Bounded Target Cascading in Hierarchical Design Optimization

Abstract

For large scale systems, as a hierarchical multilevel decomposed design optimization method, analytical target cascading coordinates the inconsistency between the assigned targets and response in each level by a weighted-sum formulation. To avoid the problems associated with the weighting coefficients, single objective functions in the hierarchical design optimization are formulated by a bounded target cascading method in this paper. In the BTC method, a single objective optimization problem is formulated in the system level, and two kinds of coordination constraints are added: one is bound constraint for the design points based on the response from each subsystem level and the other is linear equality constraint for the common variables based on their sensitivities with respect to each subsystem. In each subsystem level, the deviation with target for design point is minimized in the objective function, and the common variables are constrained by target bounds. Therefore, in the BTC method, the targets are coordinated based on the optimization iteration information in the hierarchical design problem and the performance of the subsystems, and BTC method will converge to the global optimum efficiently. Finally, comparisons of the results from BTC method and the weighted-sum analytical target cascading method are presented and discussed.

1. Introduction

The multidisciplinary design optimization (MDO) problems have become very important in most engineering designs. The key issue for MDO problems is how to coordinate coupling disciplines, and various MDO frameworks have been developed based on their problem formulations and decomposition strategies. Martins and Lambe [1] provided a general introduction for the various developed approaches and summarized the merit of different MDO frameworks. Tosserams et al. [2] discussed the general characteristics of nested formulations and alternated formulations for MDO problems. Originally, the MDO problem minimizes the cost function subject to multiple disciplinary constraints but it may be unpractical to solve the entire problem in one optimization formulation; bilevel optimization formulations have received considerable attention to solve MDO problems, such as collaborative optimization (CO), which has been widely applied to the design of aerospace systems, and the analytical and computational aspects of CO were presented [3]. For large scale engineering problems, as a method to propagate system targets and coordinate common variables of subsystems through a hierarchical multilevel system formulation, analytical target cascading (ATC) method [4 –6] has been developed and widely utilized to solve multidisciplinary engineering problems [7 –10]. In the ATC method, the all-in-one (AIO) optimization formulation is decomposed into a hierarchical structure with one system level and multiple subsystems. In order to find the optimal solution of the original MDO problem, it is essential to minimize the cost function and reduce the inconsistency between the subsystems simultaneously in ATC.

The cumulative constraint violations in the subsystems, also known as the discrepancy functions, were firstly formulated and minimized to maintain the consistency in the MDO problems [11]. Besides, the differences between the subsystem variables have been considered as the discrepancy functions [12, 13]. The discrepancy functions were then penalized in the objective functions for finding the optimal solution and diminishing the inconsistency during the optimization processes [14, 15]. The ATC method firstly considered the weighted discrepancy functions of design variables in multiobjective formulations and chose weighting coefficients to coordinate the optimal solutions of the original MDO problems [16 –19].

In the processes of solving the Analytical Target Cascading, the system level assigned its optimal solution to the subsystems as the design target points. In the subsystems, the optimal solutions were found in terms of minimizing the differences between the subsystem variables and the targets subject to the local constraints. The local optimal solutions were then updated back to the system level as the design responses. The weighted discrepancy functions in terms of the targets and the response points were diminished during the optimization processes of ATC. However, the selection of the weighting coefficients is crucial for decreasing the discrepancy functions to zero and coordinating the inconsistency between design points and assigned targets in each level during the optimization processes. In other words, the proper optimal solutions can only be obtained with an appropriate choice of weighting coefficients. Many approaches have been studied to determine the proper selection of weights [18 –22]. Michalek and Papalambros [18] proposed a weights updating method using the KKT first order necessary conditions combined user-specified inconsistency tolerances. Tosserams et al. [20] developed an augmented Lagrangian coordination method with the alternating direction method of multipliers. Kim et al. [21] formulated a Lagrangian dual coordination method to update the weighting coefficients. Li et al. [22] provided a diagonal quadratic approximation method by linearizing the cross terms of the discrepancy functions. The above methods took the discrepancy functions as equality constraints and used different penalty terms to coordinate those equality constraints. The convergence rate of these methods becomes more slowly as the discrepancy becoming smaller, and the penalty terms might lead to oscillate during the iteration process.

In this paper, single objective functions in the hierarchical design optimization are formulated to avoid the problems associated with the weighting coefficients by a new bounded target cascading (BTC) method. Instead of assigning the point targets to the subsystems in ATC, the bounded targets are introduced in the new method. The target bounds are obtained from the optimal solutions in each level and utilized to reduce the variable bounds during the optimization processes. Furthermore, the bounds of responses are also determined to avoid the weighted discrepancy functions of the responses. If the common variables exist, they are coordinated based on their sensitivities with respect to design variables. With the aids of the target bounds, the optimal solutions can be determined in the reduced variable bounds while the consistency between the variables and the targets is maintained.

