JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 198, 248261 1996 ARTICLE NO. 0080 -Pareto Otalty for Nodfferetable Multobectve Prograg va Pealty Fucto J. C. Lu Secto of Matheatcs, Natoal Uersty Prearatory School, Natoal Oerseas Chese Studet Uersty, Lkou, Tae, Hse, Tawa, 24402, Reublc of Cha Subtted by Koch Mzuka Receved Jauary 9, 1995 Necessary ad suffcet codtos wthout a costrat qualfcato for -Pareto otalty of ultobectve rograg are derved. The ecessary KuhTucker codto suggests the establshet of a Wolf-tye dualty theore for odfferetable, covex, ultobectve zato robles. The geeralzed -saddle ot for Pareto otalty of the vector Lagraga s studed. 1996 Acadec Press, Ic. 1. INTRODUCTION Several authors have bee terested recetly -otal solutos olear rograg. For detals, the readers are advsed to cosult 15. Lorda 4 derved soe roertes of -effcet ots soluto for vector zato robles ad used the Ekelad s varatoal rcle 6 to establsh the -Pareto otalty ad -quas Pareto otalty. I 7, Lu also adated the sae aroach to obta the -dualty theore of odfferetable ocovex ultobectve rograg. Recetly, several authors 812 have used a exact ealty fucto to trasfor the olear scalar rograg roble to a ucostraed roble ad derved the -otalty. I 14, Yokoyaa was cocered wth the -aroxate solutos ad exteded soe results of 13 to the vector zato robles. Slar to 13, Yokoyaa trasfored the vector robles to the ucostraed robles by usg the exact ealty fuctos ad showed the -otalty crtera by estatg the ealty araeter ters of -aroxate solutos for the assocated dual robles. 0022-247X96 $18.00 Coyrght 1996 by Acadec Press, Ic. All rghts of reroducto ay for reserved. 248
-PARETO OPTIMALITY 249 I ths aer, we are sred to use the exact ealty fucto to trasfor the equalty ultobectve rograg roble to a scalar ucostraed roble ad to derve the KuhTucker codtos whch Lagrage ultlers of obectve fuctos are oe. Soe deftos ad otatos are gve Secto 2. I Secto 3, we use scalar ealty fuctos to establsh the ecessary ad suffcet codtos of -Pareto soluto. Usg ths result, we forulate a dual roble of the Wolfe-tye for ultobectve rograg. I Secto 4, we gve soe relatoshs betwee the ral roble ad the dual roble. The geeralzed -Pareto saddle ot of the vector Lagraga s dscussed Secto 5. 2. PRELIMINARIES We cosder the followg covex ultobectve rograg roble: Ž P. ze fž x. subect to g Ž x. 0, 1,...,, where f Ž f, f,..., f. 1 2 ad each cooet fucto s a covex cotuous real-valued fucto defed o R ad where g are covex cotu- ous real-valued fuctos defed o R, 1. We deote the feasble set x R g Ž x. 0, 1 4 by F ad assue the feasble set F s oety. Let be a eleet of R. We troduce the feasble set F, ½ 5 F x R g x,1. For coveece, let. To trasfor the roble Ž P. to a scalar ucostraed roble, we use the exact ealty fucto troduced by Zagwll 8 : Ž x,. f Ž x. ax 0, g Ž x., 1 Ž. where 0. The assocated ucostraed roble whch ze Ž x,.
