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F•g•kl_jkl\h hk\•lb • gZmdb MdjZ€gb

AZihj•avdbc gZp•hgZevgbc l_og•qgbc mg•\_jkbl_l

< 1 >m[jh\•g K H Km[[hl•g

F ? L H > B H I L B F 1 A : P 1 2 L : 2 O A : K L H K M < : G G Y < A : > : Q : O G : < Q : G G Y G ? C J H G G B O F ? J ? @

J_dhf_g^h\Zgh F•g•kl_jkl\hf hk\•lb • gZmdb MdjZ€gb

yd gZ\qZevgbc ihk•[gbd

^ey klm^_gl•\ \bsbo gZ\qZevgbo aZdeZ^L\

AZihj•``y

2003

2

;;D

>

M>D

>m[jh\•g < 1 Km[[hl•g K H

> F_lh^b hilbf•aZp•€ lZ €o aZklhkm\Zggy \ aZ^ZqZo gZ\qZggy g_cjhggbo f_j_` GZ\qZevgbc ihk•[gbd AZihj•``y AGLM

k

ISBN 966-7809-24-2

=jbn gZ^Zgh FLgLkl_jkl\hf hk\Llb L gZmdb MdjZ€gb

ebkl ‹ \L^ j

J_p_ga_glb < F Ihjhogy ^hdlhj l_ogLqgbo gZmd ijhn_khj

E H =ZedLg ^hdlhj l_ogLqgbo gZmd ijhn_khj

Jha]eygmlh hkgh\g• f_lh^b c Ze]hjblfb qbk_evgh€ hilbf•aZp•€ Ijb\_^_gh hibk hjb]•gZevgbo Z\lhjkvdbo jhajh[hd • ^hke•^`_gv f_lh^•\ hilbf•aZp•€ Hkh[eb\m m\Z]m ijb^•e_gh aZklhkm\Zggx f_lh^•\ hilbf•aZp•€ \ aZ^ZqZo gZ\qZggy rlmqgbo g_cjhggbo f_j_`

IjbagZq_gh ^ey klm^_gl•\ ki_p•ZevghklL ©Ijh]jZfg_ aZ[_ai_q_ggy Z\lhfZlbah\Zgbo kbkl_fª Z lZdh` fh`_ \bdhjbklh\m\Zlbky •g`_g_jZfb Zki•jZglZfb • klm^_glZfb j•agbo ki_p•Zevghkl_c ^ey ijb^[Zggy ijZdlbqgbo gZ\bqhd \bj•r_ggy hilbf•aZp•cgbo aZ^Zq

ISBN 966-7809-24-2

‹ < 1 >m[jh\•g

 

K H Km[[hl•g

3

AF1KL

<klmi. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

Jha^•e E•g•cg_ ijh]jZfm\Zggy. . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

AZ^Zq• e•g•cgh]h ijh]jZfm\Zggy 7

<eZklb\hkl• hkgh\gh€ aZ^Zq• e•g•cgh]h ijh]jZfm\Zggy 9

Kbfie_dkgbc f_lh^ j•r_ggy aZ^Zq• e•g•cgh]h ijh]jZfm\Zggy 11

Jha^•e G_e•g•cg_ ijh]jZfm\Zggy. . . . . . . . . . . . . . . . . . . . . . . . . . 19

AZ]ZevgZ aZ^ZqZ g_e•g•cgh]h ijh]jZfm\Zggy 19

DeZkbn•dZp•y f_lh^•\ g_e•g•cgh]h ijh]jZfm\Zggy 21

Jha^•e H^gh\bf•jg• f_lh^b g_e•g•cgh€ hilbf•aZp•€. . . . . . . . . 24

F_lh^b ijyfh]h ihrmdm 24

F_lh^ jhaih^•em •gl_j\Zem gZ\i•e 24

F_lh^ N•[hgZqq• 25

Ihrmd f_lh^hf ³ahehlh]h i_j_lbgm´ 27

F_lh^ j•\ghf•jgh]h ihrmdm 28

F_lh^b ihrmdm a \bdhjbklZggyf ihe•ghf•Zevgh€ ZijhdkbfZp•€ 29

F_lh^ hp•gx\Zggy a \bdhjbklZggyf d\Z^jZlbqgh€ ZijhdkbfZp•€ 29

F_lh^ ihke•^h\gh]h hp•gx\Zggy a \bdhjbklZggyf d\Z^jZlbqgh€ ZijhdkbfZp•€ f_lh^ IZm_eeZ 31

