BMI - Kyoto U...k {V BMI) * z f ^ | '+, W BMI c d ebf ^7\ - . /0 7| 1 7W lbm \ 2 v F 3 k a min s.t....
Transcript of BMI - Kyoto U...k {V BMI) * z f ^ | '+, W BMI c d ebf ^7\ - . /0 7| 1 7W lbm \ 2 v F 3 k a min s.t....
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BMI 1�����\q���KO$��Z������[B�J3�FD�����§P=l»^�?1�O��a¨ s prb�� 1 f(x, y)� � 1 (x, y) �?°�� s P é ,�?���Z3�F§,=]»W�§(B�J���DP¡PbOBMI � ³ =�>�� u7j�f.=&�$�B�J�\^O�� 1 � 1 G�c�# 1 Æa��u���$�%j� '�)�� U����a� )�ª (���f.=&� )�ª �efag0hjik·m % 1 ¶ u�� BMI ���������a�,f��§�2(>,u��j¹(º?hj��JK��§01.==�Gb�= [1, 3, 4] DE?�&�»ZF½¾��+j=?>O BMI ���j�(�.�F¿?À�(�?��f�=P>jOPE$p& 9 < " �aD�@BA.BP1 BMI �?�����0�
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0, 1, ...,m)\3�~���u1�
β(x, y)g��iFU3h�� j a k
β(x, y) := B0,0 +n∑
i=1
xiBi,0 +m∑
j=1
yjB0,j +n∑
i=1
m∑
j=1
xiyjBi,j
{_\F|_BMI c�dTe7f ^�V�WhT! a ∈ f ^�V1K!| L1M�\'N�u BMI U'A1CFU�a3{�| w%OG
� �a [1] k �QP�WSRS7U6C%T!2� 7W BMI w EFg P�M UP�g�V1WJX3 j aT{�|�VNP� ��U`a3{�| w6O�H� �37W BMI cde1f ^F\�Y�97� c�d EFg'Z��!a{|_V�@�[����FU`a k
3 \^]^_a`^bdcfehgaij �~� e1k [2] V7W�@�[;�6����~� x�y h�z�� f ^7\�W#l�m;� c�d EFgBZ��a�{�|?gon�F|1�_�WQp�q�8%9 k vT|srSt#u�7vSwUx�yU�`va k#z�{ \�|�}�UTV�~��F\n�?�g! x_y hzT� f ^�g'EGt w WT{?\ f ^�V�~��B�T���\�~� x?y h�zT f ^�T�U�va k �W7t P�\��hF\ m U j ��~ e1k \�Y�9���%�$ wB7�H� �1a k {�\=SFUTVFW j�~ e�k g BMI cdebf ^� d ���?=��g OTjFklFm U���Fa}%A �� j a k p�q�\%Q�g 〈A,B〉 := trace(ABT ) W� 7y=g ‖ A ‖:=
(〈A,A〉) 12| h%� j a k� 3� a ≤ b U`a a, b G� t \ \��� ��1_zFg
mid(a, t, b) =
a, if t < a
t, if t ∈ [a, b]b, if t > b
| } jFk �W�n�o�pq X ∈ Sn×n g n(n+1)2 ��F\� 7T� j aJ%1 svec gsvec(X) := (X(1, 1),
√2X(1, 2), X(2, 2),
√2X(1, 3),
√2X(2, 3), X(3, 3), ...)T
� �3h%� j a k
2
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3.1 BMI ������������� S�U3h�i e �/ BMI cTd�e7f ^�g�W�� Z ∈ Sp×p g�����=W �� \;�! ������ f ^� j a k
min aT x + bT y
s.t. svec(Z − β(x, y)) = 0 (1)Z � 0
l!m \ KKT ��� g��1 j /���/ ��!�" #� U∗ ∈ Sp×p w�$&%�j a�|3�W (x∗, y∗, Z∗) g f ^(1)\ '&(&) |36 k
a −
svec(U ∗)T svec(B1,0 +∑m
j=1 y∗
jB1,j)
svec(U ∗)T svec(B2,0 +∑m
j=1 y∗
jB2,j)...
svec(U ∗)T svec(Bn,0 +∑m
j=1 y∗
jBn,j)
= 0
b −
svec(U ∗)T svec(B0,1 +∑n
i=1 x∗
i Bi,1)
svec(U ∗)T svec(B0,2 +∑n
i=1 x∗
i Bi,2)...
svec(U ∗)T svec(B0,m +∑n
i=1 x∗
i Bi,m)
= 0 (2)
svec(Z∗ − β(x∗, y∗)) = 0Z∗ � 0, U∗ � 0, 〈Z∗, U∗〉 = 0
KKT ��� (2) V1W d+* ���,�hF\ m U cd $F\ -&. �&� | � a kf ^ (1) nG�/7W#�F\2�!H�0/&1��243/_Fg c14�ej a f ^�g�5� a k
min Pα(x, y, Z) s.t. Z � 0 (3)�6 �3WPα :
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a −
svec(U ∗)T svec(B1,0 +∑m
j=1 y∗
jB1,j)
svec(U ∗)T svec(B2,0 +∑m
j=1 y∗
jB2,j)...
svec(U ∗)T svec(Bn,0 +∑m
j=1 y∗
jBn,j)
= 0
b −
svec(U ∗)T svec(B0,1 +∑n
i=1 x∗
i Bi,1)
svec(U ∗)T svec(B0,2 +∑n
i=1 x∗
i Bi,2)...
