Numerical investigation of non-hierarchical coordination for distributed multidisciplinary design optimization with fixed computational budget

Abstract

This paper presents a numerical investigation of the non-hierarchical formulation of Analytical Target Cascading (ATC) for coordinating distributed multidisciplinary design optimization (MDO) problems. Since the computational cost of the analyses can be high and/or asymmetric, it is beneficial to understand the impact of the number of ATC iterations required for coordination and the number of iterations required for disciplinary feasibility on the quality of the obtained MDO solution. At each “outer” ATC iteration, the disciplinary optimization subproblems are solved for a predefined maximum number of “inner” loop iterations. The numerical experiments consider different numbers of maximum outer iterations while keeping the total computational budget of analyses constant. Solution quality is quantified by optimality (objective function value) and consistency (violation of coordination-related consistency constraints). Since MDO problems are typically simulation-based (and often blackbox) problems, we compare implementations of the mesh-adaptive direct search optimization algorithm (a derivative-free method with convergence properties) to the gradient-based interior-point algorithm implementation of the popular Matlab optimization toolbox. The impact of the values of two parameters involved in the alternating directions updating scheme of the augmented Lagrangian penalty functions (aka method of multipliers) on solution quality is also investigated. Numerical results are provided for a variety of MDO test problems. The results indicate consistently that a balanced modest number of outer and inner iterations is more effective; moreover, there seems to be a specific combination of parameter value ranges that yield better results.

Appendices

Appendix A: Test problem formulations

In Figs. 10, 11, 12 and 13, an arrow from subproblem i to subproblem j indicates that the output of subproblem i is an input to subproblem j. Double-headed arrows denote variables shared by two subproblems.

Fig. 10

Bi-quadratic problem analysis flow

Fig. 11

Geometric programming problem analysis flow

Fig. 12

Simplified wing design problem analysis flow

Fig. 13

Supersonic Business Jet problem analysis flow

1.1 A.1 Bi-quadratic problem

Original problem:

$$ \begin{array}{ll} \min & (x-1)^{2} + (x+1)^{2} \\ \text{wrt} & x \\ \text{st} & x \in [-100~;+100] \\ \end{array} $$

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Subproblem 1:

$$ \begin{array}{ll} \min & t_{y_{21}} + t_{y_{31}} +\\ & \hspace{0cm} \phi_{y_{21}}(t_{y_{21}}-r_{y_{21}}) + \phi_{y_{31}}(t_{y_{31}}-r_{y_{31}}) \\ \text{wrt} & t_{y_{21}},t_{y_{31}} \\ \end{array} $$

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Subproblem 2:

$$ \begin{array}{ll} \min & \phi_{y_{21}}(t_{y_{21}}-r_{y_{21}}) + \phi_{s_{23}}(x_{s_{232}}-x_{s_{233}}) \\ \text{wrt} & x_{s_{232}} \\ \text{where} & r_{y_{21}} = (x_{s_{232}}-1)^{2} \\ \text{st} & x_{s_{232}} \in [-100~;+100] \\ \end{array} $$

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Subproblem 3:

$$\begin{array}{@{}rcl@{}} \min &&\phi_{y_{31}}(t_{y_{31}}-r_{y_{31}}) + \phi_{s_{23}}(x_{s_{232}}-x_{s_{233}}) \\ \text{wrt} &&x_{s_{233}}\\ \text{where} &&r_{y_{31}} = (x_{s_{233}}+1)^{2} \\ &&\text{st} x_{s_{233}} \in [-100~;+100] \\ \end{array} $$

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1.2 A.2 Geometric programming problem

