針對複雜系統可靠度最佳化設計

Mathematical Foundations and

Convergence Characteristics for

Reliability-Based Optimization

Problems with Complex Structures

Final Report Submitted to the University

Development Division, the Office of R& D,

National Cheng Kung University

Kuei-Yuan Chan,

[email protected]

Assistant Professor, Department of Mechanical Engineering

March 20, 2009

Abstract

Large-scale design problems are high dimensional and deeply-coupled in nature.

The complexity of such large-scale systems prevents designers from solving them as a whole. Analytical target cascading (ATC) provides a systematic approach in solving decomposed large-scale systems that has solvable subsystems. By coordinating between subsystems, ATC can obtain the same optima as they were undecomposed. However, series of ATC iterations are still needed to reach a convergent coordination that may hinder the efficiency of ATC. In this re- search, a sequential linear programming algorithm is proposed to improve the efficiency of ATC. The proposed linearization techniques are applied to each ATC iteration, therefore each iteration has all linear subsystems that can be solved with high efficiency. The linearization techniques are integrated within a trust-region algorithm resulting in a more efficient ATC solution approach. The global convergence of this sequential linear programming algorithm is ensured by a filter to determine the acceptance of the optima at each iteration and the corresponding trust region. A geometric programming example demonstrates the efficiency of the proposed method over standard ATC solution process with- out loss of accuracy.

Keywords : sequential linear programming, analytical target cascading, trust region algorithm, multilevel systems

0.1 Introduction

Higher consumers’ expectations and better manufacturing technologies have re- sulted in much more complex engineering products nowadays. These complex systems combines theoretical foundations from various disciplines that are cou- pled to achieve the overarching system objectives. The linkings between disci- plines means they depend highly on each other and no single elements in the system can be taken out without violating the objective. The sizes of such com- plex systems are usually large, hence complex systems are also called ‘large-scale systems’.

Design problems for large-scale systems are challenging due to their complex structures and functional relations across disciplines. Although these systems can achieve better performance requirements and functionality, they also become less intuitive for engineers to analyze. Computer simulations or various numer- ical methods are essential elements for engineers to analyze the performance of a particular design. However, large-scale systems can seldom be analyzed via a single simulation. Simulations or engineering codes from different disciplines at different platforms are necessary in predicting the system performances. Even if a single simulation does exist to analyze the performance of a large-scale system, the cost is generally too high to be practical.

An alternative approach in solving complex large-scale systems is to decom- pose the original system into ‘smaller’ (easier-to-solve) subsystems. Various ap- proaches in decomposing a system have been discussed in the literature. Three common decomposition techniques in the design literature are ‘object’, ‘aspect’, and ‘sequential’ decompositions [1]. “Object decomposition divides a system into physical components. Aspect decomposition divides a system according to the different specialties involved in its modeling, and it is the basis for multidis- ciplinary optimization (MDO). Sequential decomposition is applied to problems involving flow of elements or information.”

Interactions of solvable subsystems provide a viable approach for design and analyses of unsolvable large-scale systems. The coordinations between the in- formation of each subsystem ensure the final result satisfy the overall system requirements. Among different coordination strategies, analytical target cas- cading (ATC) is one of the common approaches in the literature that focus on studying the accuracy and efficiency of subsystem interactions.

Originated from the automotive industries, analytical target cascading (ATC) has been demonstrated in many engineering applications such as tolerance allo- cation of multistation assembly line [2], vehicle suspensions [3], military vehicle design [4], and engine piston ring/cylinder liner design. In addition to engineer- ing field, ATC is also applied in multidisciplinary field such as architecture [5], enterprise decision making [6], and marketing [7]. In addition to the variety of applications, ATC has also been proven to be convergent to the original unde- composed systems by Michelena et al. [8].

While applied in many different fields and also proven to be convergent, the actual convergence speed of ATC rely highly on the nature of the problem.

A standard ATC process requires solving a sequence of nonlinear programming

(NLP) subproblems and then coordinates them until they are consistent. There- fore a convergent ATC optimum consists of a sequence of ATC iterations of several NLP subproblems. Practical convergent can sometimes be challenging.

This research develops an efficient algorithm in solving decomposed large- scale problems using ATC. After introducing the backgrounds of analytical tar- get cascading, the mathematical formulation of ATC will be overviewed in Sec- tion 2. ATC on linear multilevel systems will be studied in Section 3. Section 4 introduces the sequential linear programming algorithm in solving ATC prob- lems. Section 5 describes the subproblem generation step and linearization process of the algorithm in more detail via a bi-level system. Section 6 discusses the convergence characteristics of the algorithm. A geometric programming ex- ample is demonstrated using the proposed algorithm in Section 7, followed by the concluding remarks on Section 8.

