On interval-valued optimization problems with generalized invex functions

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1 Ahmad t al. Jounal of Inqualitis and Alications 203, 203:33 htt:// R E S E A R C H On Accss On intval-valud otimization oblms with gnalizd inv functions Izha Ahmad,2*, Anuag Jayswal 3 and Jonaki Banj 3 * Cosondnc: dizha@kfum.du.sa Datmnt of Mathmatics and Statistics, King Fahd Univsity of Ptolum and Minals, Dhahan, 326, Saudi Aabia 2 Pmannt addss: Datmnt of Mathmatics, Aligah Muslim Univsity, Aligah, , India Full list of autho infomation is availabl at th nd of th aticl Abstact This a is dvotd to study intval-valud otimization oblms. Sufficint otimality conditions a stablishd fo LU otimal solution conct und gnalizd (, ) ρ (η, θ)-invity. Wak, stong and stict convs duality thoms fo Wolf and Mond-Wi ty duals a divd in od to lat th LU otimal solutions of imal and dual oblms. MSC: 90C46; 90C26; 90C30 Kywods: nonlina ogamming; intval-valud functions; (, ) ρ (η, θ)-invity; LU-otimal; sufficincy; duality Intoduction Many al wold dcision-making oblms nd to accomlish many objctivs: minimiz cost, maimiz liability, minimiz dviations fom dsi lvls, minimiz isk, tc. In ths cass otimization oblms hav a lag numb of alications. Th main goal of singl objctiv otimization is to find th bst solution which cosonds to th minimum o maimum valu of a singl objctiv function. In many sach filds and al wold oblms, th mthodology fo solving otimization oblms has bn usd. Th a th kinds of mthodology that a usd fo solving otimization oblms, namly dtministic otimization oblm, stochastic otimization oblm and intval-valud otimization oblm. Th otimization oblms with intval cofficints a tmd an intval-valud otimization oblm. In this oblm, th cofficint is takn as closd intvals. Th solution conct imosd uon th objctiv function is th main diffnc btwn th abov said th kinds of oblms. Intval ogamming mthods hav bn usd to tackl scific issus in multil objctiv lina ogamming: som dal with unctainty in th objctiv functions, oths handl unctainty both in th objctiv functions and in th RHS of th constaints and oths dal with unctainty in all th cofficints of th modl. Chans t al. [] oosd an ida fo solving th lina ogamming oblms in which th constaints w assumd as closd intvals. Lat, Stu [2] dvlod an algoithm to solv th lina ogamming oblm with intval objctiv functions. Uli and Nadau [3] sntd a ocss to solv th multi-objctiv lina ogamming oblms with intval cofficints. Chanas and Kuchta [4]gnalizd th solution concts of th lina ogamming oblm with intval cofficints in th objctiv function basd on fnc lations btwn intvals. By imosing a atial oding 203 Ahmad t al.; licns Sing. This is an On Accss aticl distibutd und th tms of th Cativ Commons Attibution Licns (htt://cativcommons.og/licnss/by/2.0), which mits unstictd us, distibution, and oduction in any mdium, ovidd th oiginal wok is oly citd.

