Optimality Conditions for Piecewise-Convex Maximization Problems
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(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Optimality Conditions for Piecewise-Convex Maximization Problems Ider Tsevendorj. N ˚ 3941. ISSN 0249-6399. ISRN INRIA/RR--3941--FR+ENG. ` THEME 4. apport de recherche.
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(117) 3 W© y f . xEx. ¿ O ]¤¦}jSc§ÁsO ¼VUg_!ZW\^]JO h \^] × Ä£Uga _!] L O n ¸ _ n h e½_ n \^ZgZ ª± ¤ × §Y] L O n O\ n O ³ ? X[Z`Z` ÖyZgZ`UWa[} n O efbsO k ]JU`¼EOZgiƬ2[¤ §Y¨ 1¤ × §Ë\Ea h ? ± +
(118) j¤¢À j¤ × §d§
(119) \!a h efXck L ] L \^]
(120) ½_ n \!Z`Z ± ¬2V¤ §¹] L O UWa[OCÕVXy\^ZgU`]Ãi|%; Ø ³ ? Ù ÚÓL _!Z h e ¾ K)L O a ÁVi ZgO P1P1\ vy¾ zÆ] L O n OOljUWeJ]Je
(121) ±À j¤ × §1efXc k L ] L \^] Ø ³ ? Ú·Û Ø ³
(122) Ú ¾% SV_ h XcOÔ ]d_ kL_EaV¼!Ol[U`]Ãi_^Í\^ZgZs× Xcack ]JU`_Vace) 2¬ V¤dâã § ¸ O L L \á¼!O<L |%nÓ; L Ø h ³ ? Ù h Ú-Û Ø ³
(123) Ù × ÚÓÛ ¬2j¤
(124) §F× Ù¬2V¤ §Ä ] \^]-UWP bcZ`UgO h e-1¤ × <§ ; ¬ ¤
(125) §¾ n a] O_E] O \!a Ä[ÁVi nO ÖFa[U`]JU`_Va_!ÍL À ¤ §Ä
(126) ±;n À ¤ §©UWP bcZ`UgO e
(127) ±;À \Ea 1¤ < à ib n _Ebs_EeJUg]dUg§ _Ea6; vc¬ ¾g^x
(128) ¤
(129) \!a h§½Zg_ O P1\!P1Z`Z[\°vc±¾ z ² U`]<ÓUge)'« FOC\!¶VeJ¾Ìiq]Je-_\ edO O¹OCeJ] X[L Zg]\!Ä ]
(130) ¸ ¤}[O Sc§-\áUW¼Ee(O<a[\Y_!]<ÁsO_E]Ja[]dZgO iefbsXj_!ÎUWaVk ] UgO
(131) aV]Äs±;U`]<À;Uge ¾ acO k OCeJef\ n i\!a h efXjÎk UgO aV](k _Va h Ug]dUg_Ea½_ n }EZ`_VÁc\^Z.P1\ l[UWPYXcP ]d_;¤ m)ÑÓÒm §¾ . . /. . 1 /. . 3. /. / /. . /. . ". .
(132) . Ñ _EaceJU h O n ] L O
(133) b n _VÁ[ZgO P UWa ]d_P1\ l[UgP UgÏO¹b[UgO kO ¸ UWedO
(134) k_EaV¼!OlXcack ]JU`_VaÆ 1¤¦¥§©¨ µ « ¬ ¤¥§(® °¨tx!³z[³Jvy³dQs³fýj¶V³.