svec(U ∗)T svec(B0,m +∑n
i=1 x∗
i Bi,m)
= 0
λ∗1 − λ∗2 − svec(U ∗) = 0 (5)α − µ∗l − λ∗1l − λ∗2l = 0 l = 1, ..., p̄ξ∗l ≥ 0, µ∗l ≥ 0, ξ∗l µ∗l = 0, l = 1, ..., p̄λ∗1l ≥ 0, ξ∗l − svec(Z∗ − β(x∗, y∗))l ≥ 0, λ∗1l(ξ∗l − svec(Z∗ − β(x∗, y∗))l) = 0, l = 1, ..., p̄λ∗2l ≥ 0, ξ∗l + svec(Z∗ − β(x∗, y∗))l ≥ 0, λ∗2l(ξ∗i + svec(Z∗ − β(x∗, y∗))l) = 0, l = 1, ..., p̄Z∗ � 0, U∗ � 0, 〈Z∗, U∗〉 = 0`aξ∗ ∈
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step2 rkg���i� ���D� j a k
rk :=Pα(x
k, yk, Zk) − Pα(xk + ∆xk, yk + ∆yk, Zk + ∆Zk)Pα(xk, yk, Zk) − Φα(xk,∆xk, yk,∆yk, Zk,∆Zk)���W
rk ≥ ρ1� 3J, W��
k { n \%|�}7V����� |��bW xk+1 := xk + ∆xk, yk+1 :=yk + ∆yk, Zk+1 := Zk + ∆Zk
| j a k&% U1��%�G3� k { n\�|1}�V�����|3�7Wxk+1 := xk, yk+1 := yk, Zk+1 := Zk
| j a kstep3 ck
g'�7\�� +���� j a k
ck+1 := σ2ck, if rk < ρ1
ck+1 := mid(cmin, ck, cmax), if rk ∈ [ρ1, ρ2),ck+1 := mid(cmin, σ1ck, cmax), if rk ≥ ρ2.
step4 k := k + 1|���
step1 'ku��v w x�y
3.1V1W f ^ (1) TnG���p1q�8�9 k �| rSt j ��~ e1k g d �G�?!_\U`a k ��D3BEGt � f ^ (6) VFW�� % \'|�}&)�b� /&1T�2 3 ?�g~ e �_b�\7|�����6a k �W�� f ^ (6) ∆x,∆y,∆Z \'Y1 \����Fg� |��_�A��a{�|?U6p�q18�9
k g��3'�77a�\UV��2t/W�n �T_�08/&9 : < ck g����1~ �F\��Fg! �a={|�U'pBq%89 k \�5�!"�g$#���1=�7a k {T{?U�W 8G/�9C: < ck g'Y�;t j a{�|�V'p�qB8%9�g 4 tj aT{�|��n�IG�W 8/&9 :+< ck g 4G t j a�{�|�V6p�q18�91g�Y12t j a�{�|?Tn1I �_va kp�q%819 k g&%('��1!a)" k �*6T1�W�{?\�" k V�;t P�\ *�+ � ) w `va k -, Wn�?��� a�~��7\�����&� f ^ (6) w�. �F|��/� W 3 6�p�!�"18%9�V�0FUTV���\U z7{ \�|1}�31� � @F\ c�d E�gb {| w�1�G� a k� 3 7WQp�q�8%9 k g$�=��va3|�W/ �&� \�2�3�tu w 0tu�1� a3|3I54 ��tQ���6�7 w�8 {���9 a w W � f ^(6)=7�V % \2���1{|/V 8 {73 �� k �WJ�~T x?y h�z�� f ^�U�`!a f ^ (1) n ��1W-:X);�p�q7g6�b�~�� ��1gp /{�|�V��=1U1P��?��@$A w Y1�\�U7W{�{_U�V6@%�"��Fg'�1�ba k
rkV
Pα\!B1\!C�Dz |
Pα\�E&F� |��_&GIH � F!a Φα \&C�D�z7\!*�J7U�`K�XW#p�q8�9 k 3bT'@%[�6�73 � a k {=\�z w�L�M R �7�|�FW xk+1 := xk + ∆xk, yk+1 :=
yk + ∆yk, Zk+1 := Zk + ∆Zk|��3W % JU���7|��W xk+1 := xk, yk+1 := yk, Zk+1 := Zk |�����/�8 /�9 : <
ck\��bg'YF t j a k �W step3 3� a ck \����F\!" k V7WQp1q%89 k b�&No3 � �ba_\v|&O�PFU�`a k �#P�W rk ≥ ρ2 |��/� W |�} w ���G�?1|=V����
cmax| m � cmin g'�1 ck g3h����1a�{�|_&Q�R j a k{?\3SF\!ST� U�VFW6u?7v wUx�y
3.1 w�U�V �t�h%� H� �baT{�|_g ;�j a k {=\�1�V∆xk = 0
P ∆yk = 0
P ∆Zk = 0
| � 3J���WX� W�*rk\ M�Y V y U�`a{�|?g
_j�� ,��_ k� 3 1W step1 � a ! # ��� g y * $�g OjFk-, W#�F\!Z[6� >^�P 3]\��a k^�_
3.1� f ^ (6) \ cd E�g (∆xk,∆yk,∆Zk) | j a k�% \7|3��F\�`���i w �a�)b k
Pα(xk, yk, Zk) − Φα(xk,∆xk, yk,∆yk, Zk,∆Zk) ≥
1
2ck(‖∆xk‖2 + ‖∆yk‖2 + 〈∆Zk,∆Zk〉)
5
-
���Zk � 0
��\�UbW(∆x,∆y,∆Z) = (0, 0, 0)
V�� f ^ (6) \6��p !�"�E!U�` a k (∆xk,∆yk,∆Zk)
V!{?\�� f ^F\ cd E���\TUFW lbm \3i w 9 3 � a k1
2ck(‖∆xk‖2 + ‖∆yk‖2 + 〈∆Zk,∆Zk〉) + Φα(xk,∆xk, yk,∆yk, Zk,∆Zk)
≤ Φα(xk, 0, yk, 0, Zk, 0) = Pα(xk, yk, Zk)���>^
3.1V �7� k>�^
3.1� T7W *
rk\ M�Y V��6� 5 U�`a{�| w�11GU� a k� 3�v{ � V���� �a )�q
{Pα(xk, yk, Zk)} w [��1�� FU`a{�|_g O �_�!a k���Wstep1