0.2 Analytical Target Cascading

Analytical target cascading (ATC) solves decomposed large scale optimization problems in a systematic fashion. Let a large-scale system be decomposed into subsystems of different hierarchies (levels) as shown in Fig.1, which hasi levels with j elements at each level. Design targets T are assigned to the top-level system. Design variables of the elementij, also called local variables, are x_ij. Local constraints of elementij are gij and h_ij for inequality and equality con- straints, respectively. Rⁱ_ij are the responses from the elementij to the (i − 1)th level while Rⁱ⁻¹_ij are the targets to the elementij from the (i − 1)th level where Rⁱ_ij= r_ij(x_ij). yⁱ_ij are sharing variables between elements at the same leveli.

R⁰_0l

xij yⁱ_ij yⁱ⁻¹ij

Rⁱ⁻¹ Rⁱ ij

ij

T

rij

g_ij hij

Rⁱ_(i+1)k Rⁱ⁺¹

(i+1)k

xij yⁱ_ij

Rⁱ⁻¹ Rⁱ ij

ij

rij

g_ij hij handled by the

parent level

Rⁱ_(i+1)k Rⁱ⁺¹

(i+1)k Y_ijⁱ,yⁱ_ij

Xij,xij

Rⁱ⁻¹ Rⁱ ij

ij

rij

g_ij handled by the

parent level Elements

...

...

Levels

0 1

i

0 1 j

Figure 1: Hierarchical structure of ATC

minimize

{¯xij,yⁱ_(i+1)j,ε^Rij,ε^yij}kR⁰_0l− Tk +

N−1

X

i=0

X

j∈Ei

ε^R_ij+

N−1

X

i=0

X

j∈Ei

ε^y_ij

s. t. X

k∈Cij

kw^R_(i+1)k◦ (Rⁱ_(i+1)k− Rⁱ⁺¹_(i+1)k)k²₂≤ε^R_ij X

k∈Cij

kS_kw^y_(i+1)j◦ (S_kyⁱ_(i+1)j− y₍ⁱ⁺¹_i+1)k)k²₂≤ε^y_ij (1)

g_ij(¯x_ij) ≤ 0, hij(¯x_ij) = 0, Rⁱ_ij = r_ij(¯x_ij),

¯ xij=h

xⁱ_ij, yⁱ_ij, Rⁱ_(i+1)k1, · · · , R(i+1)k_Cijⁱ

iT

∀j ∈ E_i, i = 0, 1, · · · , N

The decomposed problem in Fig.1 tries to solve each subproblem individually while at the same time ensure the consistency between subsystems. Eq.(1) shows the overall optimization formulation of ATC as a whole. The objective of Eq.(1) includes minimizing the Euclidean norm of the difference between targets and responses as well as minimizing the consistency between levels and between components. The feasibility of Eq.(1) requires satisfying the relaxed consistency constraints to the original undecomposed problem. This relaxation can be imposed as designers’ preferences by assigning different weights (w^R_(i+1)k and w^y_(i+1)j). The ◦ symbol in Eq.(1) represents element-by-element matrix product.

Figure 2 shows the information flows in and out of each subsystemij in Fig.1.

In addition to the local design variables x_ij, the responses from the (i + 1)th level, the targets passed down from the (i−1)th level , and the linking (sharing) variables are all the inputs to subsystemij. Outputs from ij are the responses to the upper level, the targets to the lower level, and the linking variables values.

The goal for the elementij is to match the target from the (i − 1)th level while keeping the consistency between itself and the (i+1)th level. After assum- ing all equality constraints are removed explicitly or implicitly, we can obtain the optimization problem for elementij as shown in Eq.(2). The consistency constraints in Eq.(2) have been moved to objective function by applying mono- tonicity principals with respect toε^R_ij andε^y_ij. ¯x_ij in Eq.(2) is the vector of all inputs for elementij.

R⁰_0l

xij yⁱ_ij

y_ijⁱ⁻¹

Rⁱ⁻¹ Rⁱ_ij ij

T

rij

gij hij

Rⁱ_(i+1)k Rⁱ⁺¹

(i+1)k

xij y_ijⁱ

Rⁱ⁻¹ Rⁱ ij

ij

rij

gij

hij handled by the

parent level

Rⁱ_(i+1)k Rⁱ⁺¹

(i+1)k Yⁱ_ij,yⁱ_ij

Xij,xij

Rⁱ⁻¹ Rⁱ ij

ij

rij

gij handled by the

parent level

Figure 2: Subsystemij within ATC

minimize

{¯xij,yⁱ_(i+1)j}kw_ij^R◦ Rⁱ_ij− Rⁱ⁻¹_ij k²₂+ kS_jw^y_ip◦ (S_jyⁱ⁻¹_ip − yⁱ_ij)k²₂

+ X

k∈Cij

kw^R_(i+1)k◦ (Rⁱ_(i+1)k− Rⁱ⁺¹_(i+1)k)k²₂ (2) +P

k∈CijkS_kw^y_(i+1)j◦ (S_kyⁱ_(i+1)j− yⁱ⁺¹_(i+1)kk²₂ subject to g_ij(¯x_ij) ≤ 0, Rⁱ_ij = r_ij(¯x_ij),