(135) . . 1. ¸ L OnO. ¬ ¤¥§¯¨7¥ ¤¥ ÜQj§ ÙÆv c³ ¬ ¤¥§¯¨ ¤¥ ÿV§ ¤¥ ÙvV§ ÙÆv c³ ¬ ¤¥§¯¨7¥ ¤¥ ÙÿV§ Ù x c³ ¬ ¤¥§¯¨ ¤¥ ÙÿV§ ¤¥ ÙvV§ Ù·QVwc³ ¬!¤¥§¯¨ ¤¥ 3Ù¯x |V§ ¤¦¥ x |j§ ÙÆQ . . . . . . . . 2. . . . 3. . . . . . . . . . Je XyÁjopO k])]d_ À ¨«C¥;±6 ®cÙ)Q Û ¥ Û x |y³ Ù Û ¥ Û ÿc³Ô¥ ÙÆ¥ Û x |c¶E´ K)L Obs_!UWaE] × ¨ ¤ c³ Ù)Qj§ »UWe \Zg_jk \^Z-P \^l[UgPXcP ¸ U`] L 1¤ × §Y¨|y¾ a[O ¸ _Va h O n e
(136) U`<Ug]YUWeY}EZ`_VÁc\^Z P1\^K)ljL UWPYXy¿ P
(137) h h × L h U !O n O O ackOÔOCÁy_!OCYeJÕj] X[¸ O_¯eJa[O_E] ayk _E aV¤¦¼E1O l¤ §ded§O ]Jke·_EaV¤¦eJ]JO\!O Ugace6Ug}y] ¸ ¾ _¯z[ac¾ã§_E¾ ackF_EXcacaca[k OC]Jk U`]J_VO ace;eJ¬ Oá]f¤Je;â §\!a\!a h _Eac¬ !O·¤Jâ _!§\ ] n OÔO P \Ek ]JUWU`e;¼EO·edO \^]p]¡ ×¾ ¿ O ](Xce<k_EaceJU h O n ] L O¹Xcack]dUg_Ea¬ ¤dâ㧾)q<k k _ n h UWa[}1]d__EX n ac_!]J\!]dUg_Eace)À ¤ × §£¨»À« ¥Æ®V¬ ¤¥§A@ |y³Æ¬ ¤¥Nn h J§ @¯n |y³Æ¬ ¤¥NJ§ L @¯|c¶
(138) edUWack O
(139) n ¬ ¤¥NJ§ @¯|½_ n \^ZgZN¥±6À$\Eh a h 1¤ × §¨ª|c¾ L Þpa _ O ]J_ × XceJOÔ ]
(140) OÆacO × k OCeJef\ i¯}!Zg_EÁc\!Z<_Ebc]dUWP L\!Z`Ug]Ãi k _Va× Ug]dUg_Ea¤¦}!ü˧ÄÓh _Va[h O \! e
(141) ]d_ k L × O k +¯] L O UWackZgXceJUg_EaÀ ᤠ§ & 3 ¤¢1¤ §d§¾× ÞÃ]-Uge-h OC\!eJi]J_eJOO5] \!]ÓÀ á¤ × §©UgL e-a[_E]ÓUWack ZWX O Ø ]J_ & 3 ¤¢1¤ §d§£Ú edUWack O \!L ] h ¨tK)¤¢|cL ³zEn § n × ¬ ¤ y§£¨Ê¬+¤ §-\!a ? ¨%¤¦|y³Jvj§ ±À ¤ §-] O
(142) UWa[O ÕjXc\!Z`Ug]Ãi 1¬ ¤ y§³ ? Ù @ | _!Z e¾ O O½_ O h n Uge)a[_E](L \ n}!Zg_EÁc\!ZP1× \^ljUWPYXyP¾ L L ü<_ ¸ Ä ¸ Ok _EayedU O \!a[_E] O by_EUgaV] n ¨%¤¦|c³fvV§ ¸ h Ugk UWe(Zg_[k\!ZÌP1\ L l[UWPYXcP ×]d_j_y¾)q3](] L Oh bs_!h UWaV]Ä Xyack ]JU`_Vace6¬ á¤dâ㧳¬ ¤Jâ §³f¬ E¤dâ㧳¬ !¤Jâ §\ OÔ\!k]dUg¼!OE¾tq<a _Ea[OÔk \!a eJOO·] \!]À ᤠ§1UWeUWack ZWX O ]d_ . . . . . . . . 2. . 1. . . . . . /. *. G10 h2i:f. . 3. -. ./.. 3. . 2. 3. . ". . . . /. 3.