�%!a ! # ����w �� � �|/�\ �bW M�YTw�� 6�1a�{3| g ��j a k step2' p t�\V M�Y�w�y \7|=�6�1��\�U7W�{ � V�u?7vSwHx�y 3.1 w�U�V �t�h%� � �baT{|�g O �/�va k^�_
3.2� f ^ (6) \ c=d Eg (∆xk,∆yk,∆Zk) | j a k % \�| Pα(xk, yk, Zk)−Φα(xk,∆xk, yk,∆yk, Zk,∆Zk) =
0 w �a��b \�V7W ∆xk = 0 W ∆yk = 0 P ∆Zk = 0 \7|=�6�U�`va k��
Φα(xk,∆xk, yk,∆yk, Zk,∆Zk)
\_hB��P3Φα(x
k, 0, yk, 0, Zk, 0) = Pα(xk, yk, Zk)
�T\U�W
∆xk = 0W∆yk = 0
P ∆Zk = 0
U`Fa_| j a_|?W Pα(xk, yk, Zk)−Φα(xk,∆xk, yk,∆yk, Zk,∆Zk) =0|#�a k�� 7W Pα(xk, yk, Zk)−Φα(xk,∆xk, yk,∆yk, Zk,∆Zk) = 0 | j a|W >�^ 3.1 � �
0 = 12ck(‖∆xk‖2 + ‖∆yk‖2 + 〈∆Zk,∆Zk〉) = 12ck(‖∆xk‖2 + ‖∆yk‖2 + ‖∆Zk‖2)|�� a/\�U7W
∆xk = 0W
∆yk = 0P
∆Zk = 0U�`va k���W
step1�� a ! # ��� \ y * $�g OTj�k � W (xk, yk, Zk) w�f ^ (1) \ y�7 �?/�1�2 3�_7\ c%4e1f ^ (3) \ '&(�)�U�`!a3|=�6%�W !$# �&��w �F H� a3{�|_g 7_j a k{�\ ; gp ���1W , � f ^ (6) �Qdn j a k (∆xk,∆yk,∆Zk) g&� f ^ (6) \ c�d E| j a k {�{�U�W
ξk := |svec[Zk−β(xk, yk)+∆Zk−n∑
i=1
{(Bi,0 +m∑
j=1
ykj Bi,j)∆xki }−
m∑
j=1
{(B0,j +n∑
i=1
xki Bi,j)∆ykj }]|
(7)| j a�|TW (∆xk,∆yk,∆Zk, ξk) VFW �� \ � f ^ (6) |C�&��� f ^F\ cd E�U�`va kmin
1
2ck(‖∆x‖2 + ‖∆y‖2 + 〈∆Z,∆Z〉) + aT xk + bT yk + aT ∆x + bT ∆y + α
p̄∑
l=1
ξl
s.t. ξl ≥ svec[Zk − β(xk, yk) + ∆Z −n∑
i=1
{(Bi,0 +m∑
j=1
ykj Bi,j)∆xi} −m∑
j=1
{(B0,j +n∑
i=1
xki Bi,j)∆yj}]l
l = 1, ..., p̄ (8)
ξl ≥ −svec[Zk − β(xk, yk) + ∆Z −n∑
i=1
{(Bi,0 +m∑
j=1
ykj Bi,j)∆xi} −m∑
j=1
{(B0,j +n∑
i=1
xki Bi,j)∆yj}]l
l = 1, ..., p̄
ξl ≥ 0 l = 1, ..., p̄Zk + ∆Z � 0
f ^ (8) V��? f ^���\�U�W KKT ��� V cd $7\�-�. L(M���� ��a k � W��F\ KKT� � g��b j / � /���!�" #� (λk1 , λk2 , µk, Uk) ∈
-
(∆xk,∆yk,∆Zk, ξk) w1f ^ (8) \ c�d EFU�`a/��\ -�. L�M&�&� U�`va k
ck∆xk + a −
svec(Uk)T svec(B1,0 +∑m
j=1 ykj B1,j)
svec(Uk)T svec(B2,0 +∑m
j=1 ykj B2,j)
...
svec(Uk)T svec(Bn,0 +∑m
j=1 ykj Bn,j)
= 0
ck∆yk + b −
svec(Uk)T svec(B0,1 +∑n
i=1 xki Bi,1)
svec(Uk)T svec(B0,2 +∑n
i=1 xki Bi,2)
...
svec(Uk)T svec(B0,n +∑n
i=1 xkjBi,m)
= 0
λ∗1 − λ∗2 − svec(Uk) = 0α − µkl − λk1l − λk2l = 0 l = 1, ..., p̄ (9)ξkl ≥ 0, µkl ≥ 0, ξkl µkl = 0, l = 1, ..., p̄λk1l ≥ 0, l = 1, ..., p̄
ξkl − svec(Zk − β(xk, yk) + ∆Zk −n∑
i=1
{(Bi,0 +m∑
j=1
ykj Bi,j)∆xki } −
m∑
j=1
{(B0,j +n∑
i=1
xki Bi,j)∆ykj })l ≥ 0,
l = 1, ..., p̄
λk1l(ξkl − svec(Zk − β(xk, yk) + ∆Zk −
n∑
i=1
{(Bi,0 +m∑
j=1
ykj Bi,j)∆xki } −
m∑
j=1
{(B0,j +n∑
i=1
xki Bi,j)∆ykj })l = 0,
l = 1, ..., p̄
λk2l ≥ 0, l = 1, ..., p̄
ξkl + svec(Zk − β(xk, yk) + ∆Zk −
n∑
i=1
{(Bi,0 +m∑
j=1
ykj Bi,j)∆xki } −
m∑
j=1
{(B0,j +n∑
i=1
xki Bi,j)∆ykj })l ≥ 0,
l = 1, ..., p̄
λk2l(ξki + svec(Z
k − β(xk, yk) + ∆Zk −n∑
i=1
{(Bi,0 +m∑
j=1
ykj Bi,j)∆xki } −
m∑
j=1
{(B0,j +n∑
i=1
xki Bi,j)∆ykj })l) = 0,
l = 1, ..., p̄
Zk + ∆Zk � 0, Uk � 0, 〈Zk + ∆Zk, Uk〉 = 0{�{_UFW
∆xk = 0,∆yk = 0,∆Zk = 0 w � f ^ (6) \ cd E�U�` � ,!W (9) V
a −
svec(Uk)T svec(B1,0 +∑m
j=1 ykj B1,j)
svec(Uk)T svec(B2,0 +∑m
j=1 ykj B2,j)
...
svec(Uk)T svec(Bn,0 +∑m
j=1 ykj Bn,j)
= 0
b −
svec(Uk)T svec(B0,1 +∑n
i=1 xki Bi,1)
svec(Uk)T svec(B0,2 +∑n
i=1 xki Bi,2)
...