In the next section, the well-known MDO algorithm and the analytical target cascading method are firstly reviewed. In Section 3, the bounded target cascading method is proposed to solve MDO problems to avoid the problems associated with the weighting coefficients. Lastly, some numerical examples are solved by the BTC method and the comparisons of the results from the proposed method and the ATC method are also presented and discussed in Section 4.

2. MDO and Analytical Target Cascading

The original formulation of the MDO problem is given by

\begin{matrix} min: f (x^{m = 1}, z^{m = 1}) \\ w . r . t . X_{i}^{m}, x_{i}^{m}, Y_{i}^{m}, y_{i}^{m}, z_{i}^{m} \\ s . t . g_{i}^{m} (X_{i}^{m}, x_{i}^{m}, Y_{i}^{m}, z_{i}^{m}) \leq 0 \\ h_{i}^{m} (X_{i}^{m}, x_{i}^{m}, Y_{i}^{m}, z_{i}^{m}) = 0 \\ {\underline{X}}_{i}^{m} \leq X_{i}^{m} \leq {\bar{X}}_{i}^{m} \\ {\underline{Y}}_{i}^{m} \leq Y_{i}^{m} \leq {\bar{Y}}_{i}^{m} \\ {\underline{x}}_{i}^{m} \leq x_{i}^{m} \leq {\bar{x}}_{i}^{m} \\ {\underline{y}}_{i}^{m} \leq y_{i}^{m} \leq {\bar{y}}_{i}^{m} \\ {\underline{z}}_{i}^{m} \leq z_{i}^{m} \leq {\bar{z}}_{i}^{m}, \end{matrix}

(1)

where f is the cost function; x_i^m stands for the ith subsystem variables in the mth level coupling with one level below; z_i^m is the local variable; and m = 1 represents the variables in the system level. In the ith subsystem, X_i^m is the variable coupling with one level above; y_i and Y_i denote the common variables from one level below and one level above, respectively; g_i^m and h_i^m are the inequality and equality constraints, respectively. The all-in-one formulation in (1) is often unpractical in engineering problems; therefore, it is decomposed into hierarchical optimization structures and solved by the MDO algorithms. The main goal of MDO methods is to minimize the cost function while the discrepancy between subsystems is diminished simultaneously.

Kim et al. [4] developed analytical target cascading (ATC) method to decompose the MDO problem into hierarchical structures with a system level and several subsystems. The system level is given as

\begin{matrix} min: f (x^{m, k}, z) + {∥ w_{x}^{m, k} \cdot (x^{m, k} - x^{m + 1, k - 1}) ∥}^{2} + {∥ w_{y}^{m, k} \cdot (y^{m, k} - y^{m + 1, k - 1}) ∥}^{2} \\ w . r . t . x^{m, k}, y^{m, k}, z^{m, k} \\ s . t . g^{m} (x^{m, k}, z^{m, k}) \leq 0 \\ h^{m} (x^{m, k}, z^{m, k}) = 0 \\ {\underline{x}}^{m, k} \leq x^{m, k} \leq {\bar{x}}^{m, k} \\ {\underline{y}}^{m, k} \leq y^{m, k} \leq {\bar{y}}^{m, k} \\ {\underline{z}}^{m, k} \leq z^{m, k} \leq {\bar{z}}^{m, k}, \end{matrix}

(2)

where w_x^m,k and w_y^m,k are the weighting coefficients. When the cost function is minimized, the weighted discrepancy terms are diminished simultaneously to maintain the consistency in the ith subsystem. Furthermore, a general formulation for each subsystem is defined as

\begin{matrix} min: w_{X, i}^{m, k} {∥ X_{i}^{m, k} - X_{i}^{m - 1, k} ∥}^{2} \\ + {∥ w_{x, i}^{m, k} \cdot (x_{i}^{m, k} - x_{i}^{m + 1, k - 1}) ∥}^{2} \\ + {∥ w_{Y, i}^{m, k} \cdot (Y_{i}^{m, k} - Y_{i}^{m - 1, k}) ∥}^{2} \\ + {∥ w_{y, i}^{m, k} \cdot (y_{i}^{m, k} - y_{i}^{m + 1, k - 1}) ∥}^{2} \\ w . r . t . X_{i}^{m, k}, x_{i}^{m, k}, Y_{i}^{m, k}, y_{i}^{m, k}, z_{i}^{m, k} \\ s . t . g_{i}^{m} (X_{i}^{m, k}, x_{i}^{m, k}, Y_{i}^{m, k}, z_{i}^{m, k}) \leq 0 \\ h_{i}^{m} X_{i}^{m, k}, x_{i}^{m, k}, Y_{i}^{m, k}, z_{i}^{m, k} = 0 \\ {\underline{X}}_{i}^{m, k} \leq X_{i}^{m, k} \leq {\bar{X}}_{i}^{m, k} \\ {\underline{x}}_{i}^{m, k} \leq x_{i}^{m, k} \leq {\bar{x}}_{i}^{m, k} \\ {\underline{Y}}_{i}^{m, k} \leq Y_{i}^{m, k} \leq {\bar{Y}}_{i}^{m, k} \\ {\underline{y}}_{i}^{m, k} \leq y_{i}^{m, k} \leq {\bar{y}}_{i}^{m, k} \\ {\underline{z}}_{i}^{m, k} \leq z_{i}^{m, k} \leq {\bar{z}}_{i}^{m, k}, \end{matrix}

(3)

where w_X,i^m,k, w_x^m,k, w_Y,i^m,k, and w_y^m,k are the weighting coefficients. The discrepancy functions associated with the target points from one level above are minimized while the functions in terms of the responses from one level below are minimized at the same time.

