A semidefinite programming approach to robust truss topology optimization under uncertainty in locations of nodes

Abstract

This paper addresses truss topology optimization taking into account robustness to uncertainty in the truss geometry. Specifically, the locations of nodes are assumed not to be known precisely and the compliance in the worst case is attempted to be minimized. We formulate a semidefinite programming problem that serves as a safe approximation of this robust optimization problem. That is, any feasible solution of the presented semidefinite programming problem satisfies the constraints of the original robust optimization problem. Since a semidefinite programming problem can be solved efficiently with a primal-dual interior-point method, we can find a robust truss design efficiently with the proposed semidefinite programming approach. A notable property of the proposed approach is that the obtained truss is guaranteed to be stable. Numerical experiments are performed to illustrate that the optimal truss topology depends on the magnitude of uncertainty.

Appendix: Technical lemmas

This section summarizes technical prerequisites that are known in literature.

We begin with three lemmas related to vector and matrix norms. These lemmas are used in Section 3.2. For p satisfying 1 ≤ p ≤ ∞, define p ^∗ by

$$\frac{1}{p} + \frac{1}{p^{*}} = 1 , $$

where we use the convention 1/∞ = 0. The ℓ _p ^∗-norm is the dual norm of the ℓ _p -norm. For any x, $\boldsymbol {y} \in \mathbb {R}^{n}$, we have that

$$ {\boldsymbol{x}}^{\top}{\boldsymbol{y}} \le \|{\boldsymbol{x}}\|_{p} \|{\boldsymbol{y}}\|_{p^{*}}, $$

(64)

which is known as the Hölder inequality (Horn and Johnson 1985; Steele 2004). For any ${\boldsymbol {x}} \in \mathbb {R}^{n}$, there exists ${\boldsymbol {y}} \in \mathbb {R}^{n}$ satisfying (64) with equality and ∥y∥ _p ^∗=1. We denote this y by ψ _p (x).

Lemma 4

Let $\mathbb {R} \ni s \ge 0$ be a constant, ${\boldsymbol {x}}\in {\mathbb {R}}^{n}$ and 1 ≤ p ≤ ∞. Then

$$\max_{{\boldsymbol{y}}\in {\mathbb{R}}^{n}} \{ {\boldsymbol{x}}^{\top} {\boldsymbol{y}} : \|{\boldsymbol{y}}\|_{p^{*}} \le s \} = s \| {\boldsymbol{x}} \|_{p}. $$

Proof

For any ${\boldsymbol {x}}\in {\mathbb {R}}^{n}$ and ${\boldsymbol {y}}\in {\mathbb {R}}^{n}$ satisfying ∥y∥ _p ^∗ ≤ s, it follows from the Hölder inequality that

$${\boldsymbol{x}}^{\top} {\boldsymbol{y}} \le \left\|{\boldsymbol{x}}\right\|_{p} \left\|{\boldsymbol{y}}\right\|_{p^{*}} \le s \left\|{\boldsymbol{x}}\right\|_{p}. $$

Here, two inequalities are satisfied with equalities at y = s ψ _p (x), which concludes the proof. □;

Lemma 4 is used in the proof of Theorem 1 in Section 3.2.

Lemma 5

Let $x \in \mathbb {R}$ , ${\boldsymbol {y}} \in \mathbb {R}^{n}$ and $\mathbb {R} \ni r > 0$ . Then

$$\max_{{\boldsymbol{z}}\in {\mathbb{R}}^{n}} \{ x {\boldsymbol{y}}^{\top} {\boldsymbol{z}} : \|{\boldsymbol{z}}\|_{2} \le r \} = r|x| \| {\boldsymbol{y}} \|_{2}. $$

Proof

The Cauchy–Schwarz inequality implies

$$x {\boldsymbol{y}}^{\top} {\boldsymbol{z}} = {\boldsymbol{y}}^{\top} (x {\boldsymbol{z}}) \le \|{\boldsymbol{y}}\|_{2} \| x{\boldsymbol{z}}\|_{2}. $$

From ∥z∥₂ ≤ r, we obtain

$$\| x{\boldsymbol{z}} \|_{2} \le |x| \|{\boldsymbol{z}}\|_{2} \le |x| r. $$

These inequalities are satisfied with equalities at

$${\boldsymbol{z}} = \left\{\begin{array}{ll} r {\boldsymbol{\psi}}_{2}({\boldsymbol{y}}) & \text{if}\;\; x \ge 0, \\ -r {\boldsymbol{\psi}}_{2}({\boldsymbol{y}}) & \text{if} \;\;x < 0, \\ \end{array}\right. $$

which concludes the proof. □;

Lemma 5 is used in the proof of Theorem 1 in Section 3.2.

