Conclusions

In this research, we solve the “Trust Region Subproblem (TRS)”, using SVD.

With the help of SVD, we enhance the TRS algorithm by proposing a better lower bound for safeguarding the Newton’s iterates and also provide a new mechanism to adjust the trust-region radius dynamically. To solve the SMOO problem, a nonlinear constrained problem, we then develop the “Generalized Reduced Trust Region (GRT)”

search method with the above modifications. We have also proved the convergence of the proposed GRT algorithm.

To verify our algorithm, a test problem and three SMOO problems were studied.

The following results were observed:

1. The GRT search method avoids the zigzagging phenomena often incurred by the GRG method and gets a better solution.

2. The GRT search combined with the Zoutendijk search method can effectively reach the optimal point in every case.

3. The GRT search method with dynamic radius adjustment can reduce the number of iterations and computing time by about 5% to 10% as compared to the conventional radius adjustment in a large scale problem such as the cases of the DFM problem and the robust semiconductor supply chain optimizations.

4. Compared against Lingo’s solution, our search algorithm usually converges

Although, the four cases lend support to this research, there are still much room to be

improved.

1. In order to deal with any kind of optimization problems, the Hessian matrix can be calculated and updated more efficiently. Moreover, the Hessian matrix indeed could be approximated for shorter computing time [4].

2. In late 1980s, many researchers try to solve the trust region problem more efficiently like the dogleg method and indefinite dogleg method [5, 14, 22].

They are all approximate techniques of the trust region problem and also lead to the same global and local convergence properties, i.e., these methods can shorten the computing time without loss of optimality conditions.

3. In this research, the SVD replaces the Cholesky factorization to compute and perform the Newton’s iterates. However the SVD is too costly for large matrices, the method is applicable only for small problems. There have been many researches on how to reduce the computational efforts of the Cholesky factorization [10].

4. Although this research propose the convergence property of the GRT algorithm but the Corollary 3.1 does not cover the Line Search method.

There have been some algorithms combine the Trust Region method and Line Search method and also provider convergence properties [8, 15, 20]

5. In this research, we propose a dynamic strategy to update the trust-region radius. In 2005, some researchers discussed about the trust region radius update [19].

6. Zoutendijk’s method sometimes incurs the zigzagging phenomenon. It may influence the search performance of the GRT search. There should be some

enhancements when searching a feasibly improving direction at the boundary of feasible set.

7. Multiple initial solutions could increase the probability to reach the global optimum, but there exists a systematic method. In Lingo’s algorithm, the

“Branch and Bound” algorithm is adopted. It divides the nonlinear programming problem into several approximate convex optimization problems, then, searches the global optimum iteratively.

REFERENCE

[1] M. Avriel,

Nonlinear Programming: Analysis and Methods, Dover

Publications, 2003.

[2] M. S. Bazaraa, H. D. Sherali, C. M. Shetty and Wiley InterScience (Online service), Nonlinear programming [electronic resource] : theory and

algorithms, Wiley-Interscience, Hoboken, N.J., 2006.

[3] A. D. Belegundu and T. R. Chandrupatla, Optimization concepts and

applications in engineering, Prentice Hall.

[4] R. H. Byrd, H. F. Khalfan and R. B. Schnabel, Analysis of a Symmetric

Rank-One Trust Region Method, SIAM, 1996, pp. 1025.

[5] R. H. Byrd, R. B. Schnabel and G. A. Shultz, Approximate Solution of the

Trust Region Problem by Minimization over Two-Dimensional Subspaces,

Mathematical Programming, 40 (1988), pp. 247-263.

[6] A. Chen, P. S. Guo and P. Lin, Statistical analysis and design of

semiconductor manufacturingsystems, 2000, pp. 335-338.

[7] N. R. Draper, Ridge analysis of response surfaces, JSTOR, 1963, pp. 469-479.

[8] J. Y. Fan, W. B. Ai and Q. Y. Zhang, A line search and trust region algorithm

with trust region radius converging to zero, Journal of Computational

Mathematics, 22 (2004), pp. 865-872.

[9] S. K. S. Fan, A different view of ridge analysis from numerical optimization, Engineering Optimization, 35 (2003), pp. 627-647.

[10] S. K. S. Fan, THE HOUSEHOLDER TRIDIAGONALIZATION STRATEGY

FOR SOLVING A CONSTRAINED QUADRATIC MINIMIZATION PROBLEM, Taylor & Francis, 2001, pp. 261-277.

