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....

32
BMI 12 16 3 16 1 30

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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  • 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

  • 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 :

  • 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∗ ∈

  • 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

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    5 0.0000 -0.7992* 557.9500*

    18

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    +αk

    p̄∑

    l=1

    svec

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    i=1

    {(Bi,0 +m∑

    j=1

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    j=1

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    j=1

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    T

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    0 · · · 0 0 0 0 0 0 0 0 0 0 svec(B0,m +∑m

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    i=1 xki Bi,n)

    T

    Al =

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    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 ������� ��� �

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    k0 5 10 15

    0

    0.1

    0.2

    0.3

    0.4

    0.5

    c k

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    101

    α k

    k

    �1:B�C����

    25

  • 0 5 10 15−5

    −4

    −3

    −2

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    x||,l

    og10

    ||dy|

    |,log

    10||d

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    k0 5 10 15

    0

    0.1

    0.2

    0.3

    0.4

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    c kk

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    99.5

    100

    100.5

    101

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    k

    �2:B�C����

    26

  • 0 10 20 30−5

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    −3

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    |,log

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    0

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    0.3

    0.4

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    �3:B�C����

    27

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    −3

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    |,log

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    0

    0.1

    0.2

    0.3

    0.4

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    �4:B�C����

    28

  • 0 5 10 15 20 25−5

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    −3

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    ||dy|

    |,log

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    k0 5 10 15 20 25

    0

    0.1

    0.2

    0.3

    0.4

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    c kk

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    29