250 J. C. LIU s called a ealzed roble wth resect to the ealty araeter. For coveece, let Ž t. axž 0, t.. Clearly, we have x, f x Žg Ž x.. 1. DEFINITION 2.1. A ot x R s called a -Pareto soluto of Ž P. f x F ad there s o x F such that fž x. fž x. ad fž x. fž x., R. DEFINITION 2.2. A ot x R s called a alost -Pareto soluto of P f x F ad there s o x F such that fž x. fž x. ad fž x. fž x., R. DEFINITION 2.3. Let 0. A ot x R s called a -soluto of f the scalar roble Ž x,. Ž x,., for all x R. DEFINITION 2.4. Let h: R R 4 be a covex fucto, fte at x. The -subdfferetal of h at x s the set hž x. defed by 4 ² : hž x. x*r hž y. hž x. x*, y x for ay y R. 3. NECESSARY AND SUFFICIENT CONDITIONS I ths secto, we reset soe KuhTucker codtos for -Pareto otalty. THEOREM 3.1. If there exsts such that x s a -soluto for Ž. 0 for ay, the x s a -Pareto soluto for Ž P. 0 ad there exst scalars 0 Ž 1., 0 Ž 1., 0 Ž 1. such that: 1 1 1 Ž. 0 f Ž x. Ž g.ž x., Ž 1. Ž. g Ž x. 0. Ž 2.
-PARETO OPTIMALITY 251 Proof. If x s a -soluto of Proble Ž., Ž x,. Ž x,., for all x R. Ž 3. Clearly, Thus, we have Ž x,. f Ž x., for all x F. f Ž x. Ž x,. f Ž x., for all x F. Ž 4. If x F, Žg Ž x.. 1 0. Choose ay feasble ot ˆx whch s also F ad let ž / ax f Ž x. f Ž x. g Ž x.,. We the have the cocluso ½ 5 ˆ Ž. 0 1 f Ž ˆx. Ž ˆx,. Ž x,. f Ž ˆx.. Ths cocluso gves a cotradcto ad hece x F. If x s ot a -Pareto soluto of Ž P., the there exsts x1 F such that fž x1. fž x., 1, wth at least oe strct equalty. Therefore, we have f Ž x. f Ž x., 1 whch cotradcts Ž. 4. Thus x s a -Pareto soluto of Ž. P. Wth Ž. 3 ad the result of 15, we have Ž ž./ 1 0 f Ž. g Ž. Ž x.. The, there exst scalars 0 Ž 1., 0 Ž 1., 0 Ž 1., ad 0 Ž 1. such that, Ž 5. 1 0 f Ž x. Ž g.ž x., Ž 6. 1
252 J. C. LIU where 1, g Ž x. g Ž x. for 1,...,. Ž 7. Ž. Ž. By 5 ad 7, we have 1 1 g Ž x. 0. Ž. Ž. Fally, we obta the results 1 ad 2 by settg,, 1, 1. REMARK 3.1. If 1, the the ecessary codto of Theore 3.1 reduces to Theore 4.1 of 13. THEOREM 3.2. If x s a feasble soluto of Ž P. ad there exst 0 Ž 1., 0 Ž 1., 0 Ž 1. such that: 1 1 1 Ž. 0 f Ž x. Ž g.ž x., Ž 8. Ž. g Ž x. 0. Ž 9. The, xsa-pareto soluto of Ž P.. Proof. If x s a feasble soluto of Ž P. ad there exst 0 Ž 1., 0 Ž 1., 0 Ž 1. whch satsfy relatos Ž. 8 ad Ž. 9, the there exst x f Ž x.,1, y Ž g.ž x.,1, such that x y 0. By usg the characterzato of the -subgradet, we obta 1 f Ž x. f Ž x. x, xx, 1, g x g x y, xx, 1. ² : ² :
-PARETO OPTIMALITY 253 Thus, we have f Ž x. g Ž x. 1 1 Ž. Wth 9, we have f Ž x. g Ž x. f Ž x., for all x F. 1 f Ž x. f Ž x., for all x F. Ž 10. If x s ot a -Pareto soluto of P, there exsts x F such that 1 fž x1. fž x., 1, wth at least oe strct equalty. Therefore, we have f Ž x. f Ž x., 1 whch cotradcts 10. Thus x s a -Pareto soluto of P. THEOREM 3.3. If for suffcetly large, xsa-soluto for Ž., the x s a alost -Pareto soluto for Ž P. ad there exst scalars 0 Ž 1., 0 Ž 1., 0 Ž 1. such that: Ž. 0 f Ž x. g Ž x., 1 Ž. g Ž x. 0. 1 1 Proof. If x s a -soluto of Proble, f Ž x. Ž x,. f Ž x., for all x F.
