OPTIMALITY CONDITIONS FOR LOCALLY LIPSCHITZ GENERALIZED B-VEX SEMI-INFINITE PROGRAMMING

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1 Mrcea cel Batra Naval Acadey Scetfc Bullet, Volue XIX 6 Issue he joural s dexed : PROQUES / DOAJ / Crossref / EBSCOhost / INDEX COPERNICUS / DRJI / OAJI / JOURNAL INDEX / IOR / SCIENCE LIBRARY INDEX / Google Scholar / Acadec Keys/ ROAD Ope Access / Acadec Resources / Scetfc Idexg Servces / SCIPIO / JIFACOR OPIMALIY CONDIIONS FOR LOCALLY LIPSCHIZ GENERALIZED B-VEX SEMI-INFINIE PROGRAMMING Vasle PREDA Veroca CORNACIU Professor, Uversty of Bucharest, Faculty of Matheatcs ad Coputer Scece Assstat Lecturer Uversty tu Maorescu, Faculty of Coputer Scece, Vacarest r. 87, Bucharest, 45, Roaa Abstract: I ths paper, by usg geeralzed drectoal dervatve ad geeralzed gradet, the class of B-vex, ( ρ, B, η) -vex, pseudo ( B, η) -vex, ad quas ( B, η) -vex fuctos for dfferetable fuctos s exteded to the class of geeralzed B-vex, ( ρ, B, η) vex, pseudo ( B, η) vex, ad quas ( B, η) vex fuctos for locally Lpschtz fuctos. he suffcet optalty codtos are obtaed for se-fte prograg probles whch volvg those fuctos. Keywords: geeralzed drectoal dervatve, geeralzed gradet,locally Lpschtz fucto, geeralzed B-vex.. Itroducto It s well kow that covexty play a portat role establshg the suffcet optalty codtos ad dualty theores for a olear prograg proble. Several class of fuctos have bee defed for the purpose of weakeg the ltatos of covexty. Bector ad Sgh exted the class of covex fuctos to the class of B-vex fuctos []. I [3], Bector ad Sueja defe the class of B-vex fuctos for dfferetable uercal fuctos. he suffcet optalty codtos ad dualty results were obtaed volvg these geeralzed fuctos. As so far ow, the study about the suffcet optalty codtos ad algorth of se-fte prograg are uder the assupto that the volvg fuctos are dfferetable. But osooth pheoea atheatcs ad optzato occur aturally ad frequetly, ad there s a eed to be able to deal wth the. I [5], the author study soe of the propertes of B- vex fuctos for locally Lpschtz fuctos, ad exted the class of B-vex, pseudo B-vex ad quas B-vex fuctos fro dfferetable uercal fuctos to locally Lpschtz fuctos. I [-3], Preda troduced soe classes of V- uvex type-i fuctos, called called (ρ, ρ')-vuvex type-i, (ρ, ρ')-quas V-uvex type-i, (ρ, ρ')- pseudo V-uvex type-i, (ρ, ρ')-quas pseudo V- uvex type-i, ad (ρ, ρ')-pseudo quas V-uvex type-i. I [8] Preda troduced the class of locally Lpschtz (B, ρ,d ) -prevex fuctos ad exted ay results of B-vexty type stated lterature. Preda troduced ( F, ρ) covex fucto as exteso of Fcovex fucto ad ρcovex fucto[9,,4]. he suffcet optalty codtos ad dualty results are obtaed for olear prograg, geeralzed fractoal prograg, ultobjectve prograg ad ax prograg proble, whch volvg those fuctos. Be al [4] obtaed soe propertes of covex fuctos based o the above geeralzed algebrac operatos. hese results were also appled to soe probles statstcal decso theory. I ths paper by usg geeralzed drectoal dervatve ad geeralzed gradet [6], the class of B-vex, B-vex, pseudo B-vex, ad quas B-vex fuctos for dfferetable fuctos [6] s exteded to the class of geeralzed B-vex, ( ρ, B, η) vex,pseudo ( B, η) vex, ad quas ( B, η) vex fuctos for locally Lpschtz fuctos. he suffcet optalty codtos are obtaed for se-fte prograg probles whch volvg those fuctos. 387 DOI:.79/ X-6-I hs work s lcesed uder the Creatve Coos Attrbuto-Nocoercal-Share Alke 4. Lcese.