The discrepancy functions can be formulated by either the L₂-norm differences [4] between the variables and target points or the squared L₂-norm functions [6]. However, the multiobjective formulations are very difficult for ATC to find the proper solutions in the MDO problems. Some researchers [16, 17] recognized the importance of the weighting coefficients in the discrepancy functions for finding the optimal solutions and diminishing the inconsistency simultaneously. Kim et al. [21] noticed that the proper selection of weighting coefficients is crucial for obtaining the reasonable optimal solutions in ATC. There are two approaches to resolving the problems associated with the weighting coefficients: one is to find the proper choice of weights and the other is not to use the weighted terms in the multiobjective formulations. Many researches have been conducted to appropriately determine the weighting coefficients, and the design problem will converge to different Pareto solution with different weighting coefficients. BTC tries to formulate the target cascading problem into a single objective design framework, and the optimization iteration information in each design level is shared to make the design points converge to the global optimum.

3. Bounded Target Cascading

In this Section, a new bounded target cascading (BTC) method is proposed to solve the MDO problem in (1) without using the weighted multiobjective formulations. Instead of assigning the point targets to the subsystems in ATC, the bounded targets are introduced in the new method. The target and response bounds are defined to avoid the weighted discrepancy formulations. Furthermore, the common variables are coordinated based on sensitivity analysis.

3.1. Target Bounds in BTC for Subsystem

In the bounded target cascading method, target bounds for the common variables in the ith subsystem are defined as

| Y_{i, j}^{m, k} - Y_{i, j}^{m - 1, k} | \leq d_{i, j}^{m, k}

(4)

which are centered at the target points Y_i,j^{m − 1,k} and ranged by d_i,j^m,k. Instead of using the weighted term of $w_{Y_{i, j}}^{m, k} {∥ Y_{i, j}^{m, k} - Y_{i, j}^{m - 1, k} ∥}^{2}$ in the multiobjective formulation, the target bounds in (4) are given to bound the common variables. In order to converge at the correct position of the common variables, the ranges of the target bound d_i,j^m,k are defined by the difference between the optimum in the previous iteration and the current target points. Therefore, the target bound is defined as

| Y_{i, j}^{m, k} - Y_{i, j}^{m - 1, k} | \leq d_{i, j}^{m, k}; d_{i, j}^{m, k} = | Y_{i, j}^{m, k - 1} - Y_{i, j}^{m - 1, k} |,

(5)

where Y_i,j^{m,k − 1} denotes the previous optimal solution and Y_i,j^{m − 1,k} stands for the target in the kth iteration.

In the coordination processes, if the discrepancy between design target and design point for any subsystem is less than one percent of the design response, the target bounds for common variables will be redefined using the bounds of the subsystem, which is calculated by

[{\underline{Y}}_{i, j}^{m, k}, {\bar{Y}}_{i, j}^{m, k}] = ⋂_{i = 1}^{n_{m}} [Y_{i, j}^{m - 1, k} - d_{i, j}^{m, k}, Y_{i, j}^{m - 1, k} + d_{i, j}^{m, k}] .

(6)

The design optimization problem in subsystem is to minimize the deviations of subsystem performances and common variables from assigned targets. The design problem in subsystem converges with the two deviations are small enough.

3.2. Target Bounds in BTC for System

In system level, the weighted discrepancy function of the design responses is replaced by the new response bound which is defined as

\begin{matrix} x_{i}^{m, k} \leq x_{i}^{m + 1, k - 1} for δ_{i}^{m, k} < 0, \\ x_{i}^{m, k} \geq x_{i}^{m + 1, k - 1} for δ_{i}^{m, k} > 0, \end{matrix}

(7)

where δ_i^m,k = x_i^{m + 1,k − 1} − x_i^{m,k − 1} is the difference between the design response and optimal solution in the previous iteration. When the response is found to be smaller than the optimal solution in the previous iteration, in order to fulfill the feasibility and optimality conditions in the subsystem, the new design point should be bounded above by the response. Contrarily, the design point should be bounded below by the response when the response is larger than the optimal solution in the previous iteration.

If the common variables of the responses exist and then constrained by their sensitivity relations with the design variables. Therefore, approximated linear equations are established by the first-order Taylor expansion, which is given by

x_{i}^{m, k} ≅ x_{i}^{m + 1, k - 1} + (y_{i}^{m, k} - y_{i}^{m + 1, k - 1}) \cdot \frac{Δ x_{i}^{m, k - 1}}{Δ y_{i}^{m, k - 1}} .

