Mixed Type Duality for Multiobjective Variational Problems

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1 Ž. ournl of Mthemtcl Anlyss nd Applctons 252, do: m , vlle onlne t http: on Mxed Type Dulty for Multoectve Vrtonl Prolems R. N. Mukheree nd Ch. Purnchndr Ro Deprtment of Appled Mthemtcs, Insttute of Technology, Bnrs ndu Un ersty, Vrns , Ind Sumtted y Wllm F. Ames Receved Septemer, 1996 The concept of mxed-type dulty hs een extended to the clss of multoectve vrtonl prolems. A numer of dulty reltons re proved to relte the effcent solutons of the prml nd ts mxed-type dul prolems. The results re otned for -convex Ž generlzed -convex. functons. These studes hve een generlzed to the cse of -nvex Ž generlzed -nvex. functons. Our results pprently generlze frly lrge numer of dulty results prevously otned for fnte-dmensonl nonlner progrmmng prolems under vrous convexty ssumptons Acdemc Press ey Words: mxed-type dul; effcent soluton; -convex; -nvex; multoectve progrmmng. 1. INTRODUCTION The dulty theory hs een studed extensvely n the nonlner progrmmng lterture. Ths theory my e regrded s the most delcte suect n the theory of nonlner progrmmng, nd ts theoretcl mportnce cnnot e questoned Že.g., n the theory of computtonl lgorthms of lner progrmmng nd n the theory of prces nd mrkets n economcs.. The mn queston whch s nvestgted n the dulty s s follows: under whch ssumptons s t possle to ssocte n equvlent mxmzton Ž dul. prolem to gven mnmzton Ž prml. prolem? For ths purpose, n the recent pst vrous dulty models hve ppered. Among mny such models, two well-known dulty models re Wolfe dul nd Mond Wer dul, whch were wdely used n the re of fnte-dmen X 00 $35.00 Copyrght 2000 y Acdemc Press All rghts of reproducton n ny form reserved.

2 572 MUEREE AND RAO sonl smooth nd nonsmooth nonlner progrmmng prolems. Qute recently, Zengkun Xu 6 ntroduced mxed-type dulty model whch contns the ove two models s specl cses nd estlshes vrous dulty results y reltng effcent solutons of hs mxed-type dul pr of prolems. On-the other hnd, nother sc concept n the theory of nonlner progrmmng s the generlzton of convexty, whch ssumes centrl role n mny spects of mthemtcl progrmmng, ncludng suffcent optmlty conons, dulty reltons, theorems of lterntves, nd convergence of the optmzton lgorthms. Vrous generlztons of convexty, for exmple, nvexty, qusnvexty, nd pseudonvexty Žsee, e.g.,. 3, re qute close to convexty n the sense tht they preserve some of the mportnt propertes of convexty. Another generlzton of convexty known s -convexty, n whch the defnng nequlty for convex holds pproxmtely, to wthn term dependng on prmeter whch my e zero Ž convex., postve Ž strongly convex., or negtve Ž wekly convex., ws ntroduced y Vl, whose role n the constructon nd convergence nlyss of lgorthms n nonlner progrmmng s well known. The noton of -convexty hs een further generlzed to the noton of -nvexty y eykumr 2. Qute recently, mny ppers hve een devoted to the study of these functons for the clss of vrtonl nd control prolems; for exmple, one my consult 1. The purpose of ths pper s to ntroduce contnuous nlog of the Ž sttc. mxed-type dul ntroduced qute recently y Zengkun Xu 6, n clss of vrtonl multoectve progrmmng prolems, nd to estlsh frly lrge numer of dulty results y reltng effcent solutons etween ths mxed-type dul pr. The results re otned for dfferentle -convex Ž generlzed -convex. functons nd -nvex Žgenerlzed -nvex. functons n ther contnuous verson. These dulty results contn s specl cses the counterprts of most well-known results orgnlly otned for conventonl nonlner progrmmng prolems wth dfferentle. 2. NOTATION AND PRELIMINARIES Let I, e rel ntervl nd let 1, 2,..., p P nd 1, 2,..., m M. In ths pper we ssume the followng: xt Ž. s n n-dmensonl pecewse smooth functon of t, nd xt Ž. s the dervtve of xt Ž. wth respect to t n,. For nottonl smplcty, we shll wrte, s nd when necessry, xt Ž. nd xž. t s x nd x, respectvely, nd so on. We denote the prtl dervtves

