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On three concepts in robust design optimization: absolute robustness, relative robustness, and less variance

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Abstract

This paper provides a clear perspective on existing several different approaches to robust design optimization of structures. We primarily consider three approaches: the worst-case optimization, the discrepancy (i.e., the maximum gap between the objective values in a nominal case and a possibly occurring case) minimization, and the variance minimization. Some other formulations can also be linked with one of these three approaches. To investigate how the solutions derived by these three approaches differ from each other, we present two numerical examples. This direct comparison clarifies different features of these approaches.

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Notes

  1. It should be clear again that, in general, \(\hat {x}_{\text {b}}^{\prime }\) is different both from \(x^{*}_{\text {b}}\) and \(\hat {x}_{\text {b}}\).

  2. Semidefinite programming (SDP) is the optimization of a linear objective function under a linear matrix inequality. Namely, it has the form

    $$ \begin{array}{@{}rcl@{}} &&{\kern-.7pc} \text{{Minimize}} {\quad} \sum\limits_{i=1}^{m} b_{i} y_{i} \\ && {\kern-.7pc}\text{{subject~to}}\quad \sum\limits_{i=1}^{m} x_{i} A_{i} + C \succeq 0 , \end{array} $$

    where \(A_{1},\dots ,A_{m}\), \(C \in \mathbb {R}^{n \times n}\) are constant symmetric matrices and \(b_{1},\dots ,b_{m} \in \mathbb {R}\) are constants.

  3. The stiffness matrix of a truss satisfies K(x + y) = K(x) + K(y) for any x, y ∈ X. Therefore, for any λ ∈ [0, 1], we have

    $$ \begin{array}{@{}rcl@{}} &&{\kern-.6pc}{\pi}(\lambda {\boldsymbol{x}} + (1-\lambda){\boldsymbol{y}};{\boldsymbol{q}} ) \\ &&{\kern-.7pc}= \sup_{{\boldsymbol{u}}} \left\{ 2 {\boldsymbol{q}}^{\top} {\boldsymbol{u}} - \left[ \lambda {\boldsymbol{u}}^{\top} K({\boldsymbol{x}}) {\boldsymbol{u}} + (1-\lambda) {\boldsymbol{u}}^{\top} K({\boldsymbol{y}}) {\boldsymbol{u}} \right] \right\} \\ &&{\kern-.7pc}\le \lambda \sup_{{\boldsymbol{u}}} \{ 2 {\boldsymbol{q}}^{\top} {\boldsymbol{u}} - {\boldsymbol{u}}^{\top} K({\boldsymbol{x}}) {\boldsymbol{u}} \} + (1-\lambda) \sup_{{\boldsymbol{u}}} \{ 2 {\boldsymbol{q}}^{\top} {\boldsymbol{u}} - {\boldsymbol{u}}^{\top} K({\boldsymbol{y}}) {\boldsymbol{u}} \} \\ &&{\kern-.7pc}= \lambda {\pi}({\boldsymbol{x}} ;{\boldsymbol{q}} ) + (1-\lambda) {\pi}({\boldsymbol{y}};{\boldsymbol{q}} ) , \end{array} $$

    which shows the convexity of π(⋅; q). See, e.g., Kanno (2011, Proposition 3.1.2) for more accounts.

  4. Using a vector-valued function as the objective function in (5a) means that this problem is a multi-objective optimization problem (particularly, problem (5a) is a bi-objective optimization problem). That is, we attempt to minimize both \({\pi }(\textit {\textbf {x}}; \tilde {\textit {\textbf {q}}})\) and πworst(x; α). We call x∗∈ X a Pareto optimal solution of (5a) if there exists no \({\boldsymbol {x}}^{\prime } \in \mathbb {R}^{n}\)\(({\boldsymbol {x}}^{\prime } \not = {\boldsymbol {x}}^{*})\) satisfying \({\pi }({\boldsymbol {x}}^{\prime }; \tilde {{\boldsymbol {q}}}) \le {\pi }({\boldsymbol {x}}^{*}; \tilde {{\boldsymbol {q}}})\) and \({\pi }^{{\text {worst}}}({\boldsymbol {x}}^{\prime }; \alpha ) \le {\pi }^{{\text {worst}}}({\boldsymbol {x}}^{*}; \alpha )\). See, e.g., Boyd and Vandenberghe (2004, Section 4.7) and Chankong et al. (1985) for fundamentals of multi-objective optimization.