(8)

If there are c_j common variables in subsystem i, the updating information for common variables with the ith subsystem is provided in system by

x_{sub, i} ≅ x_{sub, i}^{sub, k - 1} + \sum_{j = 1}^{c_{i}} (y_{sub, i j}^{sys} - y_{sub, i j}^{sub, k - 1}) \cdot \frac{\partial x_{sub, i}}{\partial y_{sub, i j}} .

(9)

In system level, for coordinating common variables between subsystem i and subsystem j, the target for common variable will be calculated based on the updating information and the sensitivity of system objective function to the subsystem performance, which is given by

y^{sys} = \frac{(\partial f_{sys} / \partial x_{sub, i}) y_{sub, i}^{sub, k} + (\partial f_{sys} / \partial x_{sub, j}) y_{sub, j}^{sub, k}}{\partial f_{sys} / \partial x_{sub, i} + \partial f_{sys} / \partial x_{sub, j}} .

(10)

Therefore, in BTC method, instead of considering the weighted discrepancy term of $w_{i}^{m, k} {∥ y_{i}^{m, k} - y_{i}^{m + 1, k - 1} ∥}^{2}$ , the common variable in system level is subject to the sensitivity relation in (8)–(10).

3.3. Solution Process

The optimization process using the proposed bounded target cascading method follows the divide-and-conquer strategy. The system level minimizes the cost function subject to the system-level constraints, the new response bounds, and the sensitivity relations of common variables, which is given by

\begin{matrix} min: f (x^{m, k}, z) \\ w . r . t . x^{m, k}, y^{m, k}, z \\ s . t . g^{m} (x^{m, k}, z) \leq 0 \\ h^{m} (x^{m, k}, z) = 0 \\ x^{m, k} ≅ x^{m + 1, k - 1} + (y_{i}^{m, k} - y_{i}^{m + 1, k - 1}) \cdot \frac{Δ x^{m, k - 1}}{Δ y_{i}^{m, k - 1}} \\ y^{m, k} = \frac{\sum (\partial f (x^{m, k}, z) / \partial x_{i}^{m, k}) y_{i}^{m, k}}{\sum (\partial f (x^{m, k}, z) / \partial x_{i}^{m, k})} \\ x^{m, k} \leq x^{m + 1, k - 1} for δ^{m, k} < 0 \\ x^{m, k} \geq x^{m + 1, k - 1} for δ^{m, k} > 0 \\ {\underline{x}}^{m, k} \leq x^{m, k} \leq {\bar{x}}^{m, k} \\ {\underline{y}}^{m, k} \leq y^{m, k} \leq {\bar{y}}^{m, k} \\ \underline{z} \leq z \leq \bar{z} . \end{matrix}

(11)

The target bounds are then formulated based on the optimal solution and assigned to the subsystems. In general, the subsystem finds the local optimal solution subject to subsystem constraints and new target bounds:

\begin{matrix} min: {∥ X_{i}^{m, k} - X_{i}^{m - 1, k} ∥}^{2} \\ w . r . t . X_{i}^{m, k}, x_{i}^{m, k}, Y_{i}^{m, k}, y_{i}^{m, k}, z_{i} \\ s . t . g_{i}^{m} (X_{i}^{m, k}, x_{i}^{m, k}, Y_{i}^{m, k}, z_{i}) \leq 0 \\ h_{i}^{m} (X_{i}^{m, k}, x_{i}^{m, k}, Y_{i}^{m, k}, z_{i}) = 0 \\ x^{m, k} ≅ x^{m + 1, k - 1} + (y_{i}^{m, k} - y_{i}^{m + 1, k - 1}) \cdot \frac{Δ x^{m, k - 1}}{Δ y_{i}^{m, k - 1}} \\ | Y_{i, j}^{m, k} - Y_{i, j}^{m - 1, k} | \leq d_{i, j}^{m, k} \\ x_{i}^{m, k} \leq x_{i}^{m + 1, k - 1} for δ_{i}^{m + 1, k} < 0 \\ x_{i}^{m, k} \geq x_{i}^{m + 1, k - 1} for δ_{i}^{m + 1, k} > 0 \\ {\underline{X}}_{i}^{m, k} \leq X_{i}^{m, k} \leq {\bar{X}}_{i}^{m, k} \\ {\underline{x}}_{i}^{m, k} \leq x_{i}^{m, k} \leq {\bar{x}}_{i}^{m, k} \\ {\underline{Y}}_{i}^{m, k} \leq Y_{i}^{m, k} \leq {\bar{Y}}_{i}^{m, k} \\ {\underline{y}}_{i}^{m, k} \leq y_{i}^{m, k} \leq {\bar{y}}_{i}^{m, k} \\ {\underline{z}}_{i} \leq z_{i} \leq {\bar{z}}_{i} . \end{matrix}

(12)