3 MULTIOBECTIVE VARIATIONAL PROBLEMS 573 t x ẋ of f wth respect to t, x nd x, respectvely, y f, f, nd f such tht f Ž f x,..., f x. nd f Ž f x,..., f x. x 1 n x 1 n. Smlrly, the prtl dervtves of the vector functon g cn e wrtten, usng mtrces wth m rows nsted of one. Let S denote the spce of n-dmensonl pecewse smooth functons x wth norm x x Dx, where the Ž. t dfferentton opertor D s u Dx xt us Ž. ds, where 0 s 0 gven oundry vlue. Therefore, D d dx, except t dscontnutes. Remrk 1. For nottonl smplcty, no nottonl dstncton s mde etween row nd column vectors. Suscrpts denote prtl dervtves, nd superscrpts denote vector components. Unless otherwse specfed, for ny ndexed set T 1, 2, 3,..., t, T mens the sum over ll T. We now gve some defntons whch wll e used susequently n our lter results. Let F x : S, denoted y F x fž t, x, x., e Frechet dfferentle. Let e rel numer. At pont u n S, we defne functonl F to e Ž. -convex f rel numer such tht x Ž u. n S, F x Fu Ž x u. f Ž t, u, u. ŽDŽ x u.. f Ž t, u, u. x u 2 u u or strctly -convex f strct nequlty holds; Ž. -pseudoconvex f rel numer such tht x Ž u. n S, Ž x u. f Ž t, u, u. ŽDŽ x u.. f Ž t, u, u. x u 2 F x u u Fu or strctly -pseudoconvex f strct nequlty holds n the rght-hnd nequlty of the ove mplcton; Ž. c -qusconvex f rel numer such tht, x Ž u. n S, F x Fu Ž x u. f Ž t, u, u. ŽDŽ x u.. f Ž t, u, u. u u x u 2. From ths pont on we use the term generlzed -convexty to ndcte -pseudoconvexty, -qusconvexty, etc. Now the most mmee wy to extend the -convexty Ž generlzed -convexty. to the vector functons requres the -convexty Ž generlzed -convexty. of the sngle components. For ths purpose let h Žh 1, h 2,...,h n. e n n-dmensonl vector functon nd ech of ts components e -convex Žgenerlzed -convex. t the sme pont u. Also, let k Ž k, k,...,k. 1 2 n e vector constnt such tht k 0 for ll 1, 2,..., n. Then Ž. h N s N -convex t u, Ž. Ech kf s k -convex t u, nd hence Ž. c ht,.,. Ž. s -convex t u. N