2 Mrcea cel Batra Naval Acadey Scetfc Bullet, Volue XIX 6 Issue he joural s dexed : PROQUES / DOAJ / Crossref / EBSCOhost / INDEX COPERNICUS / DRJI / OAJI / JOURNAL INDEX / IOR / SCIENCE LIBRARY INDEX / Google Scholar / Acadec Keys/ ROAD Ope Access / Acadec Resources / Scetfc Idexg Servces / SCIPIO / JIFACOR hs paper s orgazed as follows. I Secto, we preset soe prelares ad related results whch wll be used the rest of the paper. he defto of B-vex, ( ρ, B, η) vex, ( B, η) pseudo-vex, ( B, η) quas-vex fucto are gve sese of. I Secto 3, soe suffcet optalty codtos theores are derved.. Prelares ad soe propertes of geeralzed ( B, η) vex fuctos hroughout our presetato, we let X be a oepty covex subset of R, U be a ope ut ball R, R deote the set of oegatve real ubers, f : X R, b: X X [,] R, ad bxu (,, ) s cotuous at = for fxed xu., Be-al [4] troduced certa geeralzed operatos of addto ad ultplcato. ) Let h be a vector-valued cotuous fucto, defed o a subset X of R ad possessg a verse fucto h. Defe the h- vector addto of x X ad y X as ( ) x y h h x h y =, ad the h-scalar ultplcato of x X R as ( ) x h h x =. ad ) Let φ be a real-valued cotuous fuctos, defed o Φ R ad possessg a verse fuctos ϕ. he the φ-addto of two ubers, α Φ ad β Φ, s gve by α β = ϕ ϕ α ϕ β, ad the φ-scalar ultplcato of α Φ ad R by α = ϕ ϕ α. 3) he (h, φ)-er product of vectors, defed as ( x y) ϕ h h( y) h, ϕ =. xy X s Deote = x = x x... x, x X, =,,..., α = α α... α, α Φ, =,,..., x y x y = Θ = ( ( ) ) [ ] = [ ] ( )[ ]. α β α β By Be-al geeralzed algebrac operato, t s easy to obta the followg coclusos: α = ϕ ϕ( α) = = x = h h x = = Θ = ( ) [ ] = ( ) ( [ ] ) = ( ) = x y h h x h y α β ϕ ϕ α ϕ β ϕ α ϕ α h x h x Defto.. A real valued fucto f : X R s sad to be Lpschtz at x X f there exsts two postve costats ε, k such that [ ] [ ] Θ ( h, ) f( z) f( y) k z y, z, y Bε, f s sad to be locally Lpschtz f f s Lpschtz at every x X. Let f be a Lpschtz ad real-valued fucto defed o R. For all xv, R, the geeralzed drectoal dervatve of f wth respect to drecto v ad the geeralzed gradet of f at x, deoted by f ( xv, ) ad as follows[6]. t f ϕ, respectvely, are defed f ( x, v) = lsup f( y t v) f( y) t 388 DOI:.79/ X-6-I hs work s lcesed uder the Creatve Coos Attrbuto-Nocoercal-Share Alke 4. Lcese.,