svec(Uk)T svec(B0,n +∑n
i=1 xkj Bi,m)
= 0
λ∗1 − λ∗2 − svec(Uk) = 0α − µkl − λk1l − λk2l = 0 l = 1, ..., p̄
7
-
ξkl ≥ 0, µkl ≥ 0, ξkl µkl = 0, l = 1, ..., p̄λk1l ≥ 0, ξkl − svec(Zk − β(xk, yk))l ≥ 0, λk1l(ξkl − svec(Zk − β(xk, yk))l) = 0, l = 1, ..., p̄λk2l ≥ 0, ξkl + svec(Zk − β(xk, yk))l ≥ 0, λk2l(ξki + svec(Zk − β(xk, yk)l)) = 0, l = 1, ..., p̄Zk � 0, Uk � 0, 〈Zk, Uk〉 = 0
|��a w W�{ � V /&1��243/?F\ c%4�e1f ^ (3) \ KKT �&� (5) �����3 �� kl �7\7{�|�g |��a3|TW#�7\=h�� w 93 � a k
���3.1
` ack > 0
Tn ��7W(∆xk,∆yk,∆Zk) = (0, 0, 0) w � f ^ (6) \ c�d E�� 3s, W
(xk, yk, Zk)V�/�1�2 3/?7\ c�4�e!f ^ (3) \�'&(�)FU�` a k�� bW (xk, yk, Zk) w /�12 3 �F\ c�4Te7f ^ (3) \ '�(�)�U�`!a ��3H,1W j T1�\ ck > 0 3n�� �W (∆x,∆y,∆Z) =
(0, 0, 0)V&� f ^ (6) \ cd E�U�`va k
3.3 ���{�\�SFUV1W6u_2v w x�y3.1��T ��� H� a )�q {(xk, yk, Zk)} w� W=q7U�` a�{�|g �� | j a k {?\=SF\ n �=V�u�7vSwHx�y 3.1 \�Y%97� �%$�g OTj {�|_U�`!a k j ��H���W� R1\�����)
(x0, y0, Z0)�nG�_
{(xk, yk, Zk)}\�t��) w /&1�243/?1\ c�4�ebf ^ (3)\ '&(&)�U�`a3{�|_g OTjFkV�����FW � f ^ (6) \ KKT �&� (9) &Q n j a�|TW#�7\7{�| w HBP a k
λk1l ∈ [0, α], λk2l ∈ [0, α], µkl ∈ [0, α], l = 1, ..., p̄, k ∈ N (10){=\7{�|#PG3_W j T1�\ l = 1, ..., p̄ �n��_�W {λk1l}, {λk2l}, {µkl } V * �FU�`a3{�| w HBP a k^�_
3.3 {(xk, yk, Zk)}g�u��v w x�y
3.1�� ��� � a;)�q |��3W {(xk, yk, Zk)}k∈Kg�`ba
(x∗, y∗, Z∗)�� j a�� M )�q�U�W {ck(‖∆xk‖+ ‖∆yk‖+ ‖∆Zk‖)}k∈K → 0 g��b j�\7| j a k% \!|� (x∗, y∗, Z∗) V�/�1T�2 3/?7\ c�4ebf ^ (3) \�'&(&)FU`a k
� �Z∗V x?y hzn�o�p�q���\�U�W f ^ (3) \���p7!�"%E�U�`va k j T!�\ k ∈ K n�/
ck ≥ cmin�\TU�W
{ck(‖∆xk‖+ ‖∆yk‖+ ‖∆Zk‖)}k∈K → 0|� h�VFW
{(‖∆xk‖+‖∆yk‖ + ‖∆Zk‖)}k∈K → 0
W �∆xk → 0,∆yk → 0,∆Zk → 0
|�� a{�|�g&R��7���a k� 3 FW���1$�P 3 k → ∞ (k ∈ K) \b|?FWZk − β(xk, yk) → Z∗ − β(x∗, y∗)
svec(Bi,0 +m∑
j=1
ykj Bi,j) → svec(Bi,0 +m∑
j=1
y∗jBi,j), i = 1, ..., n
svec(B0,j +n∑
i=1
xki Bi,j) → svec(B0,j +n∑
i=1
x∗i Bi,j), j = 1, ...,m
|��a/\�U�W(7)P 3/W������t�u
KU3h%� � �� M q�nG��FW
ξkl = |svec[Zk − β(xk, yk) + ∆Zk −n∑
i=1
{(Bi,0 +m∑
j=1
ykj Bi,j)∆xki } −
m∑
j=1
{(B0,j +n∑
i=1
xki Bi,j)∆ykj }]|l
→ |svec[Z∗ − β(x∗, y∗)]|l =: ξ∗l
8
-
|�� a k� 3 (9) � (10) P 3�W#@�[�$bg�� /{�|'��t λ∗1l + λ∗2l + µ∗l = α g��! jλ∗1l, λ
∗
2l, µ∗
l ∈ [0, α]TnG�/
{λk1l}k∈K → λ∗1l, {λk2l}k∈K → λ∗2l, {µkl }k∈K → µ∗l| h j a{�| w U�FW svec(Uk) = λk1 − λk2 → λ∗1 − λ∗2 =: svec(U ∗) |�� a�\�UFW��� (9) g��7!aT{�|�I� W
k → ∞ (k ∈ K)\7|=�W
ck∆xk + a −
svec(Uk)T svec(B1,0 +∑m
j=1 ykj B1,j)
svec(Uk)T svec(B2,0 +∑m
j=1 ykj B2,j)
...
svec(Uk)T svec(Bn,0 +∑m
j=1 ykj Bn,j)
= 0
���
a −
svec(U ∗)T svec(B1,0 +∑m
j=1 y∗
jB1,j)
svec(U ∗)T svec(B2,0 +∑m
j=1 y∗
jB2,j)...
svec(U ∗)T svec(Bn,0 +∑m
j=1 y∗
jBn,j)
= 0 (11)
U�`X�XW
ck∆yk + b −
svec(Uk)T svec(B0,1 +∑n
i=1 xki Bi,1)
svec(Uk)T svec(B0,2 +∑n
i=1 xki Bi,2)
...
svec(Uk)T svec(B0,n +∑n
i=1 xkj Bi,m)
= 0
���
b −
svec(U ∗)T svec(B0,1 +∑n
i=1 x∗
i Bi,1)
svec(U ∗)T svec(B0,2 +∑n
i=1 x∗
i Bi,2)...