For $A \in \mathbb {R}^{m \times n}$, the matrix norm induced by the Euclidean vector norm is defined by

$$ \| A \|_{2} = \max_{{\boldsymbol{x}}\ne {\boldsymbol{0}}} \frac{\| A{\boldsymbol{x}} \|_{2}}{\|{\boldsymbol{x}}\|_{2}} , $$

(65)

which is equal to the largest singular value of A. From this definition we immediately obtain the following property.

Lemma 6

Let $A\in {\mathbb {R}}^{m \times n}$ and ${\boldsymbol {x}}\in {\mathbb {R}}^{n}$ . Then

$$\| A{\boldsymbol{x}} \|_{2} \le \|A\|_{2} \|{\boldsymbol{x}}\|_{2}. $$

Lemma 6 is used to show Lemma 1 in Section 3.2.

In the following we establish Lemma 9, which is used in Section 3.2 to formulate the conservative approximation of the robust optimization problem. Two lemmas, Lemma 7 and Lemma 8, prepare for Lemma 9. Firstly, we state the following fact, which is the obvious part of the well-known S-lemma; see, e.g., Boyd et al. (1987) and Pólik and Terlaky (2007).

Lemma 7

Let f ₀ , $f_{1},\dots ,f_{m}:\mathbb {R}^{n} \to \mathbb {R}$ be quadratic functions. The implication

$$ f_{1}({\boldsymbol{x}})\ge 0,\dots,f_{m}({\boldsymbol{x}})\ge 0 \quad\Rightarrow\quad f_{0}({\boldsymbol{x}})\ge 0 $$

(66)

holds if there exist t ₁ ,…,t _m ≥ 0 satisfying

$$f_{0}({\boldsymbol{x}})\ge \sum\limits_{i=1}^{m} t_{i}f_{i}({\boldsymbol{x}}) \quad (\forall {\boldsymbol{x}}\in{\mathbb{R}}^{n}). $$

Proof

Suppose that (66) does not hold, which means that there exists $\hat {{\boldsymbol {x}}} \in \mathbb {R}^{n}$ satisfying

$$f_{0}(\hat{{\boldsymbol{x}}}) < 0 , f_{1}(\hat{{\boldsymbol{x}}}) \ge 0 , {\dots} , f_{m}(\hat{{\boldsymbol{x}}}) \ge 0. $$

Then we obtain

$$f_{0}(\hat{{\boldsymbol{x}}}) < 0 \le \sum\limits_{i=1}^{m} t_{i} f_{i}(\hat{{\boldsymbol{x}}}) \quad (\forall t_{1},\dots,t_{m} \ge 0). $$

Thus the contraposition of Lemma 7 has been shown. □;

The following fact can be found, e.g., in Calafiore and El Ghaoui (2004).

Lemma 8

Let $Q\in {\mathcal {S}^{n}}$ , ${\boldsymbol {p}}\in {\mathbb {R}}^{n}$ and $r\in {\mathbb {R}}$ . Then the following two conditions are equivalent:

$$ \mathrm{(i)}\;\;\left[\begin{array}{l} 1 \\ {\boldsymbol{x}} \\ \end{array}\right]^{\top} \left[\begin{array}{l} r \;\;{\boldsymbol{p}}^{\top} \\ {\boldsymbol{p}} \;\;Q \\ \end{array}\right] \left[\begin{array}{c} 1 \\ {\boldsymbol{x}} \\ \end{array}\right] \ge 0, \quad \forall {\boldsymbol{x}}\in{\mathbb{R}}^{n}, $$

(67)

$$ \mathrm{(ii)} \;\;\left[\begin{array}{l} r \;\;{\boldsymbol{p}}^{\top} \\ {\boldsymbol{p}}\;\; Q \\ \end{array}\right] \succeq O. $$