3 Mrcea cel Batra Naval Acadey Scetfc Bullet, Volue XIX 6 Issue he joural s dexed : PROQUES / DOAJ / Crossref / EBSCOhost / INDEX COPERNICUS / DRJI / OAJI / JOURNAL INDEX / IOR / SCIENCE LIBRARY INDEX / Google Scholar / Acadec Keys/ ROAD Ope Access / Acadec Resources / Scetfc Idexg Servces / SCIPIO / JIFACOR { ξ ( ξ ) f ; f ( xv, ) v, v R. = We ote that the above deftos ca be see as geeralzatos of the deftos troduced by Zhag [6]. he relato betwee geeralzed drectoal dervatve ad Clarke drectoal dervatve ca be gve by the followg theore. heore..[6]. Let f be a real valued fucto, ϕ() t be strctly creasg ad cotuous o R, ad let f ˆ( t ) ϕ fh = () t. he (, ) ˆ f xv = ϕ f ( hx, hv ), where f s Clarke drectoal dervatve. heore..[6]. Let f be a real valued fucto, ϕ() t be strctly creasg ad cotuous o R, ad let f ˆ( t ) ϕ fh = () t. he ˆ f = h f( hx ) = h ( ξ); ξ fˆ( t). { ( t= h ) Lea..[6] Let f : X R, g: X R be local Lpschtz fuctos. he () f s a o-epty, covex ad weakly copact subset of X ; () ( ξ ) { ξ f ( xv, ) = ax v ; f, v R; () ( [ ] ) ( f ) x f x (v) ( ) (, ) (, ),, ([ ] ) = Θ f g f gx =, ; f xv = f xθv xv R f f, x R ; Lea..[6] If Ω ad Ω are two o-epty covex weakly copact of R, the () for every v R,, R, we have {( ξ v) ξ = ξ ξ ax ( v) ; ax ; [ ] ( v) { = ξ ξ Ω { Ω ax ξ ; ξ () Ω Ωff for every v R {( v) ( v) h ϕ, we have { h ϕ ax ξ ; ξ Ω ax ξ ; ξ Ω. (, ) (, ) I the followg deftos, we cosder fucto b: X X [,] R, wth bxu (,, ) cotuous at = for fxed xu., We troduce followg classes of fuctos: Defto.. he fucto f s sad to be Bvex at u X f for ad every x X we have f x ( ) u [ ] [ ] [ ] bxu (,, ) f ( bxu (,, )) f( u) () f s sad to be Bvex o X f t s Bvex at every u X. Lea.3. A locally Lpschtz fucto f : X R s sad to be Bvex wth respect to b at u X ff there exsts a fucto bxu (,, ) such that b( xu, ) f f( u) f ( ux, Θu), xu, R () where b( xu, ) = lsup bxu (,, ) = l bxu (,, ). Proof: Sce f s Bvex u X we have bxu (,, ) f ( bxu (,, )) f( u) ( ( ) ) f x u herefore ( ( ) )[ ] bxu (,, ) f bxu (,, ) f( u) f x u f u so (,, ) ( )[ ] bxu f x f u f x ( ) u f( u) whch yeld [ ] ( [ ] ) [ ] ( f ( x ( ) u)[ ] f( u) ) bxu (,, ) f f( u) Gog to the lt the above relato we have 389 DOI:.79/ X-6-I hs work s lcesed uder the Creatve Coos Attrbuto-Nocoercal-Share Alke 4. Lcese.