¯ x_ij =h

xⁱ_ij, yⁱ_ij, Rⁱ_(i+1)k1, · · · , R(i+1)k_Cijⁱ

iT

0.3 ATC with Linear Subsystems

After discussing the formulation and terminology of ATC, we will explore the performance of ATC for multilevel systems with all subsystems being linear.

The results of this section show that after initializing all required linking vari- ables and responses, ATC requires only one iterations to converge for linear subsystems. This result inspires us to take advantages of linear systems. There- fore this section can be viewed as an introductory section to the generalized approach proposed by this research.

Before discussing the solution process of ATC with linear subsystems, the definition of linear subsystems needs to be clarified. The quantity of interest includes the design variables x, the objective function f, the inequalities g, equality constraints h, the linking variables y, and the responses R for each subsystem.

A linear subsystem is a subsystem with linear objective functions and linear constraints. The source of confusion might come from the ATC target matching objective, kR − Tk²₂. Even if the objective functions R = f (x) are linear, the target matching objective function will not be linear. The nonlinearity

of the target matching objective in ATC will actually be favorable in solution efficiency. In this research we propose to use the original form of target matching in ATC with linear subsystems. Although this results in nonlinear programming problem for each subsystem, it has certain advantages.

Figure 3 illustrates a linear objective function R = f(x) versus its target matching kR − T k²2. When the linear objective is applied in addition to the corresponding linear constraints, the problem becomes an LP problem. The op- timum of the LP problem locates in the vortex of the feasible design space. Tar- get matching objective changes the original linear objective into two segments of linear functions. The location of the non-differentiable point is determined by whether the target is an achievable target. When the target is achievable, the problem with target matching objective as shown in Eq.(3) will have interior optimum. The resulting objective will beR^∗ =T . On the other hand, if the the target is unachievable, the target matching objective will be linear and the optimum of Eq.(3) will be again at the vortex.

minx kR − T )k²2 (3)

s.t. g(x) ≤ 0 R = f(x)

What have been described above can be extended to linear multilevel prob- lems. For each subsystems, the linear constraints creates a polyhedron feasible design space. If the target cascaded down from the level above is an achievable target, the subsystem will match that target and problem converged. If the cascaded down target is unachievable, the subsystem optimum will be at the vortex closest to the target. In both situations, the problem converges. Since the feasible space is a polyhedron, no hidden optima will be expected unless the cascaded down target changes its value.

Let us look at the top level subsystem in this case. If the target passed down was an achievable target by the lower level, ATC process converges. On

2 4 6 8 10

2 4 6 8 10

Target Matching R= f (x)

x

Figure 3: Linear Objective and the Matching Objective in ATC

the other hand, if the target was an unachievable target for the lower level, the optimal achievable value that is closest to the target is passed up from the lower level. Top level subsystem has to be aware of the new restrictions on its feasible design space (remains a polyhedron). The new compromised target for the lower level subsystem then becomes achievable. Similar to the lower level subsystem, the problem will converge in both scenarios.

ATC formulation for linear multilevel systems requires only one iteration to converge. This superb property will then be applied to nonlinear multilevel systems. The idea is simply to linearize the nonlinear model as linear at a local point such that the each ATC iteration converges within one iteration.

0.4 Sequential Linearization Approach in Ana-

lytical Target Cascading Problems

The proposed linearized analytical target cascading (LATC), shown as the flowchart in Fig.4, is an extension of [9] to deterministic variables. This al- gorithm, like many trust-region algorithms [10], consists of five main stages:

initialization, subproblem generation, subproblem solution, acceptance of the trial point, and trust region update. Details of each step are described below.

Initialization :

Provide initial design, trust region, and other parameter values needed in optimization process. The iteration counter is set at k = 0, and values are provided for the initial design x⁽⁰⁾, for the algorithm parameters {δ, γ, λ, ω, τ, ζ}

to be discussed further below, and for an initial trust region ∆⁽⁰⁾.