Original problem:

$$\begin{array}{@{}rcl@{}} \min && {z_{1}^{2}} + {z_{2}^{2}}\\ \text{wrt} &&z_{1}, z_{2}, z_{3}, z_{4}, z_{5}, z_{6}, z_{7}, z_{8}, z_{9}, z_{10}, z_{11}, z_{12}, z_{13}, z_{14} \\ &&\text{st} z_{i} \in [10^{-6}~;10^{6}] ~~\forall i \\ && {z_{1}^{2}} = {z_{3}^{2}}+z_{4}^{-2}+{z_{5}^{2}} \\ && {z_{2}^{2}} = {z_{5}^{2}}+{z_{6}^{2}} +{z_{7}^{2}} \\ && {z_{3}^{2}} = {z_{8}^{2}}+z_{9}^{-2}+z_{10}^{-2}+z_{11}^{2} \\ && {z_{6}^{2}} = z_{11}^{2}+z_{12}^{2}+z_{13}^{2}+z_{14}^{2} \\ && z_{3}^{-2} + {z_{4}^{2}} - {z_{5}^{2}} \leq 0 \\ && {z_{5}^{2}} + z_{6}^{-2} - {z_{7}^{2}} \leq 0 \\ && {z_{8}^{2}} + {z_{9}^{2}} - z_{11,2}^{2} \leq 0 \\ && z_{8}^{-2} + z_{10}^{2} - z_{11}^{2} \leq 0 \\ && z_{11}^{2} + z_{12}^{-2} - z_{13}^{2} \leq 0 \\ && z_{11}^{2} + z_{12}^{2} - z_{14}^{2} \leq 0 \\ \end{array} $$

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Subproblem 1:

$$\begin{array}{@{}rcl@{}} \min && t^{2}_{y_{21}} + t^{2}_{y_{31}} + z_{4}^{-2} + 2 {z_{5}^{2}} + {z_{7}^{2}} +\\ &&\phi_{y_{21}}(t_{y_{21}}-r_{y_{21}}) + \phi_{y_{31}}(t_{y_{31}}-r_{y_{31}}) \\ \text{wrt} && t_{y_{21}},t_{y_{31}}, z_{4}, z_{5}, z_{7} \in [10^{-6}~;10^{6}] \\ \text{st} && t_{y_{21}}^{-2} + {z_{4}^{2}} - {z_{5}^{2}} \leq 0 \\ && {z_{5}^{2}} + t_{y_{31}}^{-2} - {z_{7}^{2}} \leq 0 \\ \end{array} $$

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Subproblem 2:

$$\begin{array}{@{}rcl@{}} \min &&\quad \phi_{y_{21}}(t_{y_{21}}-r_{y_{21}}) + \phi_{s_{23}}(x_{s_{232}}-x_{s_{233}}) \\ \text{wrt} &&\quad z_{8},z_{9},z_{10}, x_{s_{232}} \in [10^{-6}~;10^{6}] \\ \text{st}&& \quad {z_{8}^{2}} + {z_{9}^{2}} - x_{s_{232}}^{2} \leq 0 \\ &&\quad z_{8}^{-2} + z_{10}^{2} - x_{s_{232}}^{2} \leq 0 \\ \text{where} &&r_{y_{21}} = \sqrt{{z_{8}^{2}}+z_{9}^{-2}+z_{10}^{-2}+x_{s_{232}}^{2}} \end{array} $$

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Subproblem 3:

$$ \begin{array}{ll} \min & \phi_{y_{31}}(t_{y_{31}}-r_{y_{31}}) + \phi_{s_{23}}(x_{s_{232}}-x_{s_{233}}) \\ \text{wrt} & z_{12},z_{13},z_{14},x_{s_{233}} \in [10^{-6}~;10^{6}] \\ \text{st} & x_{s_{233}}^{2} + z_{12}^{-2} - z_{13}^{2} \leq 0 \\ & x_{s_{233}}^{2} + z_{12}^{2} - z_{14}^{2} \leq 0 \\ \text{where} & r_{y_{31}} = \sqrt{x_{s_{233}}^{2}+z_{12}^{2}+z_{13}^{2}+z_{14}^{2}} \end{array} $$

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1.3 A.3 Simplified wing design problem

Subproblem 1 - aircraft:

$$ \begin{array}{ll} \min & t_{y_{21}} + t_{y_{31}} +\\ & \hspace{0cm} \phi_{y_{21}}(t_{y_{21}}-r_{y_{21}}) + \phi_{y_{31}}(t_{y_{31}}-r_{y_{31}}) \\ \text{wrt} & t_{y_{21}} , t_{y_{31}} \in [0~;10^{5}] \\ \end{array} $$