254 J. C. LIU Sce f f Ž x. f Ž x., xr we have g Ž x. f f Ž x. f f Ž x.. 1 xf xr Let Ž. f f Ž x. f f Ž x. 0 xf xr The, there exsts such that 0 Ž. g Ž x., 1. Hece, we have x F. Ths cocludes the roof of the theore. 4. -DUALITY THEOREM OF THE WOLFE TYPE The result of Theore 3.1 s used to forulate such a dual roble of the Wolfe tye for ultobectve rograg as follows: 4 Ž D. axze LŽ x,. Ž x,. F D ; here D ½ 1 gž x. 0, 1 1 0, 1, 0, 1 5, F Ž x,. R R 0 f Ž x. g Ž x.,
-PARETO OPTIMALITY 255 ad the vector Lagraga fucto L x, s defed by LŽ x,. fž x. ²², gž x. :: 1 ; 1 1 f x 1 g x,..., f x 1 g x, for all x R, R, 1. to of Ž D. f Ž x,. F ad there s o Ž x,. F, such that DEFINITION 4.1. A ot x, R R s called a -Pareto solu- D f Ž x. Ž 1. g Ž x. f Ž x. Ž 1. g Ž x., 1, 1 1 wth at least oe strct equalty. THEOREM 4.1 Ž Dualty.. If there exsts 0 such that x s a -soluto for Ž. for ay, the x s a -Pareto soluto for Ž P. 0 ad there exst scalars R such that Ž x,. s a -Pareto soluto of Ž D,for. all, 1. Proof. Wth Theore 3.1, we coclude that Ž x,. s a feasble soluto of D. Let x, R R be ay feasble soluto of Ž D.. The, there exst x f Ž x.,1, y Ž g.ž x.,1, such that 1 x y 0, g Ž x. 0. 1 1 By usg the characterzato of the -subgradet, we obta Thus, we have f Ž x. f Ž x. x, xx, 1, ² : g Ž x. g Ž x. y, xx, 1. ² : f Ž x. g Ž x. f Ž x. g Ž x. 1 1 1 f Ž x. f Ž x. g Ž x.. 1 D
256 J. C. LIU Sce ad g x 0, for all 1, 1 g Ž x. g Ž x.. 1 We the deduce that f Ž x. g Ž x. 1 f Ž x. g Ž x., for all Ž x,. F. Ž 11. D 1 If Ž x,. s ot a -Pareto soluto of the dual roble Ž D., there exsts Ž x*, *. F such that D 1 f x* 1 g x* f Ž x. Ž 1. g Ž x., 1, 1 wth at least oe strct equalty. Thus, we have 1 1 f x* g x* f x g x whch cotradcts 11. THEOREM 4.2 Ž Coverse Dualty.. Let x be a feasble soluto of Ž P.. If Ž x,. s a feasble soluto of Ž D,. xsa-pareto soluto of Ž P.. Proof. It follows fro Theore 3.2. 5. VECTOR LAGRANGIAN AND ITS -PARETO SADDLE POINT I ths secto, we cosder the -Pareto saddle ot of the vector Lagraga fucto.