The trust region ∆ is a ‘move limit’ providing the maximum step length for a given local linear approximation model. For a move limit ∆^(k) at the kth subproblem, the design variables x^(k+1) are limited:

kx^(k+1)− x^(k)k ≤ ∆^(k). (4) Although the trust region may later be reduced and updated as the design approaches the optimum, the initial trust region is important for faster conver- gence [11]. Determining an initial trust region is based on the distance from the initial design point to the the approximations of constraint boundaries g = 0.

The boundary approximations are found via linearizinggj at x⁽⁰⁾ as

ˆg_j(X) ≈g_j(x⁽⁰⁾) + ∇g^T_j(x⁽⁰⁾) · (x − x⁽⁰⁾) ≤ 0 (5)

∆⁽⁰⁾ is selected to satisfy Eq.(6) below, wheren is the number of variables and m the number of constraints.

∆⁽⁰⁾= max

j













kg_j(x⁽⁰⁾)k

n

X

i

k∂g_j

∂xik













, j = 1, · · · , m (6)

Updated Design

Acceptable?

Reduce Trust Region

Converged?

Optimum Initial Design

Current Design

Yes

Update Design

Yes No Start up step

Initialization Step

LATC Step 1

LATC Step 2

No initialization

subproblem generation

subproblem solution

acceptance of the trial point

trust region update

Figure 4: LATC Algorithm

Subproblem generation :

Based on the current design point, subproblems in ATC is linearized to form LATC. Within each iteration, the LATC consists of four main steps, namely the startup step, the initialization step, the LATC step 1 and the LATC step 2.

The functionality of each step is described as follows:

• start up step : Provide the initial starting point and the initial trust region of all subsystems. If some of the subsystems are not provided with either information, the starting point and trust region can be calculated via

• initialization step : Based on the trust region and starting point at each level, a top-down approach is employed to the LATC in obtaining the initial target values of each subsystem.

• LATC step 1 : Given the initial targets from the levels above of each

subsystem, solve for the optimal feasible responses of the linearized ATC via bottom-up approaches.

• LATC step 2 : Re-solve the LATC problem with the responses from the levels below in step 1. Once step 2 is terminated, the optimal solutions will be the optima of the LATC at iterationk.

Subproblem generation is the main step in LATC, therefore the mathemat- ical developments will be discuss in more details using a bilevel example in the next section.

Subproblem solution :

The optimum of LATC subproblem provides an updated design ‘candidate’.

Acceptance of the trial point :

To ensure convergence, a filter is used to determine whether the ‘candidate’

design from the subproblem solution step is acceptable as the next iterant point.

The criteria for acceptability are adopted from the filter algorithm introduced by Fletcher et al. [12]. In this algorithm, a filter is used to replace penalty functions in determining the acceptance of the trial point towards establishing global convergence. If this ‘candidate’ is acceptable, the algorithm will check its convergence criteria as shown in Fig.4. If the problem converged, we have the final optimum, otherwise the acceptable ‘candidate’ becomes the new design point and the algorithm iterates until convergence.

Before proceeding with the definition of the filter we define two terms for specific use in the present context: infeasibility and dominance. Infeasibility of thekth design x^(k) is the maximum constraint violation at thekth design and is represented ash(g(x^(k))):

h(g(x^(k))) = max

j (0, gj(x^(k))) (7)

With this measure of infeasibility, for every design point, a pair that indicates the infeasibility and the objective function values {h^k, f^k}, where h^k =h(g(x^(k))) andf^k =f(x^(k)), can be computed. Designk dominates design l if both h^k≤h^l andf^k≤f^l; then we say that the kth pair dominates the lth pair.

A filter is defined as a set containing only non-dominated pairs and treatsf andh as separate objectives without the need for weights. Figure 5 illustrates the concept of using a filter to determine if a trial point is acceptable. The line in the figure represents all current filter entries. Points A, B, and C are currently in the filter. A new trial point D is dominated by point B and is not accepted to the filter. The filter contains only current Pareto points in the {h, f} bi-objective optimization. A point that is not dominated by previous pairs is accepted.

In order to enhance convergence characteristics, Fletcher et al. [12] suggested that two non-zero parameters ω and γ be used, such that the condition for a point being acceptable to the filter be that its {h, f} pair satisfies either h ≤ ωh_l orf ≤ fl−γhl for all l ∈ F, where F denotes the current set of filter entries, and 1 > ω > γ > 0. An iteration is f-type if the iteration is acceptable to

h

f dominated by B

A B

C D

Figure 5: Non-dominated entries in the filter

the filter and has an actual reduction ∆f^k and a predicted reduction ∆l^k that satisfy the sufficient reduction criteria in Eq.(8) below, where h^k is the short form of h(g⁰(µ^k_X)) and ζ is a user-selected parameter with values within the range (γ, 1).