2 Ahmad t al. Jounal of Inqualitis and Alications 203, 203:33 Pag 2 of 4 htt:// on th st of all closd intvals, Wu [5] intoducd a solution conct in otimization oblms with intval-valud objctiv functions. Th a sval aoachs to modl unctainty in otimization oblms such as stochastic otimization and fuzzy otimization. H w consid an otimization oblm with intval-valud objctiv function. Stancu-Minasian and Tigan [6, 7]invstigatd this kind of otimization oblm. Wu [8] fomulatd Kaush-Kuhn-Tuck otimality conditions fo an intval-valud objctiv function. Lat, Wu [9, 0] fomulatd a Wolf-ty dual oblm latd to th intval-valud otimization oblm and stablishd duality thoms by using th conct of nondominatd solution mloyd in vcto otimization oblms. Zhou and Wang [] stablishd a sufficint otimality condition and discussd mid-ty duality fo a class of nonlina intval-valud otimization oblms. Rcntly, Bhuj and Panda [2] dvlod amthodologytostudy thistncofthsolutionsofanintval otimization oblm.vy cntly,zhangt al. [3] discussd th otimality conditions and duality sults fo intval-valud otimization oblms und gnalizd invity. Convity lays an imotant ol in oving th istnc of a solution of a gnal otimization oblm. Hnc th is a nd to study th conv oty of intval otimization oblms. On of th most usful gnalizations of convity was intoducd by Hanson [4]. Fo dtails, ads a advisd to s [5]. Aft th conct of ρ (η, θ)- inv function and (, )-inv function had bn intoducd by Zalmai [6] and Antczak [7], sctivly, Mandal and Nahak[8] oosd a nw conct of (, ) ρ (η, θ)- inv function and stablishd som symmtic duality sults. Rcntly, Jayswal t al. [9] divd sufficint otimality conditions and duality thoms fo intval-valud otimization oblms involving gnalizd conv functions. In this a, w consid an intval-valud otimization oblm in which th objctiv function is an intval-valud function and th constaint functions a alvalud, and div sufficint otimality conditions and duality thoms und gnalizd (, ) ρ (η, θ)-invity. Th oganization of th a is as follows. In Sction 2,w call som dfinitions and som basic otis latd to intval-valud otimization oblms. In Sction3, som sufficint otimality conditions fo a class of intval-valud ogamming oblms a discussd. Wolf and Mond-Wi ty duality thoms a obtaind in Sctions 4 and 5, sctivly. Conclusion and futu wok a oosd in Sction 6. 2 Notation and liminais Lt I b a class of all closd and boundd intvals in R. Thoughout this a, whn w say that A is a closd intval, w man that A is also boundd in R.IfAis a closd intval, w us th notation A =[a L, a U ], wh a L and a U man th low and u bounds of A, sctivly.ifa L = a U = a, thna =[a, a] =a is a al numb. Lt A =[a L, a U ], B =[b L, b U ] I,wdfin (i) A + B = a + b : a A and b B} =[a L + b L, a U + b U ], (ii) A = a : a A} =[ a U, a L ]. Thfo w s that A B = A +( B)=[a L b U, a U b L ]. W also hav (i) k + A = k + a : a A} =[k + a L, k + a U ],

3 Ahmad t al. Jounal of Inqualitis and Alications 203, 203:33 Pag 3 of 4 htt:// [ka L, ka U ] if k 0, (ii) ka = ka : a A} = wh k is a al numb. [ka U, ka L ] if k <0, Lt R n dnot an n-dimnsional Euclidan sac. Th function F : R n I is calld an intval-valud function, i.., F()=F(, 2,..., n ) is a closd intval in R fo ach R n. Th intval-valud function F can also b wittn as F()=[F L (), F U ()], wh F L (), F U () a al-valud functions dfind on R n and satisfy th condition F L () F U ()fo ach R n.wnotthat[f()] L = F L ()and[f()] U = F U (). In intval mathmatics, an od lation is oftn usd to ank intval numbs and it imlis that an intval numb is btt than anoth but not that on is lag than anoth. Fo A =[a L, a U ]andb =[b L, b U ], w wit A LU B if and only if a L b L and a U b U.Itisasytosthat LU is a atial oding on I.Also,wcanwitA < LU B if and only if A LU B and A B. Equivalntly, A < LU B if and only if a L < b L, a U b U o a L b L, a U < b U o a L < b L, a U < b U. Dfinition 2. [8] Ltf : R n R b a diffntiabl function and lt, b abitay al numbs. If th ist η : R n R n R n, θ : R n R n R n and ρ R such that th lations ( (f () f (y)) ) (>) f (y)( η(,y) ) + ρ θ(, y) 2 fo 0, 0, ( (f () f (y)) ) (>) f (y)η(, y)+ρ θ(, y) 2 fo =0, 0, f () f (y)(>) f (y)( η(,y) ) + ρ θ(, y) 2 fo 0, =0, f () f (y)(>) f (y)η(, y)+ρ θ(, y) 2 fo =0, =0 hold, thn f is said to b (stictly) (, ) ρ (η, θ)-inv at th oint y on R n with sct to η, θ. Dfinition 2.2 Lt f : R n R b a diffntiabl function and lt, b abitay al numbs. If th ist η : R n R n R n, θ : R n R n R n and ρ R such that th lations f (y)( η(,y) ) + ρ θ(, y) 2 0 ( (f () f (y)) ) (>) 0 fo 0, 0, f (y)η(, y)+ρ θ(, y) 2 ( 0 (f () f (y)) ) (>) 0 fo =0, 0, f (y)( η(,y) ) + ρ θ(, y) 2 0 f () f(y)(>) 0 fo 0, =0, f (y)η(, y)+ρ θ(, y) 2 0 f () f(y)(>) 0 fo =0, =0 hold, thn f is said to b (stictly) (, ) ρ (η, θ)-sudo-inv at th oint y on R n with sct to η, θ.