4 57 MUEREE AND RAO These propertes wll e used frequently throughout the pper wthout eng specfed. 3. FORMULATION OF TE MAIN PROBLEM We consder the followng multoectve vrtonl progrmmng prolem: suect to Mn fž t, x, x. Ž MP. x X x S xž., xž., gž t, x, x. 0, t I, 0 0 Ž 1 2 p. n n P where f f, f,..., f ; I, ech component functon Ž 1 2 s contnuously dfferentle rel sclr functon, nd g g, g,..., m. n n m g : I s n m-dmensonl contnuously dfferentle vector functon. Snce the oectves n multoectve progrmmng prolems generlly conflct wth one nother, n optml soluton s chosen from the set of effcent solutons n the followng sense, nd Mn mens fndng n-dmensonl pecewse smooth effcent soluton x xt, Ž. t I, for the prolem Ž MP.. DEFINITION 1. An n-dmensonl pecewse smooth functon u n the fesle regon of the prolem Ž MP. s sd to e n effcent soluton for the prolem Ž MP. f x X nd P: f Ž t, u, u. f Ž t, x, x. f Ž t, u, u. f Ž t, x, x.. DEFINITION 2. An n-dmensonl pecewse smooth functon u n the fesle regon of the prolem Ž MP. s sd to e wek mnmum for the prolem Ž MP. f there exsts no other x n X for whch fž t, u, u. fž t, x, x.. From the ove two defntons t follows tht f x n X s n effcent soluton for Ž MP., then t s lso wek mnmum for Ž MP.. Before presentng the mxed-type dul to Ž MP. we stte, n the form of the followng proposton, the contnuous verson of Theorem 2.2 of 5, whch wll e needed n the proof of the Strong Dulty Theorem. Ž. PROPOSITION 1. Let x e wek mnmum for MP t whch the uhn Tucker constrnt qulfcton s stsfed. Then there exst n p

5 MULTIOBECTIVE VARIATIONAL PROBLEMS 575 nd pecewse smooth Ž..: I m such tht f Ž t, x, x. Ž t. g Ž t, x, x. D f Ž t, x, x. Ž t. g Ž t, x, x. x x x x Ž t. gž t, x, x. 0 Ž t. 0, e 1, 0, where e s the ector of p, the components of whch re ll ones. We dvde the ndex set M of the constrnt functons of the prolem Ž MP. nto two dsont susets, nmely nd, such tht U M, nd let Ž t. g Ž t, x, x. Ž t. g Ž t, x, x. Ž t. g Ž t, x, x. Ž t. g Ž t, x, x.. Now we ntroduce the contnuous nlog of the sttc mxed-type dul Ž. 6, for the prml prolem MP. suect to Mx f t, u, u t g t, u, u e MD Ž. Ž. Ž. Ž. fuž t, u, u. Ž t. guž t, u, u. D fuž t, u, u. Ž t. guž t, u, u. Ž 1. t g t, u, u 0 2 Ž. Ž. Ž. Ž t. 0, e 1, 0 Ž 3. xž. 0, xž. 0, where e s the vector of p, the components of whch re ll ones.. DUALITY TEOREMS In ths secton, we present nd dscuss frly lrge numer of dulty results etween Ž MP. nd Ž MD. y mposng vrous -convexty Žgener- lzed -convexty. conons upon the oectve nd constrnt functons. We egn wth stuton n whch ll of the functons re -convex.

6 576 MUEREE AND RAO Susequently, we formulte more generl dulty crter n whch the generlzed -convexty requrements re plced on certn comntons of the oectve nd constrnt functons. Let Y denote the set of ll fesle solutons of Ž MD.. The theorems tht follow re wek dulty theorems n whch we prove tht f Ž t, x, x. f Ž t, u, u. Ž t. g Ž t, u, u. Ž. cnnot hold for x n X nd u n Y, for ll n P, nd for some n P, f Ž t, x, x. f Ž t, u, u. Ž t. g Ž t, u, u.. Ž 5. Ž Ž.. TEOREM 1. Let x X nd u,, t Y, nd Ž 1. for ech P, f s -con ex, nd for ech M, g s -con ex; then Ž. nd Ž 5. cnnot hold f ether of the followng hold. Ž 1. For ech P, 0 wth Ž t. P M 0 Ž 1c. Ž t. 0. P M Proof. If x u, then wek dulty theorem trvlly holds, so ssume tht x u. From the dulty constrnt Ž. 1 we hve u u u u Ž x u. f Ž t, u, u. Ž t. g Ž t, u, u. Ž x u. D f Ž t, u, u. Ž t. g Ž t, u, u.. Ž 6. Suppose, on the contrry, tht Ž. nd Ž. 5 hold. These nequltes mply, n vew of the feslty of x for Ž MP., tht Ž. Ž. Ž. f t, x, x t g t, x, x for ll P, nd for some P, f Ž t, u, u. Ž t. g Ž t, u, u. Ž. Ž. Ž. f t, x, x t g t, x, x f Ž t, u, u. Ž t. g Ž t, u, u..