∆f^k =f(µ^k+1X ) −f(µ^kX) ≥ζ∆l^k , with ζ ∈ (γ, 1) (8)

∆l^k = ∇f^T · s ≥λ(h^k)² , with λ > 0

If an iteration acceptable to the filter has a pair {h, f} that satisfies Eq.(9), then is is called an h-type.

∆l^k< δ(h^k)² (9)

An f-type iteration aims at reducing the objective function value while possibly increasing infeasibility. An h-type iteration, on the other hand, aims at decreas- ing infeasibility while possibly increasing the objective function. Following [12], not all acceptable {h, f} entries are added to the filter; indeed only h-type iter- ations are added into the filter.

Trust region update :

While an unacceptable ‘candidate’ is encountered, the trust region will be re- duced by half and the current LATC with the reduced trust region will be solved again. This trust region reduction step will be performed until an acceptable

‘candidate’ is obtained.

0.5 Subproblem Generation Process

Section 4 has provided guidelines of LATC for multilevel systems. The main difference between a standard SLP algorithm and the proposed methodology lies in the subproblem generation using sequential linearization to approximate non- linear hierarchical systems. In this section, the mathematical implementations of the subproblem generation process will be discussed in more details.

Consider the simplest multilevel problem, a bi-level system with one subsys- tem at each level as shown in Fig.6. Notations in Fig.6 are : the overall targets

to the top level, T; the local design variables and constraints of the top level, x0 and g0, respectively; the responses from the top level R⁰₀ = r0(¯x0) where

¯x0=x0, R¹1; the targets to the lower level, R⁰₁; the local design variables and constraints of the bottom level, x1and g1, respectively; and the responses from the bottom level, R¹₁.

R⁰₁ R¹

1

x1

x⁰ R⁰₀

T

r1

g₁ g0

r⁰

Figure 6: Bi-level, single element system Standard ATC formulation

The ATC formulation for the top and bottom level subsystems as adopted from Eq.(2) are Eq.(10) and (11), respectively.

¯min

x₀,εR

kR⁰₀− Tk²₂+ εR

s.t. g0(¯x0) ≤ 0 (10) R0= r0(¯x0)

kR¹₁− R⁰₁k ≤ εR

¯

x0= [x0, R¹1]

minx1

kR¹₁− R⁰₁k²₂

s.t. g1(x1) ≤ 0 (11) R¹₁= r1(x1)

LATC formulation

In the proposed LATC method, a linearized subproblem is generated at each iteration of the optimization. Applying the LATC concept on this bi-level system, we can obtain the linear approximations of Eq.(10) and Eq.(11) at the kth iteration as Eq.(12) and (13), respectively. Quantities denoted as ˆA^(k) are the linear approximations of A at iterationk.

¯min

x0,εR

k ˆR⁰₀^,(k)− Tk²₂+ ε_R s. t. ˆg⁽₀^k)(¯x0) ≤ 0 (12) kR¹₁− R⁰₁k ≤ max(εR, ∆⁽_R^k)_,0)

kx0− x⁽₀^k)k ≤ ∆⁽₀^k)

minx¯1

fˆ1^(k)(¯x1) s. t. ˆg⁽₁^k)(¯x₁) ≤ 0(13) k¯x1− ¯x⁽₁^k)k ≤ ∆⁽₁^k)

∆ in Eq.(12) and (13) are the trust regions, the essential elements in SLP for convergence. Originally only the trust regions of the bottom and top level design

variables, ∆⁽₁^k) and ∆⁽₀^k), and at each iteration are known. These trust regions determines the limits the design variables can move. Therefore the bottom level trust regions will be propagated to the top as the top level trust regions

∆⁰⁽₀^k). Top level design variables R⁰₁ is restricted by the trust region specified via the consistence toleranceεR. To allow designers specifying the trust regions of R¹₀, Eq.(12) uses the minimal value ofεR and ∆⁰⁽₀^k)as the trust region of the subproblem Eq.(12).

With the LATC formulation at the kth iteration available, the subproblem generation of the algorithm consists of the following steps :

Lower Level Top Level

1

2 3

r

R

1

R

1

min

f ˆ

(¯ x

) + ε

s. t. ˆ g

(¯ x

) ≤ 0

¯

x

= [x

, R

]

"R

− R

" ≤ max(ε

, ∆

)

"x

− x

" ≤ ∆

min¯x1

fˆ₁^(k)(¯x₁) s. t. ˆg^(k)₁ (¯x₁) ≤ 0

"¯x₁− ¯x^(k)₁ " ≤ ∆^(k)₁

x ¯

∆

∆

R

1

4

Figure 7: Bi-Level LATC iteration

• startup run 1

Provide the initial starting point and the initial trust region of all subsystems.