The local optimal solutions are then updated back to the system level as the information of the response bounds. For the subsystems at the bottom of the hierarchy structure, (12) is simplified as

\begin{matrix} min: {∥ X_{i}^{m, k} - X_{i}^{m - 1, k} ∥}^{2} \\ w . r . t . X_{i}^{m, k}, Y_{i}^{m, k}, z_{i} \\ s . t . g_{i}^{m} (X_{i}^{m, k}, Y_{i}^{m, k}, z_{i}) \leq 0 \\ h_{i}^{m} (X_{i}^{m, k}, Y_{i}^{m, k}, z_{i}) = 0 \\ | Y_{i j}^{m, k} - Y_{i j}^{m - 1, k} | \leq d_{i j}^{m, k} \\ {\underline{X}}_{i}^{m, k} \leq X_{i}^{m, k} \leq {\bar{X}}_{i}^{m, k} \\ {\underline{Y}}_{i}^{m, k} \leq Y_{i}^{m, k} \leq {\bar{Y}}_{i}^{m, k} \\ {\underline{z}}_{i} \leq z_{i} \leq {\bar{z}}_{i} . \end{matrix}

(13)

The iteration process of BTC method continues until the convergence criterion is satisfied. In this paper, the convergence criterion is defined by the absolute difference between the current and the previous design points to be smaller than the acceptable limit. In the next Section, some numerical examples solved by the proposed BTC method and the ATC method with various settings of the weighting coefficients are presented and discussed.

3.4. Convergence of BTC

The convergence properties of ATC have been discussed by [16], which proved that ATC process converges to the optimal point that satisfies the necessary optimality conditions of the original design problem. BTC solves decomposed MDO problems using the same coordinate principle with ATC in the targets cascading process, that is, hierarchical overlapping coordination, which is achieved by the exchange of information between different decompositions. The properties of BTC are presented as the following.

3.4.1. The Upper or Lower Bound for Design Point in System Level

In the system level for general ATC, the consistency constraints c_i^m,k = ∥x_i^{m + 1,k} − x_i^m,k∥ are penalized in the objective function using various penalty functions, which minimizes the deviation of design points with response from subsystems. However, these methods may lead to the optimization problem converge to a local optimum or converge earlier. Lower bounds or upper bounds based on the response from subsystems are defined to consider the performance of each subsystem and provide more flexibility for system problem.

3.4.2. The Linear Approximation for Coordinating Common Variable in System Level

The sensitivity of each subsystem to common variables is calculated to coordinate common variable between subsystems in the system level. The gradient calculation is the key issue to guarantee the accuracy of this approximation. In this paper, pseudo finite difference Δx_i^{m,k − 1}/Δy_i^{m,k − 1} and reduced gradient method is introduced to calculate the gradient.

Suppose the equality relation between the design variable, x_sub,i, and a common variable, y_sub,ij, exist:

h (x_{sub, i}, y_{sub, i j}) = 0 .

(14)

The partial derivative of the equality constraint with respect to common variables is

{\frac{\partial h (x_{sub, i}, y_{sub, i j})}{\partial y_{sub, i j}} |}_{y_{sub, i j}^{sub, k}} + {\frac{\partial h (x_{sub, i}, y_{sub, i j})}{\partial x_{sub, i}} \frac{\partial x_{sub, i}}{\partial y_{sub, i j}} |}_{x_{sub, i}^{sub, k}} = 0,

(15)

where x_sub,i^sub,k and y_sub,ij^sub,k are the design points at kth iteration, and the sensitivity relation between the design variable and common variable is derived as

{\frac{\partial x_{sub, i}}{\partial y_{sub, i j}} |}^{k} = {- \frac{\partial h (x_{sub, i}, y_{sub, i j}) / \partial y_{sub, i j}}{\partial h (x_{sub, i}, y_{sub, i j}) / \partial x_{sub, i}} |}^{k} .

(16)

If the equality relation is not available, surrogate models will be formulated by simulation, and for high dimensionality engineering problem, the efficiency of simulation is the most considerable problem.

3.4.3. Interval Convergence for Common Variables in Subsystems

During the optimization iteration in subsystems, the deviation between targets and response should be decreased:

| Y_{i, j}^{m, k} - Y_{i, j}^{m - 1, k} | \leq r | Y_{i, j}^{m, k - 1} - Y_{i, j}^{m - 1, k} | \leq \dots \leq r | Y_{i, j}^{m, 0} - Y_{i, j}^{m - 1,1} |,

(17)

where 0 ≤ r ≤ 1, in this paper, r = 1. The convergence property of subsystem will change with different r.

4. Numerical Examples

In this Section, two mathematical problems are decomposed into hierarchical structures and solved by the proposed bounded target cascading method and the analytical target cascading method with various settings of the weighting coefficients.