7 MULTIOBECTIVE VARIATIONAL PROBLEMS 577 From the strct postvty of ech component of nd the fct tht e 1, t follows tht Ž. Ž. Ž. f t, x, x t g t, x, x f t, u, u t g t, u, u. 7 Ž. Ž. Ž. Ž. Now y the defntons of -convexty of f, P, nd -convexty of g, M, we hve Ž. Ž. f t, x, x f t, u, u u Ž. u Ž x u. f Ž t, u, u. DŽ x u. f Ž t, u, u. x u for ll P Ž 8. Ž. Ž. g t, x, x g t, u, u u Ž. u Ž x u. g Ž t, u, u. DŽ x u. g Ž t, u, u. x u 2 for ll M. Ž 9. Ž. p On multplyng ech nequlty of 8 y ech of nd ech Ž. Ž. Ž. m nequlty of 9 y ech t of t, nd ddng the nequltes Ž mong P nd M. we otn fž t, x, x. Ž t. gž t, u, u. fž t, u, u. Ž t. gž t, u, u. u u Ž x u. f Ž t, u, u. Ž t. g Ž t, u, u.. Ž DŽ x u.. f Ž t, u, u. Ž t. g Ž t, u, u. u Ž P M. Ž t. x u 2. By ntegrton y prts, the rght-hnd sde reduces to the followng, Ž. v 1 : u u Ž x u. f Ž t, u, u. Ž t. g Ž t, u, u. t už. Ž. už. Ž. t f t, u, u t g t, u, u x u u u Ž x u. D f Ž t, u, u. Ž t. g Ž t, u, u.. u

8 578 MUEREE AND RAO Ž. On mkng use of the oundry conons 6, the ove yelds fž t, x, x. Ž t. gž t, x, x. fž t, u, u. Ž t. gž t, u, u. 0 Snce M U, Ž 10. Ž t. g Ž t. g Ž t. g, Ž 11. Ž. nd hence the ove nequlty mples, long wth 7, tht t g t, x, x t g t, u, u Ž. Ž. Ž. Ž. Ž. Ž Ž. Ž. Now, snce u,, t Y, from 2, Ž. t g Ž t, x, x. 0, whch s contrdcton of the fct tht x s fesle for Ž MP., nd hence Ž. nd Ž 5. cnnot hold. Ž 1c. In ths cse the multplers of the oectve functons f need not e strctly postve, nd t gves n plce of of Ž. 7. If we ssume the conon n Ž 1c., we get n plce of of Ž 10.. ence we get Ž 12. nd we conclude the theorem s n the cse of Ž 1.. Ths completes the proof. Evdently, the ove theorem hs numer of mportnt specl cses whch cn redly e dentfed y the sutle lgerc propertes of the -convex functons. We shll stte some of these s corollres. The sttc functon nlogs of these wek dulty results re well known n the re of nonlner progrmmng. Ž Ž.. COROLLARY 1. Let x X nd u,, t Y, nd Ž. for ech P, f s -con ex, nd for ech M, Ž. t g s -con ex then Ž. nd Ž. 5 cnnot hold f ether of the followng holds. Ž. For ech P, 0 wth P M 0 or Ž. c 0. P M Proof. Snce g s -convex whenever Ž t. g s Ž t. - convex nd Ž. t 0, the proof s smlr to Theorem 1. COROLLARY 2. Let x X nd Žu,, Ž t.. Y nd ssume s n Corollry 1, except tht nsted of Ž. t g eng -con ex, Ž. nd Ž c,the. functon Ž t, u, u. Ž t. g Ž t, u, u. s -con ex, Ž. M for ech P 0 wth 0 nd Ž. c 0, respect ely. Then Ž. P P nd Ž. 5 cnnot hold.