F_lh^b ihrmdm a \bdhjbklZggyf iho•^gbo 32

F_lh^ GvxlhgZ JZnkhgZ 32

F_lh^ k_j_^gvh€ lhqdb f_lh^ ;hevpZgh 33

F_lh^ k•qgbo f_lh^ ohj^ 34

F_lh^ ihrmdm a \bdhjbklZggyf dm[•qgh€ ZijhdkbfZp•€ 34

4

Jha^•e ;Z]Zlh\bf•jg• [_a]jZ^•}glg• f_lh^b g_e•g•cgh€

hilbf•aZp•€ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

F_lh^ G_e^_jZ ± F•^Z 37

F_lh^ OmdZ >`b\kZ 41

F_lh^ kijy`_gbo gZijyfd•\ IZm_eeZ 43

Jha^•e ;Z]Zlh\bf•jg• ]jZ^•}glg• f_lh^b g_e•g•cgh€

hilbf•aZp•€ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

F_lh^ Dhr• 48

F_lh^ GvxlhgZ 48

:e]hjblfb kijy`_gbo ]jZ^•}gl•\ 49

IZjlZg ± f_lh^ 50

F_lh^ AZml_g^ZcdZ 51

;Z]ZlhiZjZf_ljbqgbc ihrmd 52

D\Za•gvxlhg•\kvd• f_lh^b 52

:e]hjblf E_\_g[_j]Z FZjd\Zj^lZ 54

MaZ]Zevg_gbc ]jZ^•}glgbc Ze]hjblf 56

Hdj_f• \biZ^db maZ]Zevg_gh]h ]jZ^•}glgh]h Ze]hjblfm 57

1gl_]jZevgbc f_lh^ hilbf•aZp•€ 58

Jha^•e F_lh^b [Z]Zlh\bf•jgh€ hilbf•aZp•€ a

h[f_`_ggyfb. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

F_lh^ rljZngbo nmgdp•c 63

F_lh^ 6807 N•Zddh FZddhjf•dZ 65

Jha^•e F_lh^b hilbf•aZp•€ \ aZ^ZqZo ZijhdkbfZp•€ lZ

gZ\qZggy g_cjhggbo f_j_`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

NhjfZevgbc g_cjhg H^ghrZjh\bc i_jk_iljhg . . . . . . . . . . . . . . . . 69

;Z]ZlhrZjh\bc i_jk_iljhg 74

5

Ihj•\gyevgZ oZjZdl_jbklbdZ f_lh^•\ hilbf•aZp•€ ijb gZ\qZgg•

[Z]ZlhrZjh\bo g_cjhggbo f_j_` 76

Kihkl_j•]Zxqbc Ze]hjblf gZ\qZggy 78

JZ^•Zevgh [Zabkg• g_cjhgg• f_j_`• 85

F_lh^b hilbf•aZp•€ \ aZ^ZqZo ZijhdkbfZp•€ 87

:e]hjblf deZkl_j j_]j_kLcgh€ ZijhdkbfZpL€ lZ ch]h

g_cjhf_j_`_\Z Lgl_jij_lZpLy 92

Jha^•e Ijh]jZfg• aZkh[b qbk_evgh€ hilbf•aZp•€. . . . . . . . . . . . 107

Hilbf•aZp•y • g_cjhgg• f_j_`• \ iZd_l• 0DWODE 107

Hilbf•aZp•y • g_cjhgg• f_j_`• \ Z\lhfZlbah\Zg•c kbkl_f•

©>•Z]ghklbdZª 117

<bkgh\db. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

E•l_jZlmjZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

>h^Zlhd Iml•\gbd aZ kibkdhf e•l_jZlmjb . . . . . . . . . . . . . . . . . . 133

Summary. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

6

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Kn_jZ aZklhkm\Zggy f_lh^•\ hilbf•aZp•€ ^m`_ rbjhdZ FZl_fZlbqg_ fh^_ex\Zggy ieZgm\Zggy • ijh]gham\Zggy \ _dhghf•p• lZ l_og•p• kihemq_g• a g_h[o•^g•klx ihrmdm gZcdjZsh]h a fh`eb\bo \Zj•Zgl•\ ihe•ir_ggy ydhkl• gZ^•cghkl• lZ ^hklh\•jghkl• fh^_e_c kdhjhq_ggy €ogvh€ jhaf•jghkl•