  5. We adopt the constraint method (a.k.a. ε-constraint method) for scalarization of multi-objective optimization. Application of standard constraint method converts problem (5a) into problem (6a). See, e.g., Haimes et al. (1971), Haimes and Hall (1974), Cohon and Marks (1974), Hwang et al. (1980), Chankong et al. (1985) for details of the constraint method.

  6. We make use of the equality

    $$ \begin{array}{@{}rcl@{}} &&{\kern-.7pc}\sup\{ {\boldsymbol{x}}^{\top} {\boldsymbol{y}} \mid \| {\boldsymbol{x}} \|_{\infty} \le 1 \} = \sup\{ {\boldsymbol{x}}^{\top} {\boldsymbol{y}} \mid \| {\boldsymbol{x}} \|_{\infty} = 1 \} \\ &&~~~~~{\kern-.7pc}= \sum\limits_{i=1}^{n} \sup\{ x_{i} y_{i} \mid |x_{i}| = 1 \} = \| {\boldsymbol{y}} \|_{1} . \end{array} $$

    Here, ∥⋅∥1 is called the dual norm of \(\| \cdot \|_{\infty }\).

  7. Indeed, for any x, y ∈ X and for any λ ∈ [0, 1], we have

    $$ \begin{array}{@{}rcl@{}} &&{\kern-.7pc}{\pi}^{{\text{worst}}}(\lambda {\boldsymbol{x}} + (1-\lambda) {\boldsymbol{y}} ;\alpha) \\ &&~~~~~~{\kern-.7pc}= \sup_{{\boldsymbol{q}} \in Q(\alpha)} \{ {\pi}(\lambda {\boldsymbol{x}} + (1-\lambda) {\boldsymbol{y}} ;{\boldsymbol{q}}) \} \\ &&~~~~~~{\kern-.7pc}\le \sup_{{\boldsymbol{q}} \in Q(\alpha)} \{ \lambda {\pi}({\boldsymbol{x}} ;{\boldsymbol{q}}) + (1-\lambda) {\pi}({\boldsymbol{y}} ;{\boldsymbol{q}}) \} \\ &&~~~~~~{\kern-.7pc}\le \lambda \sup_{{\boldsymbol{q}} \in Q(\alpha)} \{ {\pi}({\boldsymbol{x}} ;{\boldsymbol{q}}) \} + (1-\lambda) \sup_{{\boldsymbol{q}} \in Q(\alpha)} \{ {\pi}({\boldsymbol{y}} ;{\boldsymbol{q}}) \} \\ &&~~~~~~{\kern-.7pc}\le \lambda {\pi}^{{\text{worst}}}({\boldsymbol{x}} ;\alpha) + (1-\lambda) {\pi}^{{\text{worst}}}({\boldsymbol{y}} ;\alpha) , \end{array} $$

    where the first inequality follows the convexity of π(⋅; q). See, e.g., Hiriart-Urruty and Lemaréchal (1993, Proposition IV.2.1.2) for more accounts.

  8. See also Calafiore and Dabbene (2008, Proposition 2).

  9. We have \(\sup \{ {\boldsymbol {x}}^{\top } {\boldsymbol {y}} \mid \| {\boldsymbol {x}} \|_{1} \le 1 \} = \| {\boldsymbol {y}} \|_{\infty }\), because dual of the dual norm is the norm itself; see, e.g., Hiriart-Urruty and Lemaréchal (1993, Proposition V.3.2.1).

  10. DC decomposition of a DC function is not unique. For example, for any convex function \(f : \mathbb {R}^{n} \to \mathbb {R}\), we see that πdisc(⋅; α) can be decomposed as

    $$ \begin{array}{@{}rcl@{}} {\pi}^{{\text{disc}}}({\boldsymbol{x}}; \alpha) = ({\pi}^{{\text{worst}}}({\boldsymbol{x}}; \alpha) + f({\boldsymbol{x}})) - ({\pi}({\boldsymbol{x}}; \tilde{{\boldsymbol{q}}}) + f({\boldsymbol{x}})) , \end{array} $$

    which is also a DC decomposition of πdisc(⋅; α).