9 MULTIOBECTIVE VARIATIONAL PROBLEMS 579 Note tht n Theorem 1, ech constrnt functon g Ž t,.,.. s ssumed to e -convex, wheres n Corollry 2 they re ggregted nto one -convex functon. We oserve tht t s lso possle to consder stuton ntermee etween these two extreme cses Žkeepng n vew the prtton of the constrnt functon n the oectve functon of the dul prolem Ž MD.., n whch some of the constrnt functons cn e comned nto -convex functon whle the rest re ndvdully -convex. Stutons of ths type re presented n the next two corollres. COROLLARY 3. Let x X nd Žu,, Ž t.. Y, nd Ž. for ech P, f s -con ex nd Ž t. g s -con- ex, wheres for ech, Ž. t g s -con ex. Then Ž. nd Ž 5. cnnot hold f ether of the followng holds. Ž. For ech P, 0 wth P 0 or Ž. c P 0. COROLLARY. Let x X nd Žu,, Ž t.. Y, nd Ž. For ech P, f s -con ex nd Ž t. g s - con ex, wheres Ž. t g s -con ex. Then Ž. nd Ž 5. cnnot hold f ether of the followng holds. Ž. For ech P, 0 wth P 0 or Ž. c P 0. The next corollry s the stuton n whch ll of the oectves nd constrnts re ggregted nto sngle one. COROLLARY 5. Let x X nd Žu,, Ž t.. Y, nd Ž. f Ž. t g s -con ex, then Ž. nd Ž. 5 cnnot hold f ether of the followng holds. Ž. For ech P, 0 wth 0 or Ž. c 0. In the rest of ths secton we use the generlzed -convexty. From ths pont on we wll try to restrct ourselves n most of the cses to stutons n whch only sclrztons of the oectve nd constrnt functons re consdered. And we remrk here tht the mmee consequences n ech of those stutons n the form of corollres cn esly e seen, ust s n the cse of Theorem 1. We do not explctly stte these corollres. TEOREM 2. Let x X nd Žu,, Ž t.. Y, nd Ž 2. Ž t. g s -quscon ex. Ž 2. For ech P, 0 nd f Ž t. g s oth -quscon ex nd -pseudocon ex wth 0. Then Ž. nd Ž. P 5 cnnot hold.

10 580 MUEREE AND RAO Proof. If x u, then wek dulty theorem trvlly holds, so ssume tht x u. Snce x X nd Žu,, Ž t.. Y, we hve t g t, x, x 0 t g t, u, u. 13 Ž. Ž. Ž. Ž. Ž. Ž. -Qusconvexty n 2, n vew of the ove, mples tht u u Ž x u. Ž t. g Ž t, u, u. DŽ x u. Ž t. g Ž t, u, u. x u 2. Ž 1. The susttuton of the dulty constrnt Ž. 1 n the frst term of the ove mplcton gves us, long wth Ž 11., u u u Ž x u. D f Ž t, u, u. Ž t. g Ž t, u, u. Ž t. g Ž t, u, u. Ž. Ž. Ž. fu t, u, u t gu t, u, u 2 u DŽ x u. Ž t. g Ž t, u, u. x u. Ž 15. On usng the oundry conons fter ntegrton y prts, u u Ž x u. f Ž t, u, u. Ž t. g Ž t, u, u. Ž DŽ x u.. f Ž t, u, u. Ž t. g Ž t, u, u. x u 2. Ž 16. u u Tht s, on mkng use of the conon 0 nd the fct tht P e 1, we hve the followng: Ž. Ž. Ž. Ž. P x u fu t, u, u t gu t, u, u Ž. Ž. Ž. Ž. D x u fu t, u, u t gu t, u, u Ž. x u 2. P Snce 0, P, t follows from the ove tht u u Ž x u. f Ž t, u, u. Ž t. g Ž t, u, u. Ž. Ž. Ž. Ž. D x u fu t, u, u t gu t, u, u x u 2 Ž 17.