4 Mrcea cel Batra Naval Acadey Scetfc Bullet, Volue XIX 6 Issue he joural s dexed : PROQUES / DOAJ / Crossref / EBSCOhost / INDEX COPERNICUS / DRJI / OAJI / JOURNAL INDEX / IOR / SCIENCE LIBRARY INDEX / Google Scholar / Acadec Keys/ ROAD Ope Access / Acadec Resources / Scetfc Idexg Servces / SCIPIO / JIFACOR lsup bxu (,, ) f f( y) l sup [ ] ( ( ) )[ ] thus ( f x y f y ) lsup bxu (,, ) f f( y) l sup [ ] ( ( Θ ))[ ] So we get ( f y x y f y ) b( xu, ) f f( u) f ( ux, Θ u). Defto.3. Let ρ be a real uber. A locally Lpschtz fucto f : X R s sad to be ( ρ, B, η) vex at u X wth respect to soe fuctos ηθ, : X X R, ( ( xu, ) wheever x u) bxu (, )[ ] ( f [ ] f( u) ) η ρ θ θ,f we have f ( u, ( xu, )) ( xu, ), x X (3) f s sad to be ( ρ, B, η) vex o X f t s ( ρ, B, η) vex at every u X. If ρ >, the f s sad to be strogly ( ρ, B, η) vex. If ρ =, the f s sad to be ( B, η) vex. If ρ >, the f s sad to be weakly ( B, η) vex. Defto.4. A locally Lpschtz fucto f : X R s sad to be ( B, η) pseudo-vex at u X, f we have f ( u, η( xu, )) bxu (, ) f bxu (, ) f( u), x X (4) f s sad to be ( B, η) pseudo-vex o X f t s ( B, η) pseudo-vex at every u X. If f ( u, η( xu, )) bxu (, ) f > bxu (, ) f( u), x X (5) the we ga the strctly ( B, η) pseudovex defto. Defto.5. A locally Lpschtz fucto f : X R s sad to be ( B, η) quasvex at u X, f we have f f( u) bxu (, ) f ( u, η( xu, )), x X (6) f s sad to be ( B, η) quas-vex o X f t s ( B, η) quas-vex at every u X. If f < f( u) bxu (, ) f ( u, η( xu, )) <, x X (7) the we ga the strctly ( B, η) quasvex defto. 3. Suffcet Optalty Codtos hrough the rest of ths paper, oe further assues that h s a cotuous oe-to-oe ad oto fucto wth h() =. Slarly, suppose that φ s a cotuous oe-to-oe strctly ootoe ad oto fucto wth φ() =. Uder the above assuptos, t s clear that α = α[ ] =. f, g, g,..., g,... be locally Let k Lpschtz fuctos defed o a o-epty ope covex subset X R. Cosder the followg se-fte prograg probles ( P ): f ( P) (8) st.. gj, j=,,... Let D { x X; g j, j,,... = = deote the feasble set of proble ( P ). Aalogously to the crtcal pot of proble ( P) defed [4], we gve a defto as follows. Defto 3.. he pot x D s sad to be a crtcal pot of proble ( P ), f the set I = ; g ( x ) =, =,,... s fte (that s { I = < ), ad for each I there exsts l R such that f( x ) l g( x ) (9) I 39 DOI:.79/ X-6-I hs work s lcesed uder the Creatve Coos Attrbuto-Nocoercal-Share Alke 4. Lcese.

5 Mrcea cel Batra Naval Acadey Scetfc Bullet, Volue XIX 6 Issue he joural s dexed : PROQUES / DOAJ / Crossref / EBSCOhost / INDEX COPERNICUS / DRJI / OAJI / JOURNAL INDEX / IOR / SCIENCE LIBRARY INDEX / Google Scholar / Acadec Keys/ ROAD Ope Access / Acadec Resources / Scetfc Idexg Servces / SCIPIO / JIFACOR I the reader of ths secto, we preset the cessary codto for a crtcal pot x to be a global optal soluto of proble ( P ). heore 3.. Let x Dbe a crtcal pot of proble ( P ). If () f be B vex ad for each I, g be