mathrm {vol}([\mathbf {v}^L, \mathbf {w}^U]) - \mathrm {vol}([\mathbf {v}^U, \mathbf {w}^L]) \le 2 n_y \alpha ^{n_y-1} \,\mathrm {diam}(X'). \end{aligned}$$

Thus assuming $[\mathbf {y}^L,\mathbf {y}^U] \subset [{\bar{\mathbf {y}}}^L, {\bar{\mathbf {y}}}^U]$ for all $(\mathbf {y}^L,\mathbf {y}^U) \in X$, this establishes that the bounding method described in § 5.1.1 is at least first-order convergent.