4.1. Convex Quadratic Programming Problem

The undecomposed convex minimization problem is stated as in

\begin{matrix} min: x_{1}^{2} + x_{2}^{2} \\ s . t . g_{1} = x_{3}^{- 2} + x_{4}^{2} - x_{5}^{2} \leq 0 \\ g_{2} = x_{5}^{2} + x_{6}^{- 2} - x_{7}^{2} \leq 0 \\ h_{1} = x_{1}^{2} - x_{3}^{2} - x_{4}^{- 2} - x_{5}^{2} = 0 \\ h_{2} = x_{2}^{2} - x_{5}^{2} - x_{6}^{2} - x_{7}^{2} = 0 \\ x_{1}, \dots, x_{7} \geq 0 . \end{matrix}

(18)

Using the analytical target cascading method, the convex problem is decomposed into the structure with two levels, constraints g₁ and h₁ in subsystem 1 and constraints g₂ and h₂ in subsystem 2. For general ATC, The system level is formulated as (19), while the first and the second subsystems are given by (20) and (21), respectively:

\begin{matrix} \min_{x_{1}, x_{2}, x_{3}} : x_{1}^{2} + x_{2}^{2} + w_{1} {(x_{1} - x_{1}^{k - 1, su b_{1}})}^{2} + w_{2} {(x_{2} - x_{2}^{k - 1, su b_{2}})}^{2} \\ + w_{3} {(x_{5} - x_{5}^{k - 1, su b_{1}})}^{2} + w_{4} {(x_{5} - x_{5}^{k - 1, su b_{2}})}^{2} \\ s . t . 0 \leq x_{1}, x_{2}, x_{5}, \end{matrix}

(19)

\begin{matrix} \min_{x_{1}, x_{3}} : w_{1} {(x_{1} - x_{1}^{k, sys})}^{2} + w_{3} {(x_{5} - x_{5}^{k, sys})}^{2} \\ s . t . h_{1} = x_{1}^{2} - x_{3}^{2} - x_{4}^{- 2} - x_{5}^{2} = 0 \\ g_{1} = x_{3}^{- 2} + x_{4}^{2} - x_{5}^{2} \leq 0 \\ 0 \leq x_{1}, x_{3}, x_{4}, x_{5}, \end{matrix}

(20)

\begin{matrix} \min_{x_{2}, x_{3}} : w_{2} {(x_{2} - x_{2}^{k, sys})}^{2} + w_{4} {(x_{5} - x_{5}^{k, sys})}^{2} \\ s . t . h_{2} = x_{2}^{2} - x_{5}^{2} - x_{6}^{2} - x_{7}^{2} = 0 \\ g_{2} = x_{5}^{2} + x_{6}^{- 2} - x_{7}^{2} \leq 0 \\ 0 \leq x_{2}, x_{5}, x_{6}, x_{7}, \end{matrix}

(21)

where $x_{1}^{k - 1, su b_{1}}$ and $x_{5}^{k - 1, su b_{1}}$ are the responses from the first subsystem in the (k − 1) th iteration; $x_{2}^{k - 1, su b_{2}}$ and $x_{5}^{k - 1, su b_{2}}$ are the ones from the second subsystem; x₁^k,sys, x₂^k,sys, and x₅^k,sys are the optimal solutions of the system level in the kth iteration as well as the targets for the both subsystems. The initial points x_{k = 0,sys} and $x^{k = 0, su b_{i}}$ follow x_{k = 0} = [1 1 1]_T as well as the initial parameters $x^{k = 1, su b_{i}}$ . In the formulation of ATC, the proper selection of the weighting coefficients w₁, w₂, w₃, and w₄ is crucial for obtaining the correct optimal solution.

In the proposed bounded target cascading method, the discrepancy terms of common variables are displaced by the sensitivity relations in the subsystems while the ones of design variables are displaced by the response bounds. Therefore, the system level is

\begin{matrix} min: f_{sys} = x_{1}^{2} + x_{2}^{2} \\ s . t . x_{i} = x_{i}^{k - 1, su b_{i}} + (y_{i} - x_{5}^{k - 1, su b_{i}}) \frac{d x_{1} (x_{i}^{k - 1, su b_{i}})}{d x_{5}} \\ y^{sys} = \frac{(\partial f_{sys} / \partial x_{sub, 1}) y_{1} + (\partial f_{sys} / \partial x_{sub, 2}) y_{2}}{\partial f_{sys} / \partial x_{sub, 1} + \partial f_{sys} / \partial x_{sub, 2}} \\ x_{i} \leq x_{i}^{k - 1, su b_{i}} for δ_{i} = x_{i}^{k - 1, su b_{1}} - x_{i}^{k - 1} < 0 \\ x_{i} \geq x_{i}^{k - 1, su b_{i}} for δ_{i} > 0 \\ 0 \leq x_{1}, x_{2}, x_{5}, y_{i}, i = 1,2 . \end{matrix}

(22)