B vex at x. Or () f be ( B, η) vex ad for each I, g be ( B, η) vex at x wth respect to the sae η, ad b( xx, ) = l b( xx,, ) >, b( xx, ) = l b( xx,, ). he x s a global optal soluto of proble ( P ). Proof: () Sce for each x D, we have g = g ( x ), I () Wthout loss of geeralty, we suppose φ s strctly ootoe creasg o R, so g g ( x ), I Usg B vexty of g ( x ) at x yelds g ( x, x) b ( x, x ) g g( x ), He x D, I () ce Σ l[ ] g ( x, x) I () (9) ad Lea. yeld ax ( ξ x) ; ξ f( x ) l ( h, ) g x ϕ = I = f ( x, x) [ ] l[ ] g ( x, x) I hs alog wth () yeld f ( x, x). Usg B vexty of f at x yelds b ( x, x ) f f( x ) f ( x, x) (3) hus, fro (3), t follows that f f( x ), x D whch copletes the () It ca be proved by followg o the les of() heore 3.. Let x Dbe a crtcal pot of proble ( P ). If oe of the followg assupto codtos s satsfed () f s ( ρ, B, η) vex ad for each I, g be ( ρ, B, η) vex at x wth respect to the sae ηθ,, ad b( xx, ) = l bxx (,, ) >, x D ad ( ρ ρ ) Σ ; l I () f s strctly ( B, η) quas-vex ad for each I, g be ( B, η) quas-vex at x wth respect to the sae η, b( xx, ) = l bxx (,, ) >,, ad b( xx, ) = l bxx (,, ) >, x D. he x s a global optal soluto of proble ( P ). Proof: Lea. ad (9) yeld f ( x, η( xx, ))[ ] l[ ] g ( x, η( xx, )), I x D (4) Wthout loss of geeralty, we ow suppose that φ s strctly ootoe creasg o R () Now, by the ( ρ, B, η) vexty assuptos of f ad g, we have b ( x, x ) f f( x ) [ ] [ ] f ( x, η( xx, )) ρ θ( xx, ) b ( x, x ) g g ( x ) g( x, η( xx, )) ρ θ( xx, ), I whch yeld b ( x, x )[ ] ( f [ ] f( x )[ ] l[ ] g ) I f ( x, η( xx, )) l g ( x, η( xx, )) I ρ Σlρ θ( xx, ) ( ) I hece, f Σ l g f( x ) (5) I proof of (). 39 DOI:.79/ X-6-I hs work s lcesed uder the Creatve Coos Attrbuto-Nocoercal-Share Alke 4. Lcese.

6 Mrcea cel Batra Naval Acadey Scetfc Bullet, Volue XIX 6 Issue he joural s dexed : PROQUES / DOAJ / Crossref / EBSCOhost / INDEX COPERNICUS / DRJI / OAJI / JOURNAL INDEX / IOR / SCIENCE LIBRARY INDEX / Google Scholar / Acadec Keys/ ROAD Ope Access / Acadec Resources / Scetfc Idexg Servces / SCIPIO / JIFACOR Sce for each x D, g, hece g l, (5) yelds that I f f( x ), x D (6) whch copletes the proof of (). () Sce for each x D, we have g = g ( x ), I, the ( B, η) quas-vexty of b ( xx, ) g ( x, η( xx, )), x D, I. Hece g yelds that (, η(, )), (7) l g x xx x D I (4) ad (7) yeld f ( x, η( xx, ) (8) If there exsts a x D, such that f f( x ) the the strctly ( B, η) quas-vexty yelds b( xx, ) f ( x, η ( xx, )) < (9) <, whch s cotradcted wth (8), the cotradcto yelds (6) holds ad copletes the proof. heore 3.3. Let x D I, there exst l R such that, I = <, for ( ) I f( x ) Σ l g ( x ) () If f s ( B, η) pseudo-vex at x ad, I l [ ] g respect to the sae η s ( ρ, B, η) vex at x wth b( xx, ) = l bxx (,, ) >, ad b( xx, ) = l bxx (,, ), x D. he x s a global optal soluto of proble ( P ). Proof: Lea. ad () yeld f ( x, η( xx, ))[ ] ( Σ l[ ] g ) ( x, η( xx, )), I x D () ρ η of l g Fro