11 MULTIOBECTIVE VARIATIONAL PROBLEMS 581 for ll P, nd for some P, u u Ž x u. f Ž t, u, u. Ž t. g Ž t, u, u. Ž. Ž. Ž. Ž. D x u fu t, u, u t gu t, u, u x u 2. Ž 18. Suppose Ž 17. holds; then the -pseudoconvexty ssumpton n Ž 2. gves, long wth the feslty of x for Ž MP., for ll P, f Ž t, x, x. f Ž t, u, u. Ž t. g Ž t, u, u.. Ž 19. Now suppose Ž 18. holds; then the equvlent form of the -qusconvexty ssumpton n Ž 2. gves, long wth the feslty of x for Ž MP., for some P, f Ž t, x, x. f Ž t, u, u. Ž t. g Ž t, u, u.. Ž 20. Ž. Ž. Ž. Ž. Ovously 19 nd 20 cn show tht nd 5 cnnot hold, whch completes the proof. The followng theorem s stted wthout proof. It wll e estlshed n mnner very smlr to tht of Theorem 2. Ž Ž.. TEOREM 3. Let x X nd u,, t Y, nd Ž 3. Ž t. g s -quscon ex. Ž 3. For ech P, 0 nd f Ž t. g s -qu- scon ex nd there exsts some k P such tht t s strctly k-pseudocon ex Ž wth the correspondng component of post e. k wth P 0. Then Ž. nd Ž. 5 cnnot hold. Ž Ž.. TEOREM. Let x X nd u,, t Y, nd Ž. Ž t. g s -quscon ex. Ž. For ech P, 0 nd f Ž t. g s - pseudocon ex wth 0. Then Ž. nd Ž. 5 cnnot hold. Ž. Proof. As n the cse of Theorem 2, ssume x u nd get 16. By mkng use of the conon 0 nd y our -pseudoconvexty

12 582 MUEREE AND RAO Ž. ssumpton n we otn Ž. Ž. Ž. f t, x, x t g t, x, x fž t, u, u. Ž t. g Ž t, u, u.. Ž. Now the feslty of x for MP nd the fct tht e 1 mply fž t, x, x. fž t, u, u. Ž t. g Ž t, u, u.. Clerly, ths concludes the theorem, snce 0 for ech P. The ssumpton tht Ž. t g s -qusconvex s very mportnt, s we see n the prevous theorems Ž 2.. Of course, to get the desred results wthout ths conon, other conons should e enforced, whch leds to the followng theorem. Ž Ž.. TEOREM 5. Let x X nd u,, t Y, nd Ž 5. For ech P, 0 nd f Ž t. g s oth -pseudocon ex nd -quscon ex wth 0. Then Ž. nd Ž. P 5 cnnot hold. Proof. Assume x u. From the dulty constrnt Ž. 1 we get Ž. 6. Now y ntegrton y prts, u u Ž x u. f Ž t, u, u. Ž t. g Ž t, u,u. t u u t Ž x u. f Ž t, u, u. Ž t. g Ž t, u, u. u u Ž DŽ x u.. f Ž t, u, u. Ž t. g Ž t, u, u.. Snce ech 0, P, from the conon e 1, we hve P u u Ž x u. f Ž t, u, u. Ž t. g Ž t, u,u. Ž. Ž. Ž. Ž. Ž. D x u fu t, u, u t gu t, u, u Gven tht P 0 nd x u 2 s lwys postve, Eq. Ž 21. x u 2. P