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>ey ihe_]r_ggy ihrmdm \bagZq_gv l_jf•g•\ • ihgylv m l_dkl• dgb]b \hgb \b^•e_g• `bjgbf rjbnlhf M ^h^Zldm ijb\_^_gbc l_fZlbqgbc iml•\gbd aZ kibkdhf e•l_jZlmjb

:\lhjb \bkeh\exxlv sbjm ih^ydm k\h€f dhe_]Zf sh ijbcfZeb mqZklv \ hdj_fbo ^hke•^`_ggyo j_amevlZlb ydbo ijb\_^_g• \ p•c dgba• j_p_ga_glZf Z lZdh` lbf olh gZ^Z\Z\ ^jm`gx i•^ljbfdm Z\lhjZf ijb gZibkZgg• dgb]b

7

JHA>1E E1G1CG? IJH=J:FM<:GGY

AZ^Zq• e•g•cgh]h ijh]jZfm\Zggy

 

AZ]Zevghx

aZ^Zq_x

e•g•cgh]h ijh]jZfm\Zggy

EI

gZab\Z}lvky aZ^ZqZ sh ihey]Z}

m \bagZq_g• fZdkbfZevgh]h

f•g•fZevgh]h agZq_ggy nmgdp•€

 

 

 

 

n

 

 

 

 

 

F = åk j

× x j ,

 

 

(1.1)

 

j=1

 

 

 

 

 

n

 

 

 

 

iSb mfh\Zo

åaij ×xj

£ bi

,

i = 1, 2, ..., k;

(1.2)

 

j=1

 

 

 

 

 

n

 

 

 

 

 

åaij × x j

³ bi

, i = k+1, k+2, ..., m;

(1.3)

 

j=1

 

 

 

 

 

x j ³ 0 , j = 1, 2, ..., n;

 

 

i = 1, 2, ..., L, L ≤ n ,

(1.4)

^_ Dij, bi, cj aZ^Zg• ihkl•cg• \_ebqbgb [j ± rmdZg• af•gg• lZ Nm. Nmgdp•y gZab\Z}lvky p•evh\hx nmgdp•}x Z[h e•g•cgh€

nhjfhx aZ^Zq• Z mfh\b h[f_`_ggyfb

^Zgh€ aZ^Zq•

KlZg^Zjlghx Z[h kbf_ljbqghx aZ^Zq_x e•g•cgh]h ijh]jZfm\Zggy gZab\Z}lvky aZ^ZqZ sh ihey]Z} m \bagZq_gg• fZdkbfZevgh]h agZq_ggy nmgdp•€ iSb \bdhgZgg• mfh\ lZ

^_ N P lZ / Q

Hkgh\ghx Z[h dZghg•qghx aZ^Zq_x e•g•cgh]h ijh]jZfm\Zggy gZab\Z}lvky aZ^ZqZ sh ihey]Z} m \bagZq_gg• fZdkbfZevgh]h agZq_ggy nmgdp•€ iSb \bdhgZgg• mfh\ lZ ^_ N lZ

L = n.

Kmdmig•klv qbk_e [ [1, x2, ..., xn sh aZ^h\hevgy} h[f_`_ggyf

gZab\Z}lvky ijbimklbfbf j•r_ggyf Z[h ieZghf .

8

IeZg [* = (x1*, x2*, ..., xn* ijb ydhfm p•evh\Z nmgdp•y aZ^Zq•

ijbcfZ} fZdkbfZevg_ f•g•fZevg_ agZq_ggy gZab\Z}lvky hilbfZevgbf.

AgZq_ggy p•evh\h€ nmgdp•€ iSb ieZg• [ [m^_fh ihagZqZlb

q_S_a ) [ Hl`_ [* hilbfZevgbc ieZg aZ^Zq• ydsh ^ey [m^v ydh]h [ \bdhgm}lvky g_j•\g•klv ) [ £ F(x* \•^ih\•^gh ) [ ³ F(x*)).