svec(U ∗)T svec(B0,n +∑n
i=1 x∗
jBi,m)
= 0 (12)
|�� a k � T KKT ��� (9) T1 k → ∞ (k ∈ K) \�� W�g!|�a�{�|_� � W (3) g�9a/\�UFW(x∗, y∗, Z∗)
V f ^ (3) \0'�(�)FU�`a kY�9�����$�g OTj ��7W� @ \���.���0�2 X�� |1�_'�7\�>�^�g 7_j a k^�_
3.4 {(xk, yk, Zk)}g1u?�v w x�y
3.4� T ��� U� a )�q |��3W {(xk, yk, Zk)}k∈Kg�`va )
(x∗, y∗, Z∗)61 j a�� M q | j a k %��W (x∗, y∗, Z∗) w /&1��2 3/_7\ c14ef ^ (3) \�'&(&)FU%���� 3U,!W lim supk→∞,k∈K ck < ∞ |��a k
� �K̄ := {k − 1|k ∈ K}
| j a�|�W {(xk+1, yk+1, Zk+1)}k∈K̄ → (x∗, y∗, Z∗) w �T� b \U�Wlim supk→∞,k∈K̄ ck+1 < ∞
g O ��,��� k {T{_U � g'h j a k {�\7|��WC-&.���3 d * � M q�g 5Fa�{�|? ����W#@1[%$Fg!� �{�| � tlim
k→∞,k∈K̄ck+1 = ∞ (13)
|�Ua k step3 \ ck \!�(��" k �Q n j a�|�W L�M �Y7� j Tb�\ k ∈ K̄ �n��_'|�}V�������B��F{�| w HBP a k ����G3U,!W�1� k { n \'|%} w �(�G��� � ,vW % \�|%}U�Vck+1 ≤ cmax
|��a P 3�U�`va k ��T �� \=i w �T�)b krk < ρ1 (14)
9
-
�W L&M �Y�= j T�\ k ∈ K̄ U=V (xk, yk, Zk) = (xk+1, yk+1, Zk+1) | �1a k {(xk+1, yk+1, Zk+1)}k∈K̄ →(x∗, y∗, Z∗)
��\�U�W{(xk, yk, Zk)}k∈K̄ → (x∗, y∗, Z∗)
�� � b �WJ|B} w �(��/�6��F|_Vck w ck+1 = σ2ck | ��� U� �ba�\�U�W?i (13) ��
limk→∞,k∈K̄
ck = ∞ (15)
g&9 a k {�{�U�W k → ∞, (k ∈ K̄) \7|=rk → 1|��a3{�|_g 7_j a�{�|? � ��� i (14) �n j a U�V g��Gt k , W
lim infk→∞,k∈K̄
ck(‖∆xk‖ + ‖∆yk‖ + ‖∆Zk‖) > 0 (16)
w �a� b {�|�&Q(R j a k ��7��3_W�%� ck(‖∆xk‖+ ‖∆yk‖ + ‖∆Zk‖) → 0 w �a�)b �3�W >�^3.3�I�
(x∗, y∗, Z∗)V7W /&1T�2 3 �F\ c14�ebf ^ (3) \ '&(�) | � � �h� U(V
j a k �� w ��W`aXh� γ > 0 w�$�% ��ck(‖∆xk‖ + ‖∆yk‖ + ‖∆Zk‖) ≥ γ k ∈ K̄
w �a��b k ��T >�^ 3.1 �� L�M BYF� k ∈ K̄ n���Pα(x
k, yk, Zk) − Φα(xk,∆xk, yk,∆yk, Zk,∆Zk) ≥1
2ck(‖∆xk‖2 + ‖∆yk‖2 + 〈∆Zk,∆Zk〉)
≥ 12γ(‖∆xk‖ + ‖∆yk‖ + ‖∆Zk‖)
w �a��b k 3�bW {‖∆xk‖ + ‖∆yk‖ + ‖∆Zk‖}k∈K̄ → 0 w � �-b {�|?(Q�R j a k � D�W���-�� b�%��F| j aT|�W�i (15) �� ck(‖∆xk‖2 + ‖∆yk‖2 + ‖∆Zk‖2) → ∞ |�� a w W�{ � V� f ^ (6) \ c�d z w t73�U QY!t�� a�{�|3g�R��;��7 a k � P�TW 3 P�%��p !"��'E
(∆x,∆y,∆Z) := (0, 0, 0) w �� 4 n �?zFg3Fa�\�U1W U�V U`a k ��{‖∆xk‖+‖∆yk‖+‖∆Zk‖}k∈K̄ → 0
|Q�!a k {?\�{�|/g?5���� � a=|�W {(xk, yk, Zk)}k∈K̄ →{(x∗, y∗, Z∗)}
� W%nH���v|β(x, y) w � ��U�� M !Q"UT`ba?{�|/g�����W k → ∞ (k ∈
K̄)\1|��W �� \0��i w 9 3 � a k
|Φα(xk,∆xk, yk,∆yk, Zk,∆Zk) − Pα(xk + ∆xk, yk + ∆yk, Zk + ∆Zk)| = o(‖∆xk‖ + ‖∆yk‖ + ‖∆Zk‖)
l �F\������ W k → ∞ (k ∈ K̄) \7|=�W
|rk − 1| =∣
∣
∣
∣
∣
Pα(xk, yk, Zk) − Pα(xk + ∆xk, yk + ∆yk, Zk + ∆Zk)
Pα(xk, yk, Zk) − Φα(xk,∆xk, yk,∆yk, Zk,∆Zk)− 1
∣
∣
∣
∣
∣
=
∣
∣
∣
∣
∣
Φα(xk,∆xk, yk,∆yk, Zk,∆Zk) − Pα(xk + ∆xk, yk + ∆yk, Zk + ∆Zk)
Pα(xk, yk, Zk) − Φα(xk,∆xk, yk,∆yk, Zk,∆Zk)
∣
∣
∣
∣
∣
≤ o(‖∆xk‖ + ‖∆yk‖ + ‖∆Zk‖)
12γ(‖∆xk‖ + ‖∆yk‖ + ‖∆Zk‖)
→ 0
|��a/\�U�W?i(14)
U�V�j a k ��T >^�V �7 � k{=\�>^�P 3�W �F\ >^ w 9 3 � a k10
-
^�_3.5 {(xk, yk, Zk)}
g�u_�v w x�y3.1��T ��� U� a )�q | j a k% \1|FW ���j aJ|�}�V � W { U�`a k
� � � g'�h j a�|�W j Tb�\ k ≥ k0 nG�_ rk < ρ1 P (xk, yk, Zk) = (xk0 , yk0 , Zk0)|�� a%�H�k0 ∈ N w&$�%�j a k � � step3 � a ck \!����" k P 3_W ck → ∞ |��a k @�" (xk0 , yk0 , Zk0) V f ^ (3) \�'�(�)�UTV��� k � �7��3�W/h�� 3.1 ��� W f ^ (3) \0'(&)��G3H,!Wstep1
U�'�������������������� >��3.4�I� U�V�� 9��������������
3.1 "!�#%$�&�'�(��")%* �+�,�-�.