4 Ahmad t al. Jounal of Inqualitis and Alications 203, 203:33 Pag 4 of 4 htt:// Dfinition 2.3 Lt f : R n R b a diffntiabl function and lt, b abitay al numbs. If th ist η : R n R n R n, θ : R n R n R n and ρ R such that th lations ( (f () f (y)) ) 0 f (y)( η(,y) ) ρ θ(, y) 2 fo 0, 0, ( (f () f (y)) ) 0 f (y)η(, y) ρ θ(, y) 2 fo =0, 0, f () f (y) 0 f (y)( η(,y) ) ρ θ(, y) 2 fo 0, =0, f () f (y) 0 f (y)η(, y) ρ θ(, y) 2 fo =0, =0 hold, thn f is said to b (, ) ρ (η, θ)-quasi-inv at th oint y on R n with sct to η, θ. Rmak 2. It should b notd that th onntials aaing on th ight-hand sids of inqualitis abov a undstood to b takn comonntwis and =(,,...,) R n. Rmak 2.2 All thoms in this a will b ovd only in th cas whn 0, 0 (oth cass can b dalt with likwis sinc th only changs ais in a fom of inquality). Moov, without loss of gnality, w shall assum that >0, >0(inthcaswhn <0, < 0, th diction of som of th inqualitis in th oof of th thoms should b changd to th oosit on). In this a, w consid th following imal otimization oblm with intvalvalud objctiv function: (IVP) min F()= [ F L (), F U () ] subjct to g j () 0, j =,2,...,m, wh F : R n I is an intval-valud function and g j : R n R is a al-valud function. Lt = R n : g j () 0, j =,2,...,m} b th st of all fasibl solutions of (IVP). W also dnot by Obj(F, )=F(): } th st of all objctiv valus of imal oblm (IVP). Dfinition 2.4 [20] Lt * b a fasibl solution of th imal oblm (IVP). W say that * is a LU otimal solution of oblm (IVP) if th ists no 0 such that F( 0 )< LU F( * ). Thom 2. (Kaush-Kuhn-Tuck ty conditions [9]) Assum that * is a LU otimal solution of imal oblm (IVP) and F, g j, j =,2,...,m, a diffntiabl at *. Suos that th constaint function g j, j =,2,...,m, satisfis th suitabl Kuhn-Tuck constaint qualification [9] at *. Thn th ist multilis 0<ξ L, ξ U Rand0 μ j R, j =,2,...,m, such that ξ L F L( *) + ξ U F U( *) + ( μ j g j * ) =0, () μ j g j ( * ) =0, j =,2,...,m. (2)

5 Ahmad t al. Jounal of Inqualitis and Alications 203, 203:33 Pag 5 of 4 htt:// 3 Sufficint otimality conditions In this sction, w shall stablish th following sufficint otimality conditions fo (IVP). Thom 3. (Sufficincy) Lt * b a fasibl solution of (IVP). Suos that th objctiv function F and th constaint function g j, j =,2,...,m, a diffntiabl at *. Assum that F L and F U a (, ) ρ (η, θ)-inv and (, ) ρ 2 (η, θ)-inv, sctivly, with sct to η, θ and m μ jg j is (, ) ρ 3 (η, θ)-inv with sct to η, θ at * with (ξ L ρ + ξ U ρ 2 + ρ 3 ) 0. If th ist (Lagang) multilis 0<ξ L, ξ U Rand