[11] D. G. Luenberger, Linear and Nonlinear Programming, Springer, 2003.

[12] D. C. Montgomery and D. C. Montgomery, Design and analysis of

experiments, Wiley New York, 1991.

[13] J. J. More and D. C. Sorensen, Computing a Trust Region Step, Siam Journal on Scientific and Statistical Computing, 4 (1983), pp. 553-572.

[14] J. Nocedal, S. J. Wright and SpringerLink (Online service), Numerical

Optimization [electronic resource], Springer Science+Business Media LLC.,

New York, NY, 2006.

[15] J. Nocedal and Y. Yuan, Combining trust region and line search techniques, Berlin: Kluwer, 1998, pp. 175.

[16] J. M. Rabaey, A. Chandrakasan and B. Nikolic, Digital integrated circuits, Prentice Hall Upper Saddle River, NJ, 2002.

[17] M. Rojas and D. C. Sorensen, A Trust-Region Approach to the Regularization

of Large-Scale Discrete Forms of Ill-Posed Problems, 2002, pp. 1843-1861.

[18] J. Semple,

Optimality conditions and solution procedures for nondegenerate dual-response systems, Springer, 1997, pp. 743-752.

[19] J. M. B. Walmag and E. J. M. Delhez, A Note on Trust-Region Radius Update, SIAM, 2005, pp. 548.

[20] R. A. Waltz, J. L. Morales, J. Nocedal and D. Orban, An interior algorithm for

nonlinear optimization that combines line search and trust region steps,

Mathematical Programming, 107 (2006), pp. 391-408.

[21] W. Wolf,

Modern VLSI Design: Systems on Silicon, 2nd Editon Prentice Hall,

Inc, 1996.

[22] J. Zhang and C. Xu, A Class of Indefinite Dogleg Path Methods for

Unconstrained Minimization, SIAM, 1999, pp. 646.

[23] Q. Zhang, K. Poolla and C. J. Spanos, Across Wafer Critical Dimension

Uniformity Enhancement Through Lithography and Etch Process Sequence:

Concept, Approach, Modeling, and Experiment, 2007, pp. 488-505.

[24] 陳彥良,

使用一般化縮減脊線搜尋與 Zoutendijk 方法於多目標統計模型最

佳化

,

工業工程學研究所

, 臺灣大學, pp. 60.

Appendix A. Proof of the solution to the Hard Case

To prove d satisfies the condition (2.7) [9], observe

( ^G

⁻

^λ

¹

^I ) ^d

⁼⁻

( ^G

⁻

^λ

¹

^I )( ( ^G

⁻

^λ

¹

^I )

⁺

^β

⁺

^{τ q}

¹

)

⁼⁻

^{( G}

⁻

^λ

¹

^{I )( G}

⁻

^λ

¹

^{I )}

⁺

^β

⁻

^{τ ( G}

⁻

^λ

¹

^{I ) q}

^1.

, where

( ^G

⁻

λ

1

^I )( ^G

⁻

λ

1

^I )

^{+ =}

^I

thus we have

( G

−

λ

1

I ) d

=−

β

−

τ ( G

−

λ

1

I ) q

1

and since

τ ^q

∈ N

( ^G

−

λ

1

^I )

we conclude

( ^G

−

λ

1

^I ) ^d

=−

^β

, which complete this proof.

For the condition (2.8) we have the squared Euclidean distance of d is decomposed as

follows

(

1

)

1 ²

T 1 2

1 2

1 1

2

( G I ) β q ( G I ) β 2 q G I β q

d = − −

λ ⁺

+

τ

= − −

λ ⁺

+ × −

λ ⁺

+

τ

, where

q

1^T

( G

−

λ

1

I )

⁺

β

=^{0 .} So we have

(

1

)

^{2 2}

2

G λ I β q

d = − −

⁺

+

τ

and then [( ( )]²

1 1 2

2

φ λ

τ

=± Δ − can be determined to meet

d

=Δ.

For the condition (2.9), it can be seen that d is a KKT point that satisfies KKT first- and second-order conditions for establishing only local optimality.

Algorithm B.1 [9]

Input:

G

F

=

μ

0 ^(where • is the Frobenius matrix norm) to ensure that _F

( ^{G + μ}

⁰

^I )

i s P.D.