-PARETO OPTIMALITY 257 ot of the vector Lagraga LŽ x,. f the followg codtos hold: DEFINITION 5.1. A ot x, R R s called a -Pareto saddle Ž. L x, Ž. L x,, for all R ; L x, Ž. L x,, for all x R. That s to say, there exst ether R or x R such that: Ž. f Ž x. Ž 1. g Ž x. 1 wth at least oe strct equalty, Ž. f Ž x. Ž 1. g Ž x. 1 wth at least oe strct equalty. f Ž x. Ž 1. g Ž x., 1, 1 f Ž x. Ž 1. g Ž x., 1, 1 THEOREM 5.1. If there exsts such that x s a -soluto for Ž. 0 for ay 0, the x s a -Pareto soluto for P ad there exst scalars R such that Ž x,. s a -Pareto-saddle ot of the ector Lagraga. Proof. Wth Theore 3.1, there exst R such that 0 f Ž x. g Ž x., Ž 12. 1 g Ž x. 0. Ž 13. 1 1 The, there exst x f Ž x.,1, y Ž g.ž x.,1, such that 1 x y 0.
258 J. C. LIU By usg the characterzato of the -subgradet, we obta Thus, we have f Ž x. f Ž x. x, xx, 1, g x g x y, xx, 1. ² : ² : fž x. gž x. fž x. gž x. 1 1 1 f Ž x. fž x. gž x., for all x R. 1 Assue that there s a x* R such that 1 1 f x* 1 g x* f Ž x. Ž 1. g Ž x., 1, wth at least oe strct equalty, we obta 1 1 f Ž x*. g Ž x*. f Ž x. g Ž x., Ž 14. whch cotradcts Ž 14.. Ths gves the frst codto of the defto for -Pareto saddle ot. Wth Ž 13., we deduce that 1 1 g x 0. Sce x F, 1g x 0, for all R. Thus, we have 1 1 g x g x for all R.
-PARETO OPTIMALITY 259 Therefore, we obta f Ž x. g Ž x. 1 f x g x, for all R. 15 1 If there s a * R such that 1 f x 1 g x f Ž x. Ž 1. g Ž x., 1, 1 wth at least oe strct equalty, we obta 1 1 f x g x f x g x whch cotradcts 15. Ths coletes the roof. THEOREM 5.2. If Ž x,. s a -Pareto saddle ot of the ector Lagraga L ad g Ž x. g Ž x.,1, for all x F, the x s a alost -Pareto soluto of Ž P.. Proof. If x, s a -Pareto saddle ot of the vector Lagraga L, there s o R such that f Ž x. 1 g Ž x. f Ž x. 1 g Ž x., 1, 1 1 wth at least oe strct equalty. If x F, for soe k ad all. Thus, we have for all. g Ž x. k gkž x. kgkž x. kgkž x. Ž 16.
260 J. C. LIU Choose 1 ad for all k for L x, ; we obta k k f Ž x. 1 g Ž x. f Ž x. 1 1 g Ž x. 1 g Ž x. k k 1 k f Ž x. 1 g Ž x. 1 f Ž x. 1 g Ž x. 1 g Ž x. k k k for all whch cotradcts Ž 16.. We coclude that x F. Now, we use the other equalty for a -Pareto saddle ot. The, there s o x R such that f Ž x. Ž 1. g Ž x. f Ž x. Ž 1. g Ž x., 1, 1 1 wth at least oe strct equalty. Fro g Ž x. g Ž x., 1, for all x F, we coclude that there s o x F such that fž x. fž x., 1, wth at least oe strct equalty. Ths cocludes the roof of the theore. ACKNOWLEDGMENTS The author s thakful to Dr. Koch Mzuka of Hrosak Uversty for hs coets ad suggestos o a earler verso of the aer, esecally o the roof of Theore 5.2. REFERENCES 1. S. S. Kutateladze, Covex -rograg, Soet. Math. Dokl 20 Ž 1979., 391393. 2. P. Lorda, Necessary codtos for -otalty, Math. Prograg Study 19 Ž 1982., 140152. 3. P. Lorda ad J. Morga, Pealty fuctos -rograg ad -ax robles, Math. Prograg 26 Ž 1983., 213231. 4. P. Lorda, -Soluto vector zato robles, J. Ot. Theory Al. 43 Ž 1984., 265267.
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