(143) xáz. . !. . . d¤ x vj§³¯¤¦1¤ L × §©¨× xCvV§¾ K)L \!]£UWe-O a[_VX[} L Äj\!k k _ n h UWa[}
(144) ]J_efXjÎk UgO aV]-}!Zg_EÁc\!Zy_Ebc]dUWP \!Z`Ug]Ãik _Va h Ug]dUg_Eace ¤¦}jSc§Äc]d_ef\ái] \^] Uge3}EZ`_VÁc\^Z.P1\ l[UWPYXcP6¾. &
(145) 3. /. /. & - ' . 2. . & - ' . E. & 1 . & 1 ' . 2. E. . . & ' . & ' . E. 2. 3 & 3 . & 3 ' E. . & ' . 2. . & E. . 2. 6879;: 4: 1D).'*UV9:P*-- =(1UV9:P*-]h ?*f .
(146) O n O ¸ Ok _VaceJU h O n \b[U`OCk O ¸ UgeJOk _EaV¼EO lÔP1\ l[UWPUgÏ \!]dUg_Ea·b n _VÁ[ZgO PÂUWa ¸ Ug] L Xyack ]JU`_Vace¹¤¦eJOO Ug}y¾©vc¾ã§ ¬ ¤¥N§ ¨7¥ ¤¦¥ zE§ ÙÆwy³ ¬ ¤¥N§ ¨ wc¤¦¥ vV§ ÈQE¥ Ùv y³ ¬ !¤¥N§ ¨ ¤¥ x᧠¤¦¥ ÙÆQj§ Ù·Qs³ ¬ ¤¥N§ ¨ wx ¤¦¥ ÙvV§ v x ¤¥ ÙÆQj§ Ù xE³ ¬ !¤¥N§ ¨ ¤¥ -ÙÈýE§ ¤¦¥ ýE§ Ù x
(147) . . . . . . . . 2. . . . . . . . . 3. . . . . . . ;. _. .<;=.
(148) ¦j ã © ¢ c
(149) 3 W© y f . xCv. eJXyÁjopO k])]d_Ás_álk _Vaced] n \!UgaV] À ¨Ê« ¥;±; ®jÙ<z Û ¥ Û ý[³¯Ù(v Û ¥ Û Qy¶E´ Ok_EaceJU h O n U`]ÌodXceJ]£]d_ e L _ ¸ eJ_EP O h U Îk X[Z`]Ãi1_!Íed_EZ`¼jUWa[}¤ m)ÑÓÒm §¾EÞÃ]-edO O P1e©]J_Xce ÄjO¼EO a1] L Uge-] ¸ _ h UgP O ayedUg_Eac\!Z.b n _EÁ[ZgO P UWe)a[_E])] n U`¼jUW\^ZN]d_eJ_!Zg¼!OE¾ ÞÃ]ËUge5OC\!eJi6]J_6eJOO ] L \^]Ë] L O n O UWe¹aVXyPYÁsO n _^£Zg_[k\!ZP1\^ljUWP1\_Va¼EO n ]dUWk O e ÄO h }EO eË\!a h UWaE]JO n U`_ n _! ] L O
(150) Ás_ál¾ÞpL a6_!] L O n ¸ _ n h e Äj}!Zg_EÁc\!Z.P \^l[UgPXcP k _VX[Z h ÁsO
(151) \!aVi n ¸ L O n O5Uga6] L O
(152) Áy_álN¾ L L n q<e5UWaÔ] O O lc\!P1b[ZgO¤¢vc¾gxá§<XceJUga[}Ƥ}Eü˧ ¸ Ok\EaOCeJk \!bsOY _EP$Zg_[k\^ZP1\ l[UWP \ ¸ UWk \ O_EaÔÁs_ál ¼EO n ]dUWk OCL e-\En a h O h }EO e ¾©q<a h ¸ O¹Z`_j-_ +1½_ n Ug])UgayedU h O¹] L n O
(153) Áy_álN¾ ú acZ`U +^O¤¢vc¾gxá§Äy] L O n OËUWe(a[_bs_!UWaE]3UWa] L O Ás_ál ¸ n O L O¹\!Z`ZXcayk ]dUg_Eaye<¬ ¤Jâ §³f¬ ¤dâ㧳¬ ¤Jâ §³f¬ ¤dâã§3\ OË\!k ]JU`¼EO!¾ L O O] O6by_EUgaV]6h ¤dÙ¹xE´ vVz^ÿ y³xEn ´ þ^v!ÿcx § n Uge}!Zg_EÁc\!n ZÓP1\ l[UWPYn XcP ¸ Ug] n ¼á\!ZgXcO6¬+¤ h× § n3¨ Ò x!´ z!z QV|c¾ K)L Uge eJ_!ZWX[]JU`_Va ¸ \!eÓ½_EXya ÁEiO ajXcP O \^]JUgac}] O O¹_^+k X ¼!OËUgaV]JO edOCk ]dUg_Ea¼!O ]JUgkO e3Xca O \Eb[Z`OE¾ . . ./.. G10 h2i:f. . . 2. . 3. .