If some of the subsystems are not provided with either information, the starting point and trust region can be calculated via

R¹⁽⁰⁾₁ = r1(¯x⁽⁰⁾₁ ) (14)

∆⁰⁽⁰⁾₀ = ∂r1

∂¯x1

   _x¯(0)

1

∆⁽⁰⁾₁ (15)

In this step, we assume that the bottom level startup information, ¯x⁽⁰⁾₁ and

∆⁽⁰⁾₁ , is known.

• initialization run 2

The top level solve the following problem

minx¯₀

fˆ0⁽⁰⁾(¯x₀) + ε_R s. t. ˆg₀⁽⁰⁾(¯x0) ≤ 0

¯

x₀= [x₀, R⁰1] (16)

kR⁰₁− R¹⁽⁰⁾₁ k ≤ max(εR, ∆⁰⁽⁰⁾0 ) kx0− x⁽⁰⁾₀ k ≤ ∆⁽⁰⁾₀

without knowing the values of R¹₁^,(0). This is a common initialization process in standard ATC. The optimal design at this stage R^0∗₁ is cascaded down to the level below.

• The LATC bottom-up iteration 3

Once received the targets from the level above, the bottom level problem then solve the problem in Eq.(17).

min¯x1

fˆ1⁽⁰⁾(¯x1)

s. t. ˆg₁⁽⁰⁾(¯x1) ≤ 0 (17) k¯x1− ¯x⁽⁰⁾₁ k ≤ ∆⁽⁰⁾₁

The optimal design of this stage R^1∗₁ is propagated up to the level above.

• The LATC top-down iteration 4

Upon receiving the responses from the lower levels, Eq.(16) is re-run with the updated R¹₁^,(k) = R^1∗₁ The LATC for iteration k will converge at this point since all subsystems are linear and no further improvements can be made. The iteration counter increments by one k = k + 1. Design point is progressed to the optimum of this iteration.

0.6 Convergence arguments

The convergence arguments of the proposed algorithm can be illustrated using Fig.8. Consider the original undecomposed all-in-one (AIO) nonlinear problem (NLP) be AIO-NLP. AIO-SLP represents using sequential linear programming (SLP) algorithm for the AIO system. ATC-NLP solves the decomposed nonlin- ear problem using standard ATC, and ATC-SLP is the approach proposed in this research, the LATC approach.

The convergence arguments from Michelena et al. [8] states : after decom- posing the problem and formulate as ATC structure, the ATC-NLP will be

AIO-NLP

AIO-SLP

ATC-NLP

ATC-SLP

1

2

3

4

Figure 8: LATC Convergence Arguments

convergent to AIO-NLP regardless of the algorithms used. Therefore we are able to link ATC-NLP with AIO-NLP via 1 .

Fletcher et al. [12] have shown that the filter-based algorithm as the one used in this research ensures that the entries in the filter will eventually satisfy the Friz-John optimality condition. In addition, the design that are initially not acceptable to the filter will eventually be accepted as the trust region goes to zero. Therefore the link between AIO-NLP and AIO-SLP 2 is also valid.

The LATC algorithm intends to link the standard ATC-NLP approach with the ATC-SLP as shown as 4 in Fig.8. With 1 and 2 being true, we need to show that 3 is true, which means for each iteration k, AIO-SLP^(k) = ATC- SLP^(k).

Without loss of generality, let us look at an NLP problem with objective functionf and constraints g ≤ 0 and its decomposed bilevel system, which has top level objective function f0, constraints g₀ ≤ 0 and bottom level objective f1 and constraints g₁≤ 0. AIO-SLP approximate objective and constraints as f and ˆg. ATC-SLP approximate the top level objective and constraints as asˆ fˆ0 and ˆg0 and bottom level objective and constraints as as ˆf1and ˆg1

∇ ˆf^(k)+ µ∇ˆg^(k)= 0

µˆg^(k)= 0 (18)

µ ≥ 0, ˆg^(k)≤ 0

The Karush-Kuhn-Tucker (KKT) conditions of kth AIO-SLP are shown in Eq.(18) with Lagrange multipliers µ. The KKT conditions of ATC-SLP top and bottom subsystems are Eq.(19) and (20) with Lagrange multipliers being µ₀ and µ₁, respectively.