In the first iteration, the sensitivity relations are omitted because no common variations have been made in $x_{i}^{k, su b_{i}}$ . When the response $x_{i}^{k - 1, su b_{i}}$ is smaller than the previous design x_i^{k − 1,sys}, that is, δ_i < 0, the new design should be smaller than or equal to the response. On the other hand, $x_{i} \geq x_{i}^{k - 1, su b_{i}}$ is considered when δ_i > 0. In the special case of $x_{i}^{k - 1, sys} = x_{i}^{k - 1, su b_{i}}$ , the direction of the response bound is determined based on the most recent nonzero δ_i. Furthermore, the ith subsystem is reformulated as

\begin{matrix} min: {(x_{1} - x_{1}^{k, sys})}^{2} \\ s . t . | x_{5} - y^{k, sys} | \leq d_{sub, 1}; d_{sub, 1} = | x_{5}^{k - 1, sub} - y^{k, sys} | \\ h_{1} = x_{1}^{2} - x_{3}^{2} - x_{4}^{- 2} - x_{5}^{2} = 0 \\ g_{1} = x_{3}^{- 2} + x_{4}^{2} - x_{5}^{2} \leq 0 \\ 0 \leq x_{1}, x_{2}, x_{3}, \\ min: {(x_{2} - x_{2}^{k, sys})}^{2} \\ s . t . | x_{5} - y^{k, sys} | \leq d_{sub, 2}; d_{sub, 2} = | x_{5}^{k - 1, sub} - y^{k, sys} | \\ h_{2} = x_{2}^{2} - x_{5}^{2} - x_{6}^{2} - x_{7}^{2} = 0 \\ g_{2} = x_{5}^{2} + x_{6}^{- 2} - x_{7}^{2} \leq 0 \\ 0 \leq x_{1}, x_{2}, x_{3} \end{matrix}

(23)

while the common variable is constrained by a target bound centered at the x₅^k,sys and ranged by the difference between the current target points and the previous optimal solution. When the current target point agrees with the previous optimal solution, the difference term d_sub,i equals zero, and the common variable will be located at the target point x₃^k,sys.

In the proposed BTC method, the weighted discrepancy terms in the multiobjective formulations have been avoided and displaced by the bounds associated with the targets and responses. The optimization process follows the divide-and-conquer strategy while target bounds are assigned to the subsystems, and the response bounds are updated back to the system level. The iteration stops when the termination criterion, the absolute difference between x_k and x_{k − 1} ≤ 10⁻³, is satisfied. The detailed iteration process of the system level is shown in Figure 1. The system-level optimal solutions of [x₁*,x₂*,x₅*] = [2.1529, 2.0725, 1.0707] are determined inside the updated bounds of responses illustrated by the solid and dash lines, respectively. Figure 2 demonstrates the iteration processes in the subsystems. The optimal solutions of x₅* = 1.0707 are found inside the updated target bounds, shown by the solid lines.

Figure 1:

System-level iteration process of solving example 1 by BTC.

Figure 2:

Iteration processes of solving example 1 by BTC in the (a) first and (b) second subsystems.

Moreover, the problem has been solved by the ATC using various settings of weighting coefficients. Two constant weights w = 1 or 150 and two increasing weights, w^{k = 0} = 1; w^k = 2w^{k − 1} or 5w^{k − 1} for k ≥ 1 are considered. Table 1 lists the results with the starting point of [3 2.2 1]_T. BTC uses only 12 iterations (Iter.) and 7569 function evaluations (FE) to find the reasonable solutions while ATC requires more to converge. The accuracy of the solutions is evaluated by the absolute difference with the results by all-in-one (AIO) methods, denoted as Error. The results show the BTC has 0.85% Error but the ATC produces Errors with more than 7%.

Table 1:

Results for example 1.

Methods	x _*	Iter.	FE	Error

AIO	[2.1491, 2.0759, 1.0746]	—	—	—
BTC	[2.1529, 2.0725, 1.0707]	12	7659	0.85%
ATC^a	Cannot converge
ATC^b	Cannot converge
ATC^c	[2.183, 2.0415, 1.0419]	15	2265	7.8%
ATC^d	[2.1177, 2.1053, 1.1055]	309	34863	7.2%

$^{a} w = 1$ .

$^{b} w = 150$ .

$^{c} w^{k = 0} = 1; w^{k} = 2^{k - 1} for k \geq 1$ .

$^{d} w^{k = 0} = 1; w^{k} = 5 w^{k - 1} for k \geq 1$ .

4.2. Nonconvex Geometric Programming Problem

The second example is a nonconvex geometric problem, which was used in [5, 18, 19, 21, 22], and the original programming problem is shown in (24).