(154) x Q. . !. . i . h
(155) . . i
(156) . h . `
(157) . `. f
(158) . f . d . d
(159) . 6879;:: 1DG:;<G =0+.' (' 1P - =:1UV9:P*-gh ? ` ;. _. .<;=.
(160) ¦j ã © ¢ c
(161) 3 W© y f . xáý. ~ ~ ~ ~ / ~ ~ ! ~ ¿ O ]©ÀÁyO<\Ëa[_Va[O P1b[]ÃiEÄEk_EP1bc\!k]ÄV\!a h k _VaV¼!O lYefXcÁceJO]_!;\Ea h Å»!ºÁyOk _EaV]JUgajX[_VXce¾ K)L UgeËeJO k ]JU`_Va h OC\^ZWe ¸ U`] L ] L O1¼á\ n UW\^]JU`_Vac\^Z+UWa[O ÕjXc\!Z`Ug]Ãib n _EÁ[ZgO P6Ä h OCa[_!]JO h ÁEi ¤¦Å(³JÀ6§ÄNU¢¾ OE¾Y]d_ ÖFa h \¼EO k]d_ n)× ±À$eJXyk L ] L \^] Ø Å
(162) ¤ × §³J¥Ù ×jÚÓÛ |y³ ½_ n \!Z`ZN¥±À;´ ¤¦Ý<Þ m ¤ K Ä ß¹§J§ K)L O k\!eJO
(163) _!ÝÞ m ¤ K Ä ß
(164) § ¸ L O a;] L OY_VbyO n \!]d_ n Ù)Å
(165) ¤dâã§-UWeP _Eac_!]d_Va[O¤¢bcedOCX h _VP_Va[_!]J_Ea[O §-_Ea;À Uge ¸ n OZgZÍeJ]JX h U`O h \!a h ] L O n n O¹O l[UWed]PO ] L _ h eedL _EZ`¼jUWa[}n Ug]Y¤¦eJn OO # \!P1ÿ^Q %§¾3Þpa;b n \!k]dUWk\!ZÍ\!bch bcZ`UWLk\!n]dUg_Ea6n Z`U +^O ] \^Îkh \!n efedUgL }Ea[OCP O aV]¹b _VÁ[ZgO P6Ä+XyeJXc\!Z`ZgiÔ] O_VbyO \!]d_ Ù)ÅY¤dâã§(UWeYac_!]
(166) P _Ea[_E]d_Va[O\!a ] O ½_ O ¸ O k_Eac eJU n O ] n Uge(k \!L eJO¹ÁyO LZ`_ ¸ ¾ n X \Ebcb _E\!k n Uga6] Uge(by\!bsO UWe)Ác\!eJO h _Ea6}!Zg_EÁy\^Z_Eb[]JUgP1\!Z`Ug]Ãik _Va h Ug]dUg_Eace-½_ n b[UgO k O ¸ UgeJO¹k _VaE¼EO l P1\^ ljUWP U`ÏC\^]JU`_Vanb _EÁcZ`OCP¾ E Ug¼!O a \ÔeJ]J\n ]JUga[}bs_!UWaE] ±ÊÀ \! a h ed_VPOUgah h O l eJOn ]² E¨ h «á|[¶E¾Èq3h ]1Ug]dO n \^]JU`_Va h ¤ ¨ |y³xE³fz[³ ´`´ã§ÓUWaj½_ P1\^]dUg_EaܤÅ
(167) ¤ §³ Å
(168) ¤ §J§)k_!ZgZ`OCk ]dO eJ_\ Uge5XceJO ]J_ÁcX[UgZ Xcb\P_ O ZÌ_!£ed_VP O UW P1b[ZgUgkU`]ÓXcnack]dUg_Ea6k _Eaya[O kh ]dO h ]d _ n ¤Å(³fÀ§¾Þpa] L OËedOCÕVXcOZ¢Ä ¸ OËXcedO ¤dâ㧽_ n Ás_!] L } n \ h U`OCaV]Ó\!a h O efedUW\!a½_ Xyack ]Jn U`_Va6\Ea ¼!O k]d_ Xcack ]JU`_VL aN¾ L ü<_ ¸»¸ O¹k_EaceJ] Xck ])eJ_EP O5Xcack]dUg_EaeJXyk ] \^])Ug]Je3} n \ h UgO aV]3O ÕjXc\^ZWe3Å»\!]