∇ ˆf0^(k)+ µ0∇ˆg⁽₀^k)= 0 µ₀gˆ⁽₀^k)= 0 (19) µ0≥ 0, ˆg⁽0^k)≤ 0

∇ ˆf1^(k)+ µ1∇ˆg⁽₁^k)= 0 µ₁gˆ⁽₁^k)= 0 (20) µ1≥ 0, ˆg⁽1^k)≤ 0

Theα-separable scheme in ATC [13] ensures the convergence of the proposed method. The function dependence table (Fig.9) decomposes the AIO-SLP into

AIO-NLP

AIO-SLP

ATC-NLP

ATC-SLP

1

2

3

4

linking variables

x

x

y

A

A

fˆ0

ˆ g0

ˆ g₁ fˆ₁

Figure 9: Bi-Level Function Dependence Table

ATC-SLP with local design variables x₀ and x₁ for top and bottom level sub- problems. After this decomposition, we can combine Eq.(19) and (20) as

∇ ˆf0^(k)+ ∇ ˆf1^(k)+ [µ₀, µ1] ·h

∇ˆg₀^(k), ∇ˆg⁽1^k)

iT

= 0 [µ0, µ1] ·h

gˆ⁽₀^k), ˆg⁽1^k)

iT

= 0 (21)

[µ0, µ1] ≥ [0, 0],h

ˆg⁽₀^k), ˆg0^(k)

i≤ [0, 0]

Therefore the α-separable additive objective functions ensure the conver- gence of ATC-SLP to AIO-SLP at each iterationk.

Summing up the statements above, ATC-SLP will be convergent to AIO- SLP, AIO-SLP will converge to AIO-NLP and then ATC-NLP will converge to AIO-NLP. The link 4 in Fig.8 is thus proven to be valid.

0.7 Case Demonstrations

geometric programming The geometric programming problem as shown in Eq.(22) has 14 design variables, six inequality constraints and four equality con- straints. This problem has been studied extensively [3, 14] and will be revisited by applying the proposed LATC algorithm.

After decomposed into a bi-level structure as shown in Fig.10, we can obtain the top-level problem (GS-PS) and two lower-level subproblems (GS-PSS1 and GS-PSS2). The optimization problems of each subsystems are Eq.(23), (24), and (25) for GP-PS, GP-PSS1, and GP-PSS2, respectively.

minx f = x²1+x²2 (22) subject to g1:x⁻²3 +x²4−x²5≤ 0, g2:x²5+x⁻²6 −x²7≤ 0

g3:x²8+x²9−x²11≤ 0, g4:x⁻²8 +x²10−x²11≤ 0 g5:x²11+x⁻²12 −x²13≤ 0, g6:x²11+x²12−x²14≤ 0 h1:x²1−x3²−x⁻²4 −x²5= 0, h2:x²2−x²5−x²6−x²7= 0

h3:x²3−x²8−x⁻²9 −x⁻²10 +x²11= 0 h4:x²6−x²11−x²12−x²13+x²14= 0

∀x = [x1, x2, · · · , x14] ≥ 0

GP-PS

GP- PSS1

GP- PSS2

Figure 10: Bi-level structure of GP

GP-PS : minimize

x1,··· ,x7,ε1,ε2,ε3w0(x²1+x²2) +

3

X

i=1

wiεi

subject to

(x11−x^L11ss1)²+ (x11−x^L11ss2)²≤ε1 (23) (x3−x^L3)²≤ε2, (x6−x^L6)²≤ε3

g1:x⁻²3 +x²4−x²5≤ 0 g2:x²5+x⁻²6 −x²7≤ 0

GP-PSS1 : minimize

x3,x8,··· ,x11 w10(x3−x3^U)²+w11(x11−x^U11)²

subject to g3:x²8+x²9−x²11≤ 0 (24) g4:x⁻²8 +x²10−x²11≤ 0

GP-PSS2 : minimize

x6,x11,··· ,x14 w20(x6−x6^U)²+w21(x11−x^U11)²

subject to g5:x²11+x⁻²12 −x²13≤ 0 (25) g6:x²11+x²12−x²14≤ 0

This problem is solved via three different methods and their results are compared. The first method is all-in-one approach where Eq.(22) is solved without decomposition. The second approach is the standard ATC solution approach as can be see in [3]. The optimization routine ‘fmincon’ in Matlab is used to solve each ATC iteration. Termination criteria, Eq.(26), of the ATC approach is set atρ = 0.01.

kx⁽ⁱ⁺¹⁾− x⁽ⁱ⁾k ≤ρ (26)

The same problem is solved using the proposed LATC approach with the following parameters : δ = 0.2, σ = 0.1, ω = 0.99, γ = 0.01. ρ in Eq.(26) for the LATC approach is also set at 0.01. Table 1 lists the results of the geometric programming problem using three different approaches. The all-in-one (AIO) results are considered as the benchmark.