13 MULTIOBECTIVE VARIATIONAL PROBLEMS 583 Agn usng the nonnegtvty of ech for P, nd -pseudocon- vexty nd the equvlent form of -qusconvexty n Ž 5., t follows from the ove nequlty tht f Ž t, x, x. Ž t. gž t, x, x. f Ž t, u, u. Ž t. gž t, u, u. for ll P, nd for some P, Ž. Ž. Ž. f t, x, x t g t, x, x f Ž t, u, u. Ž t. gž t, u, u.. Now the fesltes of x for Ž MP. nd Žu,, Ž t.. for Ž MD. led us to the desred concluson tht Ž. nd Ž. 5 cnnot hold. Next we stte wthout proof the lst wek dulty theorem. It wll e proved n smlr mnner. Ž Ž.. TEOREM 6. Let x X nd u,, t Y, nd Ž 6. For ech P, 0 nd f Ž t. g s -con- ex wth 0, or Ž 6. f Ž t. g s strctly -con ex wth 0. Then Ž. nd Ž. 5 cnnot hold. The ssumpton Ž 6. tht f Ž t. gž t,.,.. s strctly -convex cn e replced y much weker conons. Ths de leds us to the followng corollry. COROLLARY 6. Let x X nd Žu,, Ž t.. Y, nd ssume s n Theorem 6, except tht nsted of Ž 6. for ech P, f Ž t. g s -con ex, nd for t lest one k P, f k Ž t. g s strctly -con ex Ž wth the correspondng component of post e. k k wth P 0. Then Ž. nd Ž. 5 cnnot hold. We next turn our ttenton to dscusson of strong dulty. The followng lemm s for tht purpose. LEMMA 1. Assume tht wek dulty Žny of the Theorems 1 6 or ny of the Corollres 1 6 holds etween Ž MP. nd Ž MD.. If Žu,, Ž t.. s fesle for Ž MD. wth Ž t. gž t, u, u. 0 nd u s fesle for Ž MP., then u s effcent for Ž MP. nd Žu,, Ž t.. s effcent for Ž MD.. Proof. Suppose, on the contrry, tht u s not effcent for Ž MP.; then there exsts fesle x for Ž MP. such tht f Ž t, x, x. f Ž t, u, u.

14 58 MUEREE AND RAO for ll P, nd for some P, f Ž t, x, x. f Ž t, u, u.. Ž. Ž. Ž Ž. Ž.. Snce t g t, u, u 0 t g t, u, u 0, we cn wrte f Ž t, u, u. Ž t. g Ž t, u, u. n plce of f Ž t, u, u. n the rght-sde terms of the ove two equtons. Then, snce Žu,, Ž t. Y nd x X, we get contrdcton to the wek dulty. ence u s effcent for Ž MP.. In smlr wy we cn esly show tht Žu,, Ž t.. s effcent for Ž MD.. Now utlzng ths lemm n conuncton wth the necessry optmlty conons Ž Proposton 1. of Secton 3, we otn the followng strong dulty theorem. TEOREM 7. Let x e n effcent soluton for Ž MP. nd ssume tht x stsfes the uhn Tucker constrnt qulfcton for Ž MP.. Then there exst p Ž. m nd pecewse smooth functon t : I such tht Ž x,, Ž t.. s fesle for Ž MD., long wth the conon Ž t. gž t, x, x. 0. Furthermore, f ny wek dulty Žny of the theorems 1 6 or ny of the Corollres 1 6. lso holds etween Ž MP. nd Ž MD., then Žx,, Ž t.. s effcent for Ž MD.. Proof. We hve ž / Ž t. gž t, x, x. 0 Ž t. g Ž t, x, x. 0. Ž 22. Snce x s n effcent soluton for Ž MP. nd snce every effcent soluton for Ž MP. s lso wek mnmum, ll of the conons of Proposton 1 re p stsfed, nd hence there exst nd pecewse smooth functon Ž. m t : I tht f Ž t, x, x. Ž t. g Ž t, x, x. D f Ž t, x, x. Ž t. g Ž t, x, x. x x x x Ž t. gž t, x, x. 0 Ž t. 0, e 1, 0. The ove three equtons show long wth Ž 22. tht Žx,, Ž t.. s fesle for Ž MD.. Now snce Žx,, Ž t.. s fesle soluton for Ž MD., ts effcency follows from Lemm 1. Snce we eleve tht the prevous oservtons out the cse of Ž. -convexty generlzed -convexty re stll vld n the cse of -nvexty