μ =(μ, μ 2,...,μ m ), 0 μ j R, j =,2,...,m, such that ( *, ξ L, ξ U, μ) satisfis () and (2), thn * is a LU otimal solution to oblm (IVP). Poof Lt * b not a LU otimal solution of (IVP). Thn th ists a fasibl solution 0 such that F( 0 )< LU F ( *). That is, F L ( 0 )<F L ( * ), F U ( 0 )<F U ( * ), o F L ( 0 ) F L ( * ), F U ( 0 )<F U ( * ), o F L ( 0 )<F L ( * ), F U ( 0 ) F U ( * ). Sinc > 0, using th oty of an onntial function, w obtain [FL ( 0 ) F L ( * )} ]<0, [FU ( 0 ) F U ( *)} ]<0, [FL ( 0 ) F L ( *)} ]<0, o [FL ( 0 ) F L ( *)} ] 0, [FU ( 0 ) F U ( *)} ]<0, o [FU ( 0 ) F U ( * )} ] 0. Using th abov inqualitis and th (, ) ρ (η, θ)-invity of F L and th (, ) ρ 2 (η, θ)-invity of F U at *,wgt FL ( * )}( η( 0, *) )+ρ θ( 0, * ) 2 <0, FU ( * )}( η( 0, *) )+ρ 2 θ( 0, * ) 2 <0, FL ( * )}( η( 0, *) )+ρ θ( 0, * ) 2 0, FU ( * )}( η( 0, *) )+ρ 2 θ( 0, * ) 2 <0, o o FL ( * )}( η( 0, *) )+ρ θ( 0, * ) 2 <0, FU ( * )}( η( 0, *) )+ρ 2 θ( 0, * ) 2 0. Sinc ξ L >0andξ U > 0, fom th abov inqualitis, w gt ξ L F L( *) + ξ U F U( *)}( η( 0, *) ) + ξ L ρ θ ( 0, *) 2 + ξ U ρ 2 θ ( 0, *) 2 <0. (3)

6 Ahmad t al. Jounal of Inqualitis and Alications 203, 203:33 Pag 6 of 4 htt:// On th oth hand, fom th fasibility of 0 to (IVP), w hav g j ( 0 ) 0, j =,2,...,m. Sinc μ j 0, j =,2,...,m, th abov inquality togth with (2)yilds μ j g j ( 0 ) ( μ j g j * ). As > 0, using th oty of an onntial function, w gt [ m μ j g j ( 0 ) m μ j g j ( *)} ] 0, which togth with th assumtion that m μ jg j is (, ) ρ 3 (η, θ)-inv at * givs ( μ j g j * )( η( 0, *) ) + ρ 3 θ ( 0, *) 2 0. (4) On adding (3)and(4), w gt ( η( 0, *) )[ ξ L F L( *) + ξ U F U( *) + + ( ξ L ρ + ξ U ρ 2 + ρ 3 ) θ ( 0, *) 2 <0. ( μ j g j * )] Thfo, fom th hyothsis that (ξ L ρ + ξ U ρ 2 + ρ 3 ) 0 and th abov inquality, w gt ( η( 0, *) )[ ξ L F L( *) + ξ U F U( *) + ( μ j g j * )] <0, which contadicts (). Thfo * is a LU otimal solution of (IVP). This comlts th oof. Thom 3.2 (Sufficincy) Lt * b a fasibl solution of (IVP). Suos that th objctiv function F and th constaint function g j, j =,2,...,m, a diffntiabl at *. Assum that (ξ L F L + ξ U F U ) is (, ) ρ (η, θ)-sudo-inv with sct to η, θ and m μ jg j is (, ) ρ 2 (η, θ)-quasi-inv with sct to η, θ at * with (ρ + ρ 2 ) 0. If th ist (Lagang) multilis 0<ξ L, ξ U Randμ =(μ, μ 2,...,μ m ), 0 μ j R, j =,2,...,m, such that ( *, ξ L, ξ U, μ) satisfis () and (2), thn * is a LU otimal solution to oblm (IVP). Poof Lt * b not a LU otimal solution of (IVP). Thn th ists a fasibl solution 0 such that F( 0 )< LU F ( *).