δ1 = tolerance for convergence of the solution

^d ( ) ^μ

δ2 = tolerance for convergence of

μ

_k to signal the hard case ε = tolerance used in the method of iteration

μ

min= some large negative number (in our implementation, we use the minimum value of double)

k = 0 (reset the iteration index)

Begin

Repeat while d ( ) μ − Δ > δ

1

Factor

( ^{G + μ I} ) ^{= U}

^T

^U

(Cholesky Factorization) (B. 1)

If ( ^G

⁺

^{μ I} )

is P.D. then

Solve the two linear system:

( ) ^β

Ud

U

^T

μ = −

^and

^U

^T

^Uy ( ) ( ) ^{μ = d μ}

^{(B. 2)}

( ) ( )

( ) ( ) ^⎥ ⎦

⎢ ⎤

⎣

⎡ −

← μ μ μ μ μ μ

μ _d ^d

T

^d _y

T min

,

min (B. 3)

If ^d ( ) μ

<Δ (at the right of the root) then

μ

k

μ

max

←

(B. 4)

Appendix B. Trust Region Algorithm

μ

k

μ

min

←

(B. 5)

End If

( ) ( )

( ) ( ) μ μ μ μ μ

μ d y

d

d

²

~ 1 ⋅

Δ

− −

← (Newton’s iterate) (B. 6)

If μ

~<

μ

^min then

( )

2

~

μ

max

μ

min

μ

← ⁺ (safeguarding) (B. 7)

End If Else

{ ^{μ μ} }

μ

min ←^max min^, and

( )

2

~

μ

max

μ

min

μ

← ⁺ (safeguarding)

(B. 8)

End If

If μ

_max−

μ

miin <

δ

₂ then

Compute the eigenvector q via the method of inversed iteration applied to

( ^G

⁺

( ^μ

⁺

^ε ) ^I )

and then determine

π

(Problem is hard case). Return (Solutions are

^d

^*

^{= d} ( ) ^{μ + π q}

^; 1

*

μ λ

μ = ≈ −

⁾

(B. 9)

End If

End Repeat

End

Appendix C. Proof of Theorem 3.1 (Convergence to Stationary Point)

By performing some technical manipulation with the ratio

ρ ^{( )}

^k from Algorithm (3.1), we obtain

Since from Taylor’s theorem we have that

( )

⁺ ) ⁼ (

( )

) ^{+ ∇} (

( )

) ⁺ ∫

0¹

[ ^∇ (

( )

⁺ ) ^{− ∇} (

( )

)]

Suppose for contradiction that there is ε> 0 and a positive index K such that

( )^k

≥

ε

,

( )

We now derive a bound on the right-hand-side that holds for all sufficiently small values of Δ , that is, for all ^{( )k} Δ^{( )k} ≤Δ, where Δ is defined as follows:

The

R

0

γ

term in this definition ensures that the bound (C.3) is valid (because

R

0

and therefore we conclude that

( ) ^{≥ min} ( ^Δ ( ) ^{, Δ 4} )

Δ

^{k K for all}

^k ≥ ^K

. (C. 9)

Suppose now that there is an infinite subsequence κ such that

^{( )}

4

( ) ( ) ( ) ( ) ( )

[ ]

( ^, ) ^.

4 min 1

) ( ) 0 4 ( 1

) (

) ( ) ( ) (

1 1

χ ε

ε

k

k k k

k k

c

m m

f f

f f

Δ

≥

−

≥

+

−

=

−

⁺

d d x x

x x

(C. 10)

Since f is bounded below, it follows from this inequality that

( ) 0 lim, Δ =

∞

→

∈

k k

k κ , (C. 11)

contradicting (C. 9). Hence no such infinite subsequence κ can exist, and we must have

( )

4

<1

ρ

k for all k sufficiently large. In this case, Δ will eventually be multiplied by ^{( )k}

4

1 at every iteration, and we have

lim

_k→∞

Δ

^{( )k}

= 0

, which again contradicts (C. 9). Hence,

our original assertion (C. 4) must be false, giving (3.23).

Complete the proof of Theorem 3.1.