15 MULTIOBECTIVE VARIATIONAL PROBLEMS 585 Ž. generlzed -nvexty, ths topc could e the oect of further nvestgtons. 5. FURTER EXTENSIONS In the prevous secton t hs een shown tht, y mens of generlztons Ž.e., -convexty nd generlzed -convexty., t s possle to cheve the unfcton of the most well-known dulty results Žnmely Wolfe nd Mond Wer. n the clss of vrtonl prolems. In ths secton our purpose s to show tht the requrements of oectve nd constrnt functons Ž of the prml nd dul prolems of Secton 3., e.g., to e -convex, -pseudoconvex, or -qusconvex, cn e further wekened to e requred to e -nvex, -pseudonvex, nd -qusnvex respectvely. For ths purpose we reconsder here the prolems Ž MP. nd Ž MD. nd recll few defntons nd concepts pertnng to certn types of -nvex Ž generlzed -nvex. functons whch re used frequently throughout ths secton. If there exst vector functons Ž t, x, u.: I S S Žwth Ž t, x, u. 0t x u. nd : I S S nd rel numer such tht the functonl F x Ž s n Secton 2. stsfes u u F x F u Ž t, x, u. f Ž t, u, u. D Ž t, x, u. f Ž t, u, u. Ž t, x, u. 2, then F s sd to e -nvex t u S wth respect to nd. ere D Ž t, x, u. s the vector whose th component s D Ž t, x, u..in smlr wy the defntons of strct -nvexty, -pseudonvex, strct pseudonvex, nd -qusnvex cn esly e otned. From ths pont on we use the term generlzed -nvexty to ndcte -pseudonvexty, -qusnvexty, etc. From the ove defntons t s cler tht every -nvex functon Ž generlzed -nvex. s -convex functon Žgenerlzed -convex. wth Ž t, x, u. Ž x u. Ž t, x, u.. In prllel wth the results presented n Secton, we cn esly estlsh nlogous theorems Žtwo of them wthout proof re presented next s n exmple. for -nvex Ž generlzed -nvex. functons, snce only renterpretton of -convexty s nvolved. WEA DUALITY TEOREM. Let x X nd Žu,, Ž t.. Y, nd Ž. t g s -qusn ex, nd for ech P, 0 nd f Ž. t g s oth -qusn ex nd -pseudon ex wth P 0. Then Ž. nd Ž. 5 cnnot hold.

16 586 MUEREE AND RAO STRONG DUALITY TEOREM. Let x e n effcent soluton for Ž MP. nd ssume tht x stsfes the uhn Tucker constrnt qulfcton for Ž MP.. p Ž. m Then there exst nd pecewse smooth functon t : I such tht Žx,, Ž t.. s fesle for Ž MD. long wth the conon Ž. t gž t, x, x. 0. Furthermore, f wek dulty lso holds etween Ž MP. nd Ž MD. then Žx,, Ž t.. s effcent for Ž MD.. REFERENCES 1. D. Bht nd P. umr, Multoectve control prolem wth generlzed nvexty,. Mth. Anl. Appl. 189 Ž 1995., V. eykumr, Strong nd wek nvexty n mthemtcl progrmmng, Mth. Oper. Res. 55 Ž 1985., R. N. ul nd S. ur, Optmlty crter n nonlner progrmmng nvolvng nonconvex functons,. Mth. Anl. Appl. 105 Ž P. Vl, Strong nd wek convexty of sets nd functons, Mth. Oper. Res. 8 Ž 1983., T. Wer nd B. Mond, Generlzed convexty nd dulty n multple oectve progrmmng, Bull. Austrl. Mth. Soc. 39 Ž 1989., Zengkun Xu, Mxed type dulty n multoectve progrmmng prolems,. Mth. Anl. Appl. 198 Ž 1996.,