\begin{matrix} min: x_{1}^{2} + x_{2}^{2} \\ s . t . g_{1} = (x_{3}^{- 2} + x_{4}^{2}) \times x_{5}^{- 2} - 1 \leq 0 \\ g_{2} = (x_{6}^{- 2} + x_{5}^{2}) \times x_{7}^{- 2} - 1 \leq 0 \\ g_{3} = (x_{8}^{2} + x_{9}^{- 2}) \times x_{11}^{- 2} - 1 \leq 0 \\ g_{4} = (x_{10}^{2} + x_{8}^{- 2}) \times x_{11}^{- 2} - 1 \leq 0 \\ g_{5} = (x_{11}^{2} + x_{12}^{- 2}) \times x_{13}^{- 2} - 1 \leq 0 \\ g_{6} = (x_{11}^{2} + x_{12}^{2}) \times x_{14}^{- 2} - 1 \leq 0 \\ h_{1} = x_{1} - {(x_{3}^{2} + x_{4}^{- 2} + x_{5}^{2})}^{1 / 2} = 0 \\ h_{2} = x_{2} - {(x_{6}^{2} + x_{7}^{2} + x_{5}^{2})}^{1 / 2} = 0 \\ h_{3} = x_{3} - {(x_{8}^{2} + x_{9}^{- 2} + x_{10}^{- 2} + x_{11}^{2})}^{1 / 2} = 0 \\ h_{4} = x_{6} - {(x_{12}^{2} + x_{13}^{2} + x_{14}^{2} + x_{11}^{2})}^{1 / 2} = 0 \\ x_{1}, x_{2}, \dots, x_{14} \geq 0 . \end{matrix}

(24)

The geometric problem is decomposed into the structure two levels, with constraints g₁, g₂, h₁, and h₂ in the system level, constraints g₃, g₄, and h₃ in subsystem 1, and constraints g₅, g₆, and h₄ in subsystem 2. The formulations for system level and subsystem level of the decomposed design problem in general ATC method and the proposed BTC method are provided in Figures 3 and 4.

Figure 3:

Design problem in general ATC method.

Figure 4:

Design problem in BTC method.

The BTC method converges with 15 iterations and 7632 function evaluations. The optimal solution and comparisons with other methods are shown in Table 2. Figure 5 presents the iteration process of solving the problem using the proposed BTC in the system level. On the other hand, Figure 6 demonstrates the iteration processes in the subsystems using the BTC. Similarly, the optimal common variables are found inside the target bounds, illustrated by the solid lines.

Table 2:

Results for example 2.

Solution	AIO	BTC	LATC [21]
Solution	AIO	BTC	m = 100	m = 10

x ₁	2.84	2.82	2.84	2.84
x ₂	3.09	3.10	3.07	3.09
x ₃	2.36	2.34	2.36	2.36
x ₄	0.76	0.76	0.76	0.76
x ₅	0.87	0.87	0.87	0.87
x ₆	2.81	2.82	2.80	2.81
x ₇	0.94	0.94	0.94	0.94
x ₈	0.97	0.97	0.97	0.97
x ₉	0.87	0.88	0.86	0.87
x ₁₀	0.80	0.80	0.79	0.80
x ₁₁	1.30	1.31	1.29	1.30
x ₁₂	0.84	0.84	0.84	0.84
x ₁₃	1.76	1.77	1.75	1.76
x ₁₄	1.55	1.56	2.84	1.55
Iter.	—	15	101	556
FE	—	7632	8124	136162
Function value	17.61	17.57	17.49	17.61
Error	—	0.56%	0.49%	0.15%

Figure 5:

System-level iteration process of solving example 2 by BTC.

Figure 6:

Iteration processes of solving example 2 by BTC in the (a) first and (b) second subsystems.

The comparisons of computational cost between the BTC method and methods proposed in [4, 18, 20 –22] is shown in Figure 7, including the quadratic penalty method (QP), the quadratic penalty method with block coordinate descent method (QP-BCD), the augmented Lagrangian method (AL), the augmented Lagrangian method with BCD (AL-BCD), the augmented Lagrangian with alternating direction method of multipliers (ALAD), and truncated diagonal quadratic approximation method (TDQA).

Figure 7:

Computational cost versus solution accuracy.

5. Conclusions

MDO problems have been decomposed into hierarchical structures with one system level and multiple subsystems. The main task in the decomposed multidisciplinary design optimization algorithm is to minimize the cost function and diminish the discrepancy between subsystems simultaneously. The well-known method, analytical target cascading, formulates the multiobjective formulations in terms of the weighted discrepancy functions in order to maintain the consistency between subsystems. However, the choice of the weighting coefficients is very problem dependent and improper selections of the weighting coefficients will lead to incorrect solutions.

In this paper, a new bounded target cascading method is proposed to solve the MDO to avoid the problems associated with the weighting coefficients. Instead of the point targets assigned for design variables in the analytical target cascading method, bounded targets are introduced in the new method. The target bounds are obtained from the optimal solutions in each level. Furthermore, the response bounds are established based on the optimal solutions from the subsystems and utilized to replace the weighted discrepancy functions of responses in the analytical target cascading method. If the common variables exist, they are coordinated based on their sensitivities with respect to design variables. The numerical examples validate that the single-objective bounded target cascading can efficiently and accurately find the optimal solutions.

How to efficiently formulate a hybrid framework of hierarchical ATC and nonhierarchical ATC for large scale MDO problems is the further work.

Footnotes

Nomenclature

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This research was partially supported by the National Natural Science Foundation of China under the Grant no. U1330130 and Fundamental Research Funds for the Central Universities under the Grant no. ZYGX2012J098.

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