AZagZq_g• \bs_ ljb nhjfb aZ^Zq• EI aZ]ZevgZ klZg^ZjlgZ c hkgh\gZ _d\•\Ze_glg• \ lhfm af•kl• sh dh`gZ a gbo aZ ^hihfh]hx klZg^Zjlgbo i_j_l\hj_gv fh`_ [mlb i_j_ibkZgZ m nhjf• •grh€ aZ^Zq• P_ hagZqZ} sh ydsh } kihk•[ agZoh^`_ggy j•r_ggy h^g•}€ a aZagZq_gbo aZ^Zq lh lbf kZfbf fh`_ [mlb \bagZq_gbc hilbfZevgbc ieZg dh`gh€ a ljvho aZ^Zq

Sh[ i_j_clb \•^ h^g•}€ nhSfb aZibkm aZ^Zq• EI ^h •grh€ ihlj•[gh \ aZ]Zevghfm \biZ^dm \f•lb ih i_jr_ a\h^blb aZ^Zqm f•g•f•aZp•€ nmgdp•€ ^h aZ^Zq• fZdkbf•aZp•€ ih ^jm]_ i_j_oh^blb \•^ h[f_`_gv g_j•\ghkl_c ^h h[f_`_gv j•\ghkl_c • gZ\iZdb ih lj_l} aZf•gylb af•gg• yd• g_ \•^ih\•^Zxlv mfh\• g_\•^}fghkl•

M lhfm \biZ^dm dheb ihlj•[gh agZclb f•g•fmf nmgdp•€

F = c1x1+c2x2+...+cnxn,

fh`gZ i_j_clb ^h agZoh^`_ggy fZdkbfmfm nmgdp•€

F1 = -F = -c1x1 - c2x2 - ... -cnxn,

hkd•evdb PLQ ) PD[ )

H[f_`_ggy g_j•\g•klv \bo•^gh€ aZ^Zq• EI sh fZ} \b]ey^ £ ",

fh`gZ i_j_l\hjblb \ h[f_`_ggy j•\g•klv ^h^Z\Zggyf g_\•^ }fgh€ af•ggh€ Z h[f_`_ggy g_j•\g•klv \b]ey^m ³ \ h[f_`_ggy j•\g•klv

\bjZom\Zggyf a ch]h e•\h€ qZklbgb ^h^Zldh\h€ g_\•^}fgh€ af•ggh€ LZdbf qbghf h[f_`_ggy g_j•\g•klv

ai1x1 + ai2x2 +...+ ainxn £ bi

i_j_l\hjx}lvky \ h[f_`_ggy j•\g•klv

ai1x1 + ai2x2 +...+ ainxn + xn+i = bi (xn+i ³ 0),

9

Z h[f_`_ggy g_j•\g•klv

ai1x1 + ai2x2 +...+ ainxn ³ bi

\ h[f_`_ggy j•\g•klv

ai1x1 + ai2x2 +...+ ainxn - xn+i = bi (xn+i ³ 0).

M lhc `_ qZk dh`g_ j•\gyggy kbkl_fb h[f_`_gv

ai1x1 + ai2x2 +...+ ainxn = bi

fh`gZ aZibkZlb m \b]ey^• g_j•\ghkl_c

ìai1x1 + ai2 x 2

+ ... + ain x n £ bi ,

 

(1.5)

í

 

 

- ... - a in x n £ - bi .

 

î ai1x1

- ai2 x 2

 

 

Qbkeh g_\•^ }fgbo af•ggbo sh \\h^ylvky

iSb i_j_l\hj_gg•

h[f_`_gv g_j•\ghkl_c

\

h[f_`_ggy j•\ghkl•

^hj•\gx}

qbkem

g_j•\ghkl_c sh i_j_l\hjxxlvky

 

 

<•^agZqbfh gZj_rl•

sh

ydsh af•ggZ [k g_

\•^ih\•^Z}

mfh\•

g_\•^ }fghkl• lh €€ \Zjlh aZf•gblb ^\hfZ g_\•^}fgbfb af•ggbfb Xk lZ Yk ihdeZ\rb [k = uk - vk.

<eZklb\hkl• hkgh\gh€ aZ^Zq• e•g•cgh]h ijh]jZfm\Zggy

AZibr_fh hkgh\gm aZ^Zqm e•g•cgh]h ijh]jZfm\Zggy m \_dlhjg•c nhjf• AgZclb fZdkbfmf nmgdp•€

F = cx

(1.6)

iSb mfh\Zo

 

x1p1+x2p2+...+xnpn = p0;

(1.7)

x ³ 0,

(1.8)

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