  11. Indeed, with the first-order approximation of \({\pi }(\tilde {{\boldsymbol {q}}}+{\boldsymbol {\zeta }})\), (18) follows

    $$ \begin{array}{@{}rcl@{}} {\mathrm{E}}[{\pi}(\tilde{{\boldsymbol{q}}}+{\boldsymbol{\zeta}})] \simeq {\mathrm{E}}[ {\pi}(\tilde{{\boldsymbol{q}}}) + \nabla{\pi}(\tilde{{\boldsymbol{q}}})^{\top} {\boldsymbol{\zeta}} ] = {\pi}(\tilde{{\boldsymbol{q}}}) + \nabla{\pi}(\tilde{{\boldsymbol{q}}})^{\top} {\mathrm{E}}[{\boldsymbol{\zeta}}] \end{array} $$

    and (20) follows

    $$ \begin{array}{@{}rcl@{}} &&{\kern-.7pc}{\text{Var}}[{\pi}(\tilde{{\boldsymbol{q}}}+{\boldsymbol{\zeta}})] \\ &&{\kern-.7pc}~~~~~~~= {\mathrm{E}} \left[ ({\pi}(\tilde{{\boldsymbol{q}}}+{\boldsymbol{\zeta}}) - {\mathrm{E}}[{\pi}(\tilde{{\boldsymbol{q}}}+{\boldsymbol{\zeta}})])^{2} \right] \\ &&{\kern-.7pc}~~~~~~~\simeq {\mathrm{E}} \left[ ({\pi}(\tilde{{\boldsymbol{q}}}) + \nabla{\pi}(\tilde{{\boldsymbol{q}}})^{\top} {\boldsymbol{\zeta}} - {\pi}(\tilde{{\boldsymbol{q}}}))^{2} \right] \\ &&{\kern-.7pc}~~~~~~~= {\mathrm{E}}\left[ \nabla{\pi}(\tilde{{\boldsymbol{q}}})^{\top} ({\boldsymbol{\zeta}} {\boldsymbol{\zeta}}^{\top}) \nabla{\pi}(\tilde{{\boldsymbol{q}}}) \right] \\ &&{\kern-.7pc}~~~~~~~= \nabla{\pi}(\tilde{{\boldsymbol{q}}})^{\top} {\mathrm{E}}\left[ {\boldsymbol{\zeta}} {\boldsymbol{\zeta}}^{\top} \right] \nabla{\pi}(\tilde{{\boldsymbol{q}}}) . \end{array} $$

    Here, the explicit dependency of π on x has been suppressed for notational simplicity.

  12. It is clear that \(\max \limits \{ {\pi }(\textit {\textbf {x}};\textit {\textbf {q}}) \mid \textit {\textbf {q}} \in Q(\alpha ) \}\), as well as \(\min \limits \{ {\pi }({\boldsymbol {x}};{\boldsymbol {q}}) \mid {\boldsymbol {q}} \in Q(\alpha ) \}\), is attained at a point on the boundary of Q(α).

  13. For α = 10kN, there exists almost no difference between the solution with minimal nominal-case compliance and the solution with minimal worst-case compliance.

  14. In the following we assume without loss of generality that the original design optimization problem is formulated as a minimization problem, as the case in the previous sections.

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Funding

The work described in this paper is partially supported by Research Grant from the Maeda Engineering Foundation and JSPS KAKENHI 17K06633.

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  1. Mathematics and Informatics Center, The University of Tokyo, Hongo 7-3-1, Tokyo, 113-8656, Japan

    Yoshihiro Kanno

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Replication of results

Source codes for solving the optimization problems presented in Section 5 are available online at https://github.com/ykanno22/relative_robust/.

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Kanno, Y. On three concepts in robust design optimization: absolute robustness, relative robustness, and less variance. Struct Multidisc Optim 62, 979–1000 (2020). https://doi.org/10.1007/s00158-020-02503-9

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