3.2 {(xk, yk, Zk)} � �/�%�0�1�2� 3.1 3 ��45�7698�:A@CBD�+�"��E >9@ "F�G "H�I >�J,K�L �AM0NPO/Q ,R9S�TVU � (3) "W�X >29�Y���Z�[
(x∗, y∗, Z∗) �"F�G� �H�I >YB�\�] {(xk, yk, Zk)}k∈K � (x∗, y∗, Z∗) 3"&^' ���`_�a^@B����Db9ck d�e�f \hg�B�i 3 J (xk, yk, Zk) = (xk+1, yk+1, Zk+1) 9�j 0 35| \h�
ck ≤ γ k ∈ K (17)B�o94"� w ���5}7g2]D�^u�� b�c k ∈ K J^8�v~\P� w � rk ≥ ρ1 B�oj�], ,� ] k�
3.1�j�7uV� k ∈ K 35| \P�5% �" ���
Pα(xk, yk, Zk) − Pα(xk+1, yk+1, Zk+1) ≥ ρ1(Pα(xk, yk, Zk) − Φα(xk,∆xk, yk,∆yk, Zk,∆Zk))
≥ 12ρ1ck(‖∆xk‖2 + ‖∆yk‖2 + 〈∆Zk,∆Zk〉) (18)
≥ 12ρ1cmin(‖∆xk‖2 + ‖∆yk‖2 + ‖∆Zk‖2)
{Pα(xk, yk, Zk)}J�"A9���]
Pα(x∗, y∗, Z∗) 3 �/4^� 3 mA%o �] k → ∞ B"i2]
Pα(xk, yk, Zk) − Pα(xk+1, yk+1, Zk+1) → 0
B,oj 3 oj�]�x�y 35�� �,����4^� )�* :;Cg��
3.4 YYYYy^
3.2J�] �h�����9�V�5E9B �"�) \P� w ����\P�\"]/� 35 �¡ g w J"¢ 9U � (1) �W�X >���Y�/��¢ 9U � (1) ,W^X > �" � 3 J]1K�L ��MN£~¤5¥D¦§ α "¨�©��"ª«r¬�®�d ����E "¯ ^J�]hV � r
-
step0 k := 0B9\"]������ � % � r3���� �
α0 > 0, δ > 0, 0 < ρ1, < ρ2 < 1, 0 < σ1 < 1 < σ2,
cmax ≥ cmin > 0, c0 ∈ [cmin, cmax], x0, y0, Z0 � 0
step1{ ��VU � � p �� ∆xk ∈ �@ {ck(∆xk + ∆yk + ∆Zk)} J5m9 *JI��/�F�-LM
:�/������@ {(∆xk,∆yk,∆Zk)} J (0, 0, 0) 3^&�'^8�/�F�-3N
: {(x∗, y∗, Z∗)} ,F�G ,H�I > (x∗, y∗, Z∗) 37| \212]/� ��O g (λ∗1, λ∗2, U∗) ∈
-
svec(U ∗)T svec(B0,1 +∑n
i=1 x∗
i Bi,1)
svec(U ∗)T svec(B0,2 +∑n
i=1 x∗
i Bi,2)...
svec(U ∗)T svec(B0,n +∑n
i=1 x∗
jBi,m)
= 0 (20)
λ∗1 − λ∗2 − svec(U ∗) = 0λ∗1 ≥ 0, λ∗2 ≥ 0, U∗ � 0, 〈Z∗, U∗〉 = 0
x�y�� J���� $�*�I$�D� x�y�� ?3;� x^y�J�]� C��^>�@ {ck(∆xk +∆yk +∆Zk)} d���3"&'^3��;Yr o���� 3 80j`z/{ d ]^E
-
λk1l ≥ 0, l = 1, ..., p̄
ξkl − svec(Zk − β(xk, yk) + ∆Zk −n∑
i=1
{(Bi,0 +m∑
j=1
ykj Bi,j)∆xki } −
m∑
j=1
{(B0,j +n∑
i=1
xki Bi,j)∆ykj })l ≥ 0,
l = 1, ..., p̄
λk1l(ξkl − svec(Zk − β(xk, yk) + ∆Zk −
n∑
i=1
{(Bi,0 +m∑
j=1
ykj Bi,j)∆xki } −
m∑
j=1
{(B0,j +n∑
i=1
xki Bi,j)∆ykj })l = 0,
l = 1, ..., p̄
λk2l ≥ 0, l = 1, ..., p̄
ξkl + svec(Zk − β(xk, yk) + ∆Zk −
n∑
i=1
{(Bi,0 +m∑
j=1
ykj Bi,j)∆xki } −
m∑
j=1
{(B0,j +n∑
i=1
xki Bi,j)∆ykj })l ≥ 0,
l = 1, ..., p̄
λk2l(ξki + svec(Z
k − β(xk, yk) + ∆Zk −n∑
i=1
{(Bi,0 +m∑
j=1
ykj Bi,j)∆xki } −
m∑
j=1
{(B0,j +n∑
i=1
xki Bi,j)∆ykj })l) = 0,
l = 1, ..., p̄
Zk + ∆Zk � 0, Uk � 0, 〈Zk + ∆Zk, Uk〉 = 0� ÀA3$; 4�1,y^ �)�*,�,ª«r7� E�E * ] uQ1 k ∈ K 35| \ 1 ξk 6= 0 *JI 0 *I.��;Yr o��>C�� lk d'D*E^���� ; 4�1�] ¬�® o)� ��� �� C � E�B * ]Dp�q (2�"esr E�B�ot u
-
B,o ��� :�� 3 ] step2 αk 5¨9©��AÀ B u.1 k ∈ K 3�| \�1 ξk 6= 0 B,o � EAB %8�{αk} → ∞
B�o �/ �* ]hx9y 1 ?�; � x9y 2 %��h] k ∈ K 37| \�1 k → ∞ B�i
1
αk(ck∆x
k + a) − 1αk
svec(Uk)T svec(B1,0 +∑m
j=1 ykj B1,j)
svec(Uk)T svec(B2,0 +∑m
j=1 ykj B2,j)
...