Appendix D. Problem Formulation of DFM Case Minimize:

2

subject to:

3

Appendix E. Expected Cycle Times and Raw Process Time of Supply Chain

The estimated cycle time with raw process time for products, plants and priorities in FAB:

The estimated cycle time with raw process time for products, plants and priorities in Assembly:

FAB Priority Expect Cycle Time

Row Process

Time

Expect Cycle Time

Row Process

Time

Expect Cycle Time

Row Process

Time FAB1 Priority1

107786.3 43545.6 101322.4 55065.6 106238.2 65491.2

(Min) Priority2

138157.9 46425.6 143855.6 59745.6 154811.1 69393.6

Priority3

198175.9 49305.6 200123.4 63705.6 191290.5 72720

FAB2 Priority1

106754.1 46569.6 110731.8 55209.6 112257.4 66974.4

Priority2

140035.7 49449.6 144816.8 56433.6 155978.6 70905.6

Priority3

203083.4 52329.6 196421.5 63849.6 193376.6 75643.2

FAB3 Priority1

116164.1 45576 112373 57096 not not

Priority2

138316.2 48456 147136.1 59587.2 not not

Priority3

202811.6 51336 206241.8 62856 not not

FAB4 Priority1

140333.1 29966.4 117665.1 55886.4 not not

Priority2

138654 47246.4 146215.2 58348.8 not not

Priority3

194274.4 50126.4 211231.9 63086.4 not not

FAB5 Priority1

112853.2 45748.8 not not not not

Priority2

139512.7 48628.8 not not not not

Priority3

198760.6 51508.8 not not not not

FAB6 Priority1

136227 43027.2 not not not not

Priority2

137850.2 45907.2 not not not not

Priority3

206452.7 48787.2 not not not not

Product Produc1 Produc2 Produc3

Fab Priority Expect Cycle Time

Row Process

Time

Expect Cycle Time

Row Process

Time

Expect Cycle Time

Row Process

Time Asse1 Priority1

16819.8 8523.07 17240.97 9001.94 17598.24 9403.24

(Min) Priority2

21227.3 9963.07 21754.13 10585.94 22099.07 10987.24

Priority3

26258.9 12123.07 25022.45 12745.94 28684.21 13147.24

Asse2 Priority1

16852.1 8560.02 17368.85 9146.06 17727.48 9547.3

Priority2

19715.8 10000.02 20231.61 10586.06 20588.89 10987.3

Priority3

24732.4 12160.02 25239.6 12746.06 25590.77 13147.3

Product Product1 Product2 Product3

The estimated cycle time with raw process time for products, plants and priorities in Final test:

Fab Priority Expect Cycle Time

Row Process

Time

Expect Cycle Time

Row Process

Time

Expect Cycle Time

Row Process

Time FT1 Priority1

22869.36 15051.3 24142.48 16376.1 24261.33 16499.34

(Min) Priority2

27098.66 16491.3 28347.25 17816.1 28463.96 17939.34

Priority3

33953.16 22251.3 35223.73 23576.1 35342.23 23699.34

FT2 Priority1

22014.74 15170.94 23511 16713.24 23352.4 16550.16

Priority2

27210.93 16610.94 28666.72 18153.24 28512.11 17990.16

Priority3

33093.73 22370.94 34583.09 23913.24 34425.29 23750.16

Product Product1 Product2 Product3

Appendix F. Problem Formulation of Supply Chain Case Minimize:

2

subject to:

1

50

1.023555 2 4.7

0.929101 1.7 4.7

1.101756 2 4.7

1.023555 1.7 4.7

1.111142 2 4.7

1.014317 1 4.7

0.93226 1.7 4.7

0.93226 2 4.7

0.953016 1.7 4.7

1.062922 2 4.7

1.018262 1.7 4.7

0.953016 1.7 4.7

0.959422 1.7 4.7

1.0585 2 4.7

1.013327 1 4.7

959422 1.7 . 0.845686 1.7 4.7

0.8 2

13.57879 1.122413

4.7 0.895949 1.7 4.7

0.861823 1.7 4.7

0.8 2

21.62003 1.250269

4.7 1.051238 1 4.7

0.831313 1.7 4.7

4.2553191 2

33.8298 4.7

1.7 4.7

7.8723 2

10.8510638 4.7

3.6170213 1.7

8.5106383 4.7

4.2553191 2

32.1277

16.383 2

12.7659574 4.7

8.5106383 1.7

10.8510638 4.7

3.6170213 1.7

8.5106383 4.7

4.2553191 2

7 85

100

在文檔中 一般化縮減信賴域搜尋及其在多目標統計模型最佳化之應用 (頁 130-147)