7 Ahmad t al. Jounal of Inqualitis and Alications 203, 203:33 Pag 7 of 4 htt:// That is, F L ( 0 )<F L ( * ), F U ( 0 )<F U ( * ), o F L ( 0 ) F L ( * ), F U ( 0 )<F U ( * ), o F L ( 0 )<F L ( * ), F U ( 0 ) F U ( * ). Sinc ξ L >0andξ U > 0, fom th abov inqualitis, w hav ξ L F L ( 0 )+ξ U F U ( 0 )<ξ L F L( *) + ξ U F U( *). As > 0, using th oty of an onntial function, w gt [ (ξ L F L ( 0 )+ξ U F U ( 0 )) (ξ L F L ( * )+ξ U F U ( *))} ] <0, which togth with th assumtion that ξ L F L + ξ U F U is (, ) ρ (η, θ)-sudo-inv at * givs ξ L F L( *) + ξ U F U( *)}( η( 0, *) ) + ρ θ ( 0, *) 2 <0. (5) On th oth hand, fom th fasibility of 0 to (IVP), w hav g j ( 0 ) 0, j =,2,...,m. Sinc μ j 0, j =,2,...,m, th abov inquality togth with (2)yilds μ j g j ( 0 ) ( μ j g j * ). As > 0, using th oty of an onntial function, w gt [ m μ j g j ( 0 ) m μ j g j ( *)} ] 0, which togth with th assumtion that m μ jg j is (, ) ρ 2 (η, θ)-quasi-inv at * givs ( μ j g j * )( η( 0, *) ) ( + ρ 2 θ 0, *) 2 0. (6) On adding (5)and(6), w gt ( η( 0, *) )[ ξ L F L( *) + ξ U F U( *) + +(ρ + ρ 2 ) θ ( 0, *) 2 <0. Sinc (ρ + ρ 2 ) 0, w hav ( η( 0, *) )[ ξ L F L( *) + ξ U F U( *) + ( μ j g j * )] ( μ j g j * )] <0,

8 Ahmad t al. Jounal of Inqualitis and Alications 203, 203:33 Pag 8 of 4 htt:// which contadicts (). Thfo * is a LU otimal solution of (IVP). This comlts th oof. 4 Wolf-tyduality In this sction, w consid th following Wolf-ty dual oblm: (WD) ma F(y)+ subjct to μ j g j (y) ξ L F L (y)+ξ U F U (y)+ μ j g j (y)=0, (7) ξ L >0,ξ U >0,μ j 0, j =,2,...,m, (8) wh F(y)+ m μ jg j (y)=[f L (y)+ m μ jg j (y), F U (y)+ m μ jg j (y)] is an intval-valud function. Dfinition 4. Lt (y *, ξ *L, ξ *U, μ * ) b a fasibl solution of dual oblm (WD). W say that (y *, ξ *L, ξ *U, μ * ) is a LU otimal solution of dual oblm (WD) if th ists no (y, ξ *L, ξ *U, μ * )suchthatf(y * )+ m μ* j g j(y * )< LU F(y)+ m μ* j g j(y). Thom 4. (Wak duality) Lt X 0 b an on subst of R n. Lt F and g j, j =,2,...,m, b diffntiabl on X 0. Suos that and (y, ξ L, ξ U, μ) a th fasibl solutions to (IVP) and (WD), sctivly. Futh assum that ξ L >0,ξ U >0and μ j 0, j =,2,...,m, such that ξ L F L + ξ U F U + m μ jg j is (, ) ρ (η, θ)-inv with sct to η, θ at y with ξ L + ξ U = and ρ 0. Thn F() LU F(y)+ μ j g j (y). Poof Suos contay to th sult that F()< LU F(y)+ μ j g j (y). That is, F L ()<F L (y)+ m μ jg j (y), F U ()<F U (y)+ m μ jg j (y), F L ()<F L (y)+ m μ jg j (y), F U () F U (y)+ m μ jg j (y). o F L () F L (y)+ m μ jg j (y), F U ()<F U (y)+ m μ jg j (y), o Sinc ξ L >0,ξ U >0andξ L + ξ U =, th abov inqualitis togth with th fasibility of to (IVP) bcom ξ L F L ()+ξ U F U ()+ μ j g j ()<ξ L F L (y)+ξ U F U (y)+ μ j g j (y). (9)