the (, B, ) vexty I we have ( Σ l[ ] g ) ( x, η( xx, ))[ ] ρ[ ] θ( xx, ) I b ( x, x )[ ] ( Σ l[ ] g [ ] l[ ] g( x )), I I x D () hece, ( Σ l[ ] g ) ( x, η( xx, )), x D. I hs wth () yeld f ( x, η( xx, )), x D (3) Followg the B η x, we coplete the proof. (, ) pseudo-vexty of f at BIBLIOGRAPHY [] Avrel, M., Nolear Prograg: Aalyss ad Methods, Pretce Hall, Eglewood Clffs, NJ, 976. [] Bector, C.R., Sgh, C., B-Vex Fuctos, Joural of Optzato heory ad Applcatos, 7():37-53, 99. [3] Bector, C.R., Sueja, S.K., Laltha, C.S., Geeralzed B-vex Fuctos ad Geeralzed B-vex Prograg, Joural of Optzato heory ad Applcatos, 76(3):56-576,, 993. [4] Be-al A., O geeralzed eas ad geeralzed covex fuctos, J.Opt.heory Appl., :-3, 977. [5] Clarke, F.H., Optzato ad Nosooth Aalyss. New York: Wely-Iterscece, 983. [6] Dehu Yua, Altaar Chchuluu,Xaolg Lu,Paos M. Pardalos, Geeralzed covextes ad geeralzed gradets based o algebrac operatos, J. Math. Aal. Appl., 3():675-69,,6 [7] Jeyakuar, V., Equvalece of Saddle-pots ad Opta, ad Dualty for a Class of No-sooth o-covex probles. Joural of Matheatcal Aalyss ad Applcatos, 3(): ,, 988. [8]. Preda, V., O Effcecy ad Dualty for Mult-objectve Progras, Joural of Matheatcal Aalyss ad Applcatos, 66(8), , 99. [9] Preda, V., Soe optalty codtos for ultobjectve prograg probles wth set fuctos, Revue Rouae de Mathéatques Pures et Applquées 39 (3), 33-48, 994. [] Preda, V., Dualty for ultobjectve fractoal prograg probles volvg -set fuctos, Aalyss ad opology, , DOI:.79/ X-6-I hs work s lcesed uder the Creatve Coos Attrbuto-Nocoercal-Share Alke 4. Lcese.

7 Mrcea cel Batra Naval Acadey Scetfc Bullet, Volue XIX 6 Issue he joural s dexed : PROQUES / DOAJ / Crossref / EBSCOhost / INDEX COPERNICUS / DRJI / OAJI / JOURNAL INDEX / IOR / SCIENCE LIBRARY INDEX / Google Scholar / Acadec Keys/ ROAD Ope Access / Acadec Resources / Scetfc Idexg Servces / SCIPIO / JIFACOR [] Preda, V, Stacu-Masa, I.M., Koller, E.: O optalty ad dualty for ultobjectve prograg probles volvg geeralzed d-type-i ad related -set fuctos. J. Math. Aal. Appl. ;83: 4-8, 3 [] Preda, V. Stacu-Masa, I., Belda, M., Stacu, A. M., Optalty ad dualty for ultobjectve prograg wth geeralzed V-type I uvexty ad related -set fuctos, Proceedgs of the Roaa Acadey, Seres A, Volue 6( 3): 83-9, 5. [3] Preda, V. Stacu-Masa, I., Belda, M., Stacu, A. M., Geeralzed V-uvexty type-i for ultobjectve prograg wth -set fuctos, Joural of Global Optzato, 44(): 3 48,, 9. [4] Preda, V., Stacu, D.E., New suffcet codtos for B-prevexty ad soe extesos, Proceedgs of the Roaa Acadey, Seres A, Volue( 3): 97,,. [5] Zhag,C.K., Optalty ad Dualty for A Class of No-sooth Geeralzed B-vex Prograg,OR rasactos, 3():6-8, 999. [6] Zhag Chegke, Optalty codtos for locally Lpschtz geeralzed B-vex se-fte prograg, Proceedgs of 4 Iteratoal Coferece o Maageet Scece ad Egeerg, DOI:.79/ X-6-I hs work s lcesed uder the Creatve Coos Attrbuto-Nocoercal-Share Alke 4. Lcese.