As can be seen from Table 1, AIO, ATC, and LATC all have comparable re- sults. The differences between optimal objective function values are within 5%.

However the efforts in obtaining these results are significantly different. AIO and ATC both requires about five times more function evaluations than LATC.

Standard ATC requires much more ATC iterations to converge comparing to LATC due to the nonlinearity of the problem.

The optimization history of both ATC and LATC for objective function and linking variablesx3,x6, andx11are shown in Fig.11 and 12, respectively. As can be seen, although take much less iteration to converge, the objective function of LATC oscillates. This is due primarily to the linear approximations to nonlinear functions.

Figure 13 illustrates all the filter entries in the LATC approach. The filter starts with a large constraint violation and small objective function value. As iteration goes, filter accepts design points that will reduce the constraint vio- lation values. Consequently, the objective function value increases. The upper right diagram of Fig.13 zooms in the area where design is convergent. Since the filter entries have decreasinghk values,hk will eventually be zero meaning that no constraint violations are allowed. The algorithm will then accept design points that reduce objective function valuefk.

0.8 Concluding Remarks

A sequential linear programming algorithm is introduced for analytical target cascading in solving design problems of large-scale systems. By sequentially linearize nonlinear multilevel problems, this algorithm is capable of improving the efficiency in finding the optima without loss of accuracy.

0 50 100 150 200 250 300 20

40 60 80 100

Optimal outputs from the TOP

0 50 100 150 200 250 300 2

2.2 2.4 2.6 2.8

x3

0 50 100 150 200 250 300 2.5

3 3.5 4 4.5 5

x6

0 50 100 150 200 250 300 1

1.5 2 2.5 3

x11

Figure 11: Optimization history of standard ATC

0 50 100 150 200

20 40 60 80 100

Optimal outputs from the TOP

0 50 100 150 200

2.5 2.6 2.7 2.8 2.9 3 3.1 3.2

x3

0 50 100 150 200

2.6 2.8 3 3.2 3.4 3.6

x6

0 50 100 150 200

1 1.5 2 2.5

x11

Figure 12: Optimization history of LATC

A filter consists of constraint violations and objective function values is used in the algorithm to determine where the ‘candidate’ design point at each it- eration is acceptable. This filter also ensures the convergence of the proposed algorithm to the original undecomposed system.

Although proven to be convergent and also shown to be efficient in the example, the practical convergence of the LATC method might encounter dif- ficulties. The initial design point is an important factor in the convergence of the algorithm. Infeasible design point requires a large step to be feasible in the subsequent iteration. Eq.(6) is important in providing an efficient convergent result. Whenever possible, a feasible starting point is suggested for the proposed LATC algorithm to ensure feasible direction is found.

This filter-based algorithm has parameters that need to be determined by users. Although the values of these parameters will not change a convergent result, they might affect efficiency significantly. The suggested ranges of each

Table 1: Geometric programming example results comparisons AIO Standard ATC LATC

x1 2.84 2.70 2.95

x2 3.09 3.08 3.11

x3 2.36 2.19 2.50

x4 0.76 0.76 0.76

x5 0.87 0.89 0.86

x6 2.81 2.81 2.84

x7 0.94 0.91 0.93

x8 0.97 0.94 1.02

x9 0.87 1.03 0.75

x10 0.79 0.91 0.77

x11 1.30 1.36 1.32

x12 0.84 0.84 0.89

x13 1.77 1.78 1.73

x14 1.55 1.56 1.59

ATC ite. N/A 112 77

Func. evals 25, 173^[9] 26,651 5772

parameter as mentioned in Section 4 is strongly recommended.

The proposed approach linearizes functions of each subsystems. However, the target matching objectives are still nonlinear. Even the subproblems are solved using nonlinear programming, the overall solution approach resembles that of the sequential linear programming as shown in [9].

0.9 Acknowledgments

This research was partially supported by the National Science Council (NSC96- 2221-E-006-215) in Taiwan and the Office of R& D at the National Cheng Kung University. The supports are gratefully acknowledged.

0 10 20 30 40 50 60 70 80 90 100 15

20 25 30 35 40 45 50 55 60 65

!2 !1 0 1 2 3 4 5

16 17 18 19 20 21 22 23 24

fk

hk

Figure 13: Filter entries

0.10 Self-Evaluation

The current status of the algorithm can be generalized as a software package that can be readily available for public. Results and the progressing of the research resembles what have been proposed in the initial proposal. The sec- ond phase of the algorithm requires adding random variables in the multilevel structures and study the impacts to the algorithm. ASME design community is highly interested in any methodology that can handle complex problems un- der uncertainty and the results of this research can be significant after further investigations.

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