(169) j¤¥N§©¨ x Ø Å
(170) ¤ §÷¥Í³J¥ Ú Ø Å
(171) ¤ §ÌÙ Å
(172) ¤ § ³J¥ Ú ³ ¤¦vj§ z . . . ". . . . . ". . . -. ". . . . . . j¤¥N§£¨ YÅ ¤ § YÅY¤ §¤¦¥Ù §³
(173) ¤ §©¨ Å
(174) ¤ § ´
(175). . . . . . . 4. . ÃÞ ]L Uge ¸ L OZgZ +ja[_ ¸ a] L \^]] L O n O \ n O
(176) ] ¸ _by_VedUg]dUg¼!O edOCPU h OÖsa[Ug]dOP1\!] n UgkO e Å
(177) ¤ § \Ea h Å
(178) ¤ § # efXck ] \!] Å
(179) ¤ §©¨ YÅY¤ § Ù YÅY¤ § # ´ ¤ Q[§ ú eJUgac}] L UWe)_EacO ¸ n Ug]dO e-Xyack ]JU`_Va
(180) [¤¥§)\!e h ¾ k^¾)¤ h U !O n O ayk O¹_^+] ¸ _k_EaV¼!Oly§©Xcayk ]dUg_Ea
(181) ¤¥§£ ¨ ¤¦¥§Ù ¤¥§³ ¢¤ ýV§ ¸ L OnO ¤¥§£¨ zx Ø Å
(182) ¤ § ¥Í³d¥ Ú Ø Å
(183) ¤ §ÌÙ Å
(184) ¤ § ³d¥ Ú ³ x ¤¦¥§©¨ z Ø Å
(185) ¤ § # ¥Ì³d¥ Ú ´ . . . . . . ./.. G10 h2i:f. 5. ". . . ". . . ". . ". . .
(186) x. . !. . . ßËX[O¹]d_k _VaV¼!O l[Ug]Ãi_^ [¤¥§Äyk _VaV¼!O l]f\!a[}EO aV](\!byb n _ál[UWP1\^]dUg_EaÁEi6\!Ás_ ¼!O5_^
(187) j¤¥N§3\^] UWe 2¬ j¤¥N§©¨ [¤¥§Ù # .¤ § Ø [¤ §³J¥Ù Ú % edUWackO ¬ ¤¥ §
(188) ¤¦¥§³3½_ n \^ZgZ.¥;±; ³ 2¬ V¤ § ¨
(189) j¤ §´ ü<_ ¸Ê¸ O¹k_EX[Z hh O Ösa[O
(190) \1b[UgO kO ¸ UWedO¹k_EaV¼!OlXcack]dUg_Ea\!e)eJ_EP O¹P _ h OZ`¡¦Xcayk ]dUg_Ea\!]-OC\!k L U`]JO n \^]JU`_Va j¤¥N§©¨ P UWa«á¬C!¤¥§(® °±;² ¹¨Ê«á|c³xE³fzc³´g´`´g ³ F¶E¶E´ ¤ j§ ÁV¼jU`_VXcedZgi c¤dâ㧠h _jO e)a[_E]3k_EbVi] L O h O eJU n O h UgP1b[ZgUgkU`]3Xyack ]JU`_VaN¾ K _P1\!eJ]dO n ] L UWe3Zg\Ek+1_^+Ugaj½_ n ¡ P1\!]dUg_Ea_EacOYO a n UWk L O e<] L O P _ h OZÁELiÖsa h Uga[}_Ea[O ½ P æ _ [n OYêby_EUgaV] \Ea h ÁViUWackZgX h Uga[}_Ea[O P _ n O Xyack ]JU`_Va.¬ ¤Jâ §¾-SV_sÄ ¸ O¹_VÁ[]J\!Uga] O˽_EZ`Zg_ ¸ UWa[} Ù ¾ . -. ". . . 3. '. ". ". . . . . . .