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Subproblem 2 - structures:

$$ \begin{array}{ll} \min & \phi_{y_{21}}(t_{y_{21}}-r_{y_{21}}) + \phi_{s_{23}}(x_{s_{232}}-x_{s_{233}})\\ \text{wrt} & \mathbf{x}_{2},\mathbf{x}_{s_{232}} \in [0~;10]^{2} \\ \text{where} & r_{y_{21}} = 4,000 \big(1+\|\mathbf{x}_{s_{232}}-1\|_{2}^{2}\big)\big(1+\|\mathbf{x}_{2}-1\|_{2}^{2}\big) \end{array} $$

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Subproblem 3 - aerodynamics:

$$ \begin{array}{ll} \min & \phi_{y_{31}}(t_{y_{31}}-r_{y_{31}}) + \phi_{s_{23}}(x_{s_{232}}-x_{s_{233}})\\ \text{wrt} & \mathbf{x}_{3},\mathbf{x}_{s_{233}} \in [0~;10]^{2} \\ \text{where} & r_{y_{31}} = 20,000 \,+\, 380,952\text{\;Drag} \,+\, 9,523,809\text{\;Drag}^{2}\\ & \text{Drag} = 0.025+0.004 \log_{10}(\omega) \\ & \omega \!= \big(1\,+\,\|\mathbf{x}_{s_{233}}\,-\,2\|^{2}_{2}\big) \big(1\,+\,\frac{\|\mathbf{x}_{3}-2\|^{2}_{2}}{1,000}\big) \big(1\,+\,1,000 |EH|\big) \\ & \mathbf{u} = 10(\mathbf{x}_{s_{233}}+\mathbf{x}_{3}) \\ & EH = f_{EH}(\mathbf{u} ) \\ & f_{EH}(\mathbf{u} ) = -(u_{2}+47) \sin \left( \sqrt{ \left| u_{2}+\frac{u_{1}}2+47 \right| } \right) \\ & \hspace{2cm} - u_{1} \sin \left( \sqrt{ \left| u_{1}-u_{2}-47 \right| } \right) \end{array} $$

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1.4 A.4 Supersonic business jet problem

Box constraints apply to each design variable of each problem. See Tosserams et al. (2010) for details.

Subproblem 1 - aircraft:

$$ \begin{array}{ll} \min & W_{T}+\phi_{y_{21}}(\mathbf{t}_{y_{21}}-\mathbf{r}_{y_{21}})+\phi_{y_{31}}(\mathbf{t}_{y_{31}}-\mathbf{r}_{y_{31}}) +\\ & \hspace{0cm} \phi_{y_{41}}(\mathbf{t}_{y_{41}}-\mathbf{r}_{y_{41}})\\ \text{wrt} & \textbf{t}_{y_{21}}, \textbf{t}_{y_{31}}, \textbf{t}_{y_{41}}\\ \text{st} & {g}_{\text{aircraft}}(\textbf{t}_{y_{21}}, \textbf{t}_{y_{31}}, \textbf{t}_{y_{41}}) \leq \mathbf{0}\\ \text{where} & W_{T} = f(\textbf{t}_{y_{21}}, \textbf{t}_{y_{31}}, \textbf{t}_{y_{41}}) \\ \end{array} $$

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Subproblem 2 - propulsion:

$$ \begin{array}{ll} \min & \phi_{y_{21}}(\mathbf{t}_{y_{21}}-\mathbf{r}_{y_{21}})+\phi_{y_{23}}(t_{y_{23}}-r_{y_{23}})+\phi_{y_{32}}(t_{y_{32}}-r_{y_{32}})\\ \text{wrt} & t_{y_{32}}, x_{2}=T \\ \text{st} & \mathbf{g}_{\text{prop}}(t_{y_{32}}, x_{2}) \leq \mathbf{0}\\ \text{where} &\mathbf{r}_{y_{21}} = f_{21}(t_{y_{32}}, x_{2}) \\ & r_{y_{23}} = f_{23}(t_{y_{32}}, x_{2}) \end{array} $$