(68)

Proof

It is trivial that (ii) implies (i). We show that (i) implies (ii) by contradiction. Suppose that (ii) does not hold, which means that there exists $(\hat {\xi }, \hat {{\boldsymbol {x}}}) \in \mathbb {R} \times \mathbb {R}^{n}$ satisfying

$$ \left[\begin{array}{c} \hat{\xi} \\ \hat{{\boldsymbol{x}}} \\ \end{array}\right]^{\top} \left[\begin{array}{c} r \;\;{\boldsymbol{p}}^{\top} \\ {\boldsymbol{p}} \;\;Q \end{array}\right] \left[\begin{array}{c} \hat{\xi} \\ \hat{{\boldsymbol{x}}} \\ \end{array}\right] < 0. $$

(69)

If $\hat {\xi } \not = 0$, then (69) is reduced to

$$\left[\begin{array}{c} \hat{{\boldsymbol{x}}}/\hat{\xi} \\ 1 \\ \end{array}\right]^{\top} \left[\begin{array}{l} r \;\;{\boldsymbol{p}}^{\top} \\ {\boldsymbol{p}} \;\;Q \\ \end{array}\right] \left[\begin{array}{c} \hat{{\boldsymbol{x}}}/\hat{\xi} \\ 1 \\ \end{array}\right] < 0 , $$

which contradicts with (i). Hence, suppose $\hat {\xi }=0$. Then (69) is reduced to

$$ \hat{{\boldsymbol{x}}}^{\top} Q \hat{{\boldsymbol{x}}} < 0. $$

(70)

Let $\gamma \in \mathbb {R}$ be a parameter and choose ${\boldsymbol {x}} = \gamma \hat {{\boldsymbol {x}}}$ in (i) to reduce the left-hand side to

$$ \left(\hat{{\boldsymbol{x}}}^{\top} {Q} \hat{{\boldsymbol{x}}}\right) \gamma^{2} + 2 \left({\boldsymbol{p}}^{\top} \hat{{\boldsymbol{x}}}\right) \gamma + r. $$

(71)

The left-hand side of (71) is a quadratic function of γ and (70) implies that (71) is not bounded below. Therefore, there exists γ such that (71) becomes negative, which contradicts (i). □;

The following fact is obtained as a straightforward corollary of Lemma 7 and Lemma 8.

Lemma 9

Let f ₀ , $f_{1},\dots ,f_{m} : \mathbb {R}^{n} \to \mathbb {R}$ be quadratic functions defined by

$$f_{i}({\boldsymbol{x}}) = \left[\begin{array}{c} 1 \\ {\boldsymbol{x}} \\ \end{array}\right]^{\top} \left[\begin{array}{c} r_{i} \;\;{\boldsymbol{p}}_{i}^{\top} \\ {\boldsymbol{p}}_{i}\;\; {Q}_{i} \end{array}\right] \left[\begin{array}{c} 1 \\ {\boldsymbol{x}} \\ \end{array}\right] , \quad i=0,1,\dots,m , $$

where ${Q}_{i} \in {\mathcal {S}}^{n}$ , ${\boldsymbol {p}}_{i} \in \mathbb {R}^{n}$ , $r_{i} \in \mathbb {R}\,(i=0,1,\dots ,m)$ . Then the implication

$$f_{1}({\boldsymbol{x}}) \ge 0,\dots,f_{m}({\boldsymbol{x}}) \ge 0 \quad\Rightarrow\quad f_{0}({\boldsymbol{x}}) \ge 0 $$

holds if there exist t ₁ ,…,t _m ≥ 0 satisfying

$$\left[\begin{array}{c} r_{0} \;\;{\boldsymbol{p}}_{0}^{\top} \\ {\boldsymbol{p}}_{0}\;\; {Q}_{0} \end{array}\right] \succeq\sum\limits_{i=1}^{m} t_{i} \left[\begin{array}{c} r_{i} \;\;{\boldsymbol{p}}_{i}^{\top} \\ {\boldsymbol{p}}_{i} \;\;{Q}_{i} \end{array}\right]. $$

Lemma 9 plays a key role in proving Theorem 1 in Section 3.2.