svec(Uk)T svec(Bn,0 +∑m
j=1 ykj Bn,j)
→
svec(U ∗)T svec(B1,0 +∑m
j=1 ykj B1,j)
svec(U ∗)T svec(B2,0 +∑m
j=1 ykj B2,j)
...
svec(U ∗)T svec(Bn,0 +∑m
j=1 ykj Bn,j)
1
αk(ck∆y
k + b) − 1αk
svec(Uk)T svec(B0,1 +∑n
i=1 xki Bi,1)
svec(Uk)T svec(B0,2 +∑n
i=1 xki Bi,2)
...
svec(Uk)T svec(B0,n +∑n
i=1 xkj Bi,m)
→
svec(U ∗)T svec(B0,1 +∑n
i=1 xki Bi,1)
svec(U ∗)T svec(B0,2 +∑n
i=1 xki Bi,2)
...
svec(U ∗)T svec(B0,n +∑n
i=1 xkjBi,m)
B�o ���Q; 4�12]�� "8���
svec(U ∗)T svec(B1,0 +∑m
j=1 ykj B1,j)
svec(U ∗)T svec(B2,0 +∑m
j=1 ykj B2,j)
...
svec(U ∗)T svec(Bn,0 +∑m
j=1 ykj Bn,j)
= 0
svec(U ∗)T svec(B0,1 +∑n
i=1 xki Bi,1)
svec(U ∗)T svec(B0,2 +∑n
i=1 xki Bi,2)
...
svec(U ∗)T svec(B0,n +∑n
i=1 xkjBi,m)
= 0
}7g2]/x�y3B
(21) �R��� �� %�� ½ 5��/�0 =
1
αk〈Zk + ∆Zk, Uk〉 = 〈Z∗, U∗〉
:,� 3 ] svec(Z∗ − β(x∗, y∗))l 6= 0 �O g � u$1 l ∈ 1, ..., p̄ 3^| \�1 λ∗1l = 0, λ∗2l = 0o �* ] (λ∗1, λ∗2, U∗) J7 (20) O g ��Q; 4�1�]�x�y 4 ; j λ∗1 = 0, λ∗2 = 0 B�o �/ �* ]λ∗1h + λ
∗
2h = 1B ������� � 3.1 ; jV �� 5����
�3.1
x�y �–� ,� B * % d 8=j`z�{ �
(a)K�L ��M=N£¤^¥�¦�§ @ {αk} J R 5 $53 J7p�y 3 o �/�
(b) {(xk, yk, Zk)} ,F�G� "H�I > (x∗, y∗, Z∗) J5¢ 9U�� (1) ª*JI����Z�[
(a)J � � 3.1 B step2 P¨5©��5À %=� �53� % �2� ; 4 1 (b) /)�*�=�D� (x∗, y∗, Z∗)
>7@ {(xk, yk, Zk)} �H^I >YB9\/] {(xk, yk, Zk)}k∈K (x∗, y∗, Z∗) 3"&7'7)� _7a^@B 3���� � 3.1 %8�P] F a 35! i w u
-
d 80j�z���]�x�y 3 %��h]2E �-�� �� n B � E�B 3�; j1]
svec(Z∗ − β(x∗, y∗))l ≤ 0, l = 1, ..., p̄,svec(Z∗ − β(x∗, y∗))l ≥ 0, l = 1, ..., p̄,
o �svec(Z∗ − β(x∗, y∗)) = 0
d 8 jDz"{ � :�� 3� u�1 k ∈ N 3�| \�1 Zk � 0 * I)� % � Z∗ � 0 B"o ��Q; 41(x∗, y∗, Z∗)
J^¢ �U � (1) ª,#/%���*JI��/�� 3.1 ; j`]/x9y � – �
-
rk a�� ������� A] Pα(xk, yk, Zk) max
l=0,1,...,10(Pα(x
k−l, yk−l, Zk−l))
B�\hg � E1;YJ���� 3 K�L �AMN�O�Q / 3���� : �P �* J�ot�] O�Q � � �I.���� ���� *�I.�Dd ]2E ���� "ª 4�1 �?!A#�$�&�'�( J�� �?o w E�B d�� � ;�1 w � [?] �}7g�]���� M ' �1U,� %
(x0, y0) �! C ���step1 ½� �U � ,R���� xk+1 5 �¡��/�
min aT x
s.t. β(x, yk) � 0
step2 ½� �U � ,R���� yk+1 5 �¡��/�min bT y
s.t. β(xk+1, y) � 0
step3 ‖xk+1 − xk‖ = 0, ‖yk+1 − yk‖ = 0o��4��5$6 � r7* o w o3�/] step1 ()�
Q ���� 3? w 1]2E ����� �I g�419JSDPT3-Solver(Version 3.0) [6] �� w g � }5g�] �U,�� �'�½�( ����)�i ¾�¿�*/± y�� ��+�U��(� ��B �,�- J�)�. A 3�E \�g �� *�� w � M ' �lU,� J 4.1 ¯Q*�/ uVg �AÀ�* 698�\�g BMI R3�ATVU�� 1032�4 �9G \g � �; � �U,� � �
-
= 1: 5� 3�� w g U,� �
U,� � p n m -¤
1 -0.0000 -3.6168 12.5600
2 0.0000 -0.7290* 243.8400*
3 0.0000 -1.6031* 243.3400*
4 -0.0000 -1.0837* 573.4600*
5 0.0000 -0.7992* 557.9500*
18
-
4.3 ���= ��] = : � B5] »9½�¾�¿�T�ÀA3�; 41 �;g "`$ O�Q ��J�]DK9L � M0N�O�Q �B Fa��+tPo+j1]�� R*��T>*5 �;g "`$ O�Q �B6�7u�� �D1 ; j ; w �~B�oA4�1 w � E�B d��* i ��� : � 3 ] �;Vg �� ª* i ��� }5g�]»�½�¾�¿�T�À b�c ! Q J U ��3=; 4�1����9²%,J ����h 9 ]����BA\D1 U,�� ����Ad7! itPo � B ! iVtPo �����"d I j�]� ª 9�>"J U,�� ���9d^! itDo � B��+tPoA4�1 w � E�B d