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Subproblem 3 - aerodynamics:

$$ \begin{array}{ll} \min & \phi_{y_{31}}(\mathbf{t}_{y_{31}}-\mathbf{r}_{y_{31}}) + \phi_{y_{32}}(t_{y_{32}}-r_{y_{32}}) +\\ & \hspace{0cm} \phi_{y_{34}}(t_{y_{34}}-r_{y_{34}})+\phi_{y_{23}}(t_{y_{23}}-r_{y_{23}}) +\\ & \hspace{0cm} \phi_{y_{43}}(t_{y_{43}}-r_{y_{43}})+\phi_{s_{34}}(\mathbf{x}_{s_{343}}-\mathbf{x}_{s_{344}})\\ \text{wrt} & t_{y_{23}},t_{y_{43}}, \textbf{x}_{3} = \left[ {\Lambda}_{\text{ht}},L_{\text{w}},L_{\text{ht}} \right]^{T},\\ & \hspace{0cm} \textbf{x}_{s_{343}} = [t/c,AR_{\text{w}},{\Lambda}_{\text{w}},S_{\text{ref}} ,S_{\text{ht}},AR_{\text{ht}}]^{T} \\ \text{st} & \mathbf{g}_{\text{prop}}(t_{y_{23}},t_{y_{43}}, \textbf{x}_{3}, \textbf{x}_{s_{343}}) \leq \mathbf{0}\\ \text{where} &\mathbf{r}_{y_{31}} = f_{31}(t_{y_{23}},t_{y_{43}}, \textbf{x}_{3}, \textbf{x}_{s_{343}})\\ &r_{y_{32}} = f_{32}(t_{y_{23}},t_{y_{43}}, \textbf{x}_{3}, \textbf{x}_{s_{343}})\\ &r_{y_{34}} =f_{34}(t_{y_{23}},t_{y_{43}}, \textbf{x}_{3}, \textbf{x}_{s_{343}})\\ \end{array} $$

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Subproblem 4 - structures:

$$ \begin{array}{ll} \min & \phi_{y_{41}}(\mathbf{t}_{y_{41}}-\mathbf{r}_{y_{41}})+\phi_{y_{43}}(t_{y_{43}}-r_{y_{43}}) +\\ & \hspace{0cm} \phi_{y_{34}}(t_{y_{34}}-r_{y_{34}})+\phi_{s_{34}}(\mathbf{x}_{s_{343}}-\mathbf{x}_{s_{344}})\\ \text{wrt} & t_{y_{34}}, \textbf{x}_{4} = [\mathbf{t},\mathbf{t}_{\text{s}},\lambda]^{T},\\ & \hspace{0cm} \textbf{x}_{s_{344}} = [t/c,AR_{\text{w}},{\Lambda}_{\text{w}},S_{\text{ref}} ,S_{\text{ht}},AR_{\text{ht}}]^{T} \\ \text{st} & \mathbf{g}_{\text{struc}}(t_{y_{34}}, \textbf{x}_{4}, \textbf{x}_{s_{344}} ) \leq \mathbf{0}\\ \text{where} &\mathbf{r}_{y_{41}} = f_{41}(t_{y_{34}}, \textbf{x}_{4}, \textbf{x}_{s_{344}})\\ &r_{y_{43}} = f_{43}(t_{y_{34}}, \textbf{x}_{4}, \textbf{x}_{s_{344}}) \\ \end{array} $$

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Appendix B: Complete numerical results

Table 4 Inconsistency and discrepancy from best known solution for all test problems with β = 2.2 and γ = 0.4

Table 5 Inconsistency and discrepancy from best know solution for the bi-quadratic problem with NI = NO = 64

Table 6 Inconsistency and discrepancy from best know solution for the geometric programming problem with NI = NO = 64

Table 7 Inconsistency and discrepancy from best know solution for the simplified wing design problem with NI = NO = 64

Table 8 Inconsistency and discrepancy from best know solution for the supersonic business jet problem with NI = NO = 64