(191) . $ . ". . . .
(192) '. Ex ¾¯½_ Ñ n n _Vn P| ¸ L UgZ`O ± \ n }©P1\^lF«á V¤¦¥§)®^¥;±;À¶ zc¾ _EaceJ] Xck ])b[UgO kO ¸ UWedO
(193) k_EaV¼!OlXcack]dUg_Ea ¤¥§£¨ P UWa«C # ^¤¥N§³¬ ¤¥N§f¶ vy¾ Ñ _EP1bcX[]JO ¨ \ n }ÓP \^ls«á [¤¥§(®á¥±À¶ Qs¾ ² h n 㨠² « x^¶ ýc¾ O a ½_ ä ¿ O ]£Xce©\EeJefXcP O)] L \^]£b[UgO kO ¸ UWedO(k_EaV¼!OlP1\ l[UWPUgÏ \!]dUg_Ea1b n _EÁ[ZgO P%k\Ea ÁsOOCaed_EZ`¼EO h \^]OC\!k L Ug]dO n \ ¡ ]JU`_VaN¾ K)L O aXca h O n ] L UWe-\EeJefXcP1b[]JU`_Va ¸ O5\ n O<UWabs_EeJU`]JU`_Va]d_½_ n PYXcZg\!]dO<eJ_EP Ok_EaV¼!O n }EO ackO n OCeJXcZ`]fe¾ J 6c, L % N ByF
(194) 5«C j¤ §¶ 6I ãj @JC
(195) U n eJ]Ä ¸ O
(196) b n _ ¼!O5] L \!]5«C ¤ §f¶
(197) UWe h OCk n O \!eJUWa[}y¾ à i h OÖsa[Ug]dUg_Ea6_^] L OËXyack ]JU`_Va6 j¤Jâ § ¸ O L \á¼!O ¤ §¨ªPUWa« ¬ ¤ §)® °±² ¶<¨ªP UgaN«C # ^¤ §³¬ ¤ §¶ Û # ^¤ §´ SVUWack O
(198) # ^¤¥N§ Û # ¤ §£½_ n \!Z`Z.¥;±À;Äy\Ee)\1bc\ n ]dUWkX[ZW\ n k \!eJO ¸ O L \á¼EO # !¤ § Û # ¤ § \!a h ] L O n O ½_ n O¹ ¤ § Û # ¤ §¾ . . . . .
(199) /. . ". . . . . . . . . . . . . . . . ". . . . . . ". ". ;. _. .<;=.
(200) ¦j ã © ¢ c
(201) 3 W© y f . x þ. K)L OCa ¸ O(b n _ ¼!O3] L \^]] L UgeÓedOCÕVXcO ackO-UWe©Ás_EXca h O h ÁViÁsOZg_ ¸ ¾ n _EP h O Ösa[Ug]dUg_Ea _!Nk_EaV¼!Ol
(202) ]J\Ea[}!OCaE] \Ebcb n _ál[UgP1\^]JU`_Va ¬ !¤¥N§
(203) V¤¥§£½_ n \^ZgZ.¥;±; ³°±² \Ea h [¤¥§£¨ P UWa«á¬ ^¤¦¥§(®°±;² E¶ ¯P UWa«
(204) !¤¦¥§)®°±;² V¶ ¤÷þE§ à i ] L O OUgO n ed] n \EeJe e©] L O _ n O P%] L O5k _EaV]JUgajX[_VXceXcayk ]dUg_EaP Uga«
(205) ¤¥N§)® °±² ¶ L \!e-\
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