% ��� � R*�9T�* J U�� � � – 0l3�? w 1 500 ! b9c 7ª 4�1 � 5$6 B�C�d�O gY:1;�1 w ow �* ]? ª 9 > ��2o7����J ª C�o w d ]�p !5 b^c 3 %�% � 9�>�J0� R���T� �Ad�� w �E1;�J] »�½�¾�¿�T^À� �*�%]6)�. A 3E \/g ;Yr3���U � ��B �� 3 ] '�¤ � � � Q o�A � Q "© g 3 !" \21 w � g ¡ ]
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���
��� dNe�������
-
���
A �������
�BMI OPQ BWC (1) J���������� u 3.2 Z%P\ L M���J6�7Q�� D�k�� �� BnC Z�� �! #"p���M cmin
1
2ck(‖∆x‖2 + ‖∆y‖2 + 〈∆Z,∆Z〉) + aT (xk + ∆x) + bT (yk + ∆y)
+αk
p̄∑
l=1
∣
∣
∣
∣
∣
∣
svec
Zk − β(xk, yk) + ∆Z −n∑
i=1
{(Bi,0 +m∑
j=1
ykj Bi,j)∆xi} −m∑
j=1
{(B0,j +n∑
i=1
xki Bi,j)∆yj}
l
∣
∣
∣
∣
∣
∣
s.t. Zk + ∆Z � 0, ∆x ∈
-
vvW57%;�XD��[J [a Um�M�D�5F7 t2θ Z 8���B�] sθ 5 OH�PH7 t2θ ≤ sθ g m.� j�4Z�� M c b 7�v�D j�4 6 XD? j54 J � 3PH M�v g p�5�M c(
sθ tθ
tθ 1
)
� 0
e�JWθ g m.�KJ0
-
ξl + svec
Zk − β(xk, yk) + ∆Z −n∑
i=1
{(Bi,0 +m∑
j=1
ykj Bi,j)∆xi} −m∑
j=1
{(B0,j +n∑
i=1
xki Bi,j)∆yj}
l
− wl = 0
∆z ∈
-
Aq2 =
0 · · · 0 0 0 0 0√
2 0 0 0 0 0 · · · 0 0 · · · 00 · · · 0 0 0 0 0 0 0 0 0 0 svec(B1,0 +
∑mj=1 y
kj B1,j)
T −svec(B1,0 +∑m
j=1 ykj B1,j)
T
......
......
......
......
......
......
0 · · · 0 0 0 0 0 0 0 0 0 0 svec(Bn,0 +∑m
j=1 ykj Bn,j)
T −svec(Bn,0 +∑m
j=1 ykj Bn,j)
T
Aq3 =
0 · · · 0 0 0 0 0 0 0 0√
2 0 0 · · · 0 0 · · · 00 · · · 0 0 0 0 0 0 0 0 0 0 svec(B0,1 +
∑mi=1 x
ki Bi,1)
T −svec(B,10 +∑m
i=1 xki Bi,1)
T
......
......
......
......
......
......
0 · · · 0 0 0 0 0 0 0 0 0 0 svec(B0,m +∑m
i=1 xki Bi,n)
T −svec(B0,m +∑m
i=1 xki Bi,n)
T
Al =
0 · · · 0 1 0 0 0 0 0 0 0 0 0 · · · 0 0 · · · 00 · · · 0 0 0 0 1 0 0 0 0 0 0 · · · 0 0 · · · 00 · · · 0 0 0 0 0 0 0 1 0 0 0 · · · 0 0 · · · 0
0 0 0 0 I I
0 0 0 0 −I 00 0 0 0 0 −I
b =(
−svec(Zk)T 0 0 − 1 0 0 − 1 0 0 − 1 svec(Zk − β(xk, yk))T −svec(Zk − β(xk, yk))T)T
24
-
B ������� ��� �
��������
0 5 10 15−5
−4
−3
−2
−1
0
1
max
(log1
0||d
x||,l
og10
||dy|
|,log
10||d
Z||)
k0 5 10 15
0
0.1
0.2
0.3
0.4
0.5
c k
k
0 5 10 1599
99.5
100
100.5
101
α k
k
�1:B�C����
25
-
0 5 10 15−5
−4
−3
−2
−1
0
1
2
max
(log1
0||d
x||,l
og10
||dy|
|,log
10||d
Z||)
k0 5 10 15
0
0.1
0.2
0.3
0.4
0.5
c kk
0 5 10 1599
99.5
100
100.5
101
α k
k
�2:B�C����
26
-
0 10 20 30−5
−4
−3
−2
−1
0
1
2
max
(log1
0||d
x||,l
og10
||dy|
|,log
10||d
Z||)
k0 10 20 30
0
0.1
0.2
0.3
0.4
0.5
c kk
0 10 20 3099
99.5
100
100.5
101
α k
k
�3:B�C����
27
-
0 5 10 15 20−5
−4
−3
−2
−1
0
1
2
max
(log1
0||d
x||,l
og10
||dy|
|,log
10||d
Z||)
k0 5 10 15 20
0
0.1
0.2
0.3
0.4
0.5
c kk
0 5 10 15 2099
99.5
100
100.5
101
α k
k
�4:B�C����
28
-
0 5 10 15 20 25−5
−4
−3
−2
−1
0
1
2
max
(log1
0||d
x||,l
og10
||dy|
|,log
10||d
Z||)
k0 5 10 15 20 25
0
0.1
0.2
0.3
0.4
0.5
c kk
0 5 10 15 20 2599
99.5
100
100.5
101
α k
k
�5:B�C����
29