9 Ahmad t al. Jounal of Inqualitis and Alications 203, 203:33 Pag 9 of 4 htt:// Fom th assumtion that ξ L F L + ξ U F U + m μ jg j is (, ) ρ (η, θ)-inv at y,whav [ (ξ L F L ()+ξ U F U ()+ m μ j g j ()) (ξ L F L (y)+ξ U F U (y)+ m μ j g j (y))} ] [ ( ξ L F L (y)+ξ U F U (y)+ μ j g j (y)] η(,y) ) + ρ θ(, y) 2. Th abov inquality togth with (7)andρ 0yilds [ (ξ L F L ()+ξ U F U ()+ m μ j g j ()) (ξ L F L (y)+ξ U F U (y)+ m μ j g j (y))} ] 0. Sinc > 0, using th oty of an onntial function, w gt ξ L F L ()+ξ U F U ()+ μ j g j () ξ L F L (y)+ξ U F U (y)+ μ j g j (y), which contadicts th inquality (9). This comlts th oof. Thom 4.2 (Stong duality) Lt * b a LU otimal solution to (IVP) at which Kuhn- Tuck constaints qualification a satisfid. Thn th ist ξ *L >0,ξ *U >0and μ * 0 such that ( *, ξ *L, ξ *U, μ * ) is fasibl fo (WD) and th two objctivs hav th sam valu. Futh, if th hyothsis of wak duality Thom 4. holds fo all fasibl solutions (y *, ξ *L, ξ *U, μ * ), thn ( *, ξ *L, ξ *U, μ * ) is a LU otimal solution to (WD). Poof Sinc * is a LU otimal solution to (IVP) and th Kuhn-Tuck constaints qualification a satisfid at *, thn by Thom 2.,thistmultilisξ *L >0,ξ *U >0and μ * j 0, j =,2,...,m,suchthat ξ *L F L( *) + ξ *U F U( *) + μ * j g j( * ) =0, * ) =0, which yilds that ( *, ξ *L, ξ *U, μ * ) is a fasibl solution fo (WD) and cosonding objctiv valus a th sam. Futh, if ( *, ξ *L, ξ *U, μ * ) is not a LU otimal solution of (WD), thn th ists a fasibl solution (y *, ξ *L, ξ *U, μ * )fo(wd)suchthat F ( *) < LU F ( y *) + y * ), which contadicts wak duality Thom 4..Hnc( *, ξ *L, ξ *U, μ * ) is a LU otimal solution to (WD). Thom 4.3 (Stict convs duality) Lt X 0 b an on subst of R n. Lt F and g j, j =,2,...,m, b diffntiabl on X 0. Suos that * and (y *, ξ *L, ξ *U, μ * ) a th fasibl solutions to (IVP) and (WD), sctivly. Assum that ξ *L F L + ξ *U F U + m μ* j g j is

10 Ahmad t al. Jounal of Inqualitis and Alications 203, 203:33 Pag 0 of 4 htt:// stictly (, ) ρ (η, θ)-inv with sct to η, θ at y * with ρ 0 and ξ *L F L( *) + ξ *U F U( *) + * ) ξ *L F L( y *) + ξ *U F U( y *) + y * ). (0) Thn * = y *. Poof Now w assum that * y * and hibit a contadiction. Fom th assumtion that ξ *L F L + ξ *U F U + m μ* j g j is stictly (, ) ρ (η, θ)-inv at y *,whav [ (ξ *L F L ( * )+ξ *U F U ( * )+ m * )) (ξ *L F L (y * )+ξ *U F U (y * )+ m y *))} ] [ > ξ *L F L( y *) + ξ *U F U( y *) + μ * j g j( y * )] ( η( *,y *) ) + ρ ( θ *, y *) 2. Fom th fasibility of (y *, ξ *L, ξ *U, μ * ) to (WD), th abov inquality togth with th hyothsis ρ 0yilds [ (ξ *L F L ( * )+ξ *U F U ( * )+ m * )) (ξ *L F L (y * )+ξ *U F U (y * )+ m y *))} ] >0. Sinc > 0, using th oty of an onntial function, w gt ξ *L F L( *) + ξ *U F U( *) + * ) > ξ *L F L( y *) + ξ *U F U( y *) + y * ), which contadicts (0). This comltsth oof. 5 Mond-Wi ty duality In this sction, w consid th following Mond-Wi ty dual oblm fo (IVP): (MWD) ma F(y)= [ F L (y), F U (y) ] subjct to ξ L F L (y)+ξ U F U (y)+ μ j g j (y)=0, () μ j g j (y) 0, j =,2,...,m, (2) ξ L >0,ξ U >0,μ j 0, j =,2,...,m. (3) Dfinition 5. Lt (y *, ξ *L, ξ *U, μ * ) b a fasibl solution of dual oblm (MWD). W say that (y *, ξ *L, ξ *U, μ * ) is a LU otimal solution of dual oblm (MWD) if th ists no (y, ξ *L, ξ *U, μ * )suchthatf(y * )< LU F(y). Thom 5. (Wak duality) Lt X 0 b an on subst of R n. Lt F and g j, j =,2,...,m, b diffntiabl on X 0. Suos that and (y, ξ L, ξ U, μ) a th fasibl solutions to (IVP) and (MWD), sctivly. Futh assum that th ist ξ L >0and ξ U >0and μ j 0, j =

11 Ahmad t al. Jounal of Inqualitis and Alications 203, 203:33 Pag of 4 htt:// such that (ξ L F L +ξ U F U ) is (, ) ρ (η, θ)-sudo-inv with sct to η, θ and m μ jg j is (, ) ρ 2 (η, θ)-quasi-inv with sct to η, θ at y with (ρ + ρ 2 ) 0. Thn F() LU F(y). Poof Suos contay to th sult that F()< LU F(y). That is, F L ()<F L (y), F U ()<F U (y), o F L () F L (y), F U ()<F U (y), o F L ()<F L (y), F U () F U (y). Sinc ξ L >0andξ U > 0, fom th abov inqualitis, w hav ξ L F L ()+ξ U F U ()<ξ L F L (y)+ξ U F U (y). (4) On th oth hand, sinc μ j 0, j =,2,...,m, fom th fasibility of and (y, ξ L, ξ U, μ)to (IVP) and (MWD), sctivly, w obtain μ j g j () μ j g j (y). Sinc > 0, using th oty of an onntial function, w gt [ m μ j g j () m μ j g j (y)} ] 0, which togth with th assumtion that m μ jg j is (, ) ρ 2 (η, θ)-quasi-inv at y, givs μ j g j (y) ( η(,y) ) + ρ 2 θ(, y) 2 0. Thfo, fom () and th hyothsis that (ρ + ρ 2 ) 0, th abov inquality yilds ( η(,y) )[ ξ L F L (y)+ξ U F U (y) ] + ρ θ(, y) 2 0, which togth with th assumtion that ξ L F L + ξ U F U is (, ) ρ (η, θ)-sudo-inv at y givs [ (ξ L F L ()+ξ U F U ()) (ξ L F L (y)+ξ U F U (y))} ] 0. Sinc > 0, using th oty of an onntial function, w gt ξ L F L ()+ξ U F U () ξ L F L (y)+ξ U F U (y), which contadicts (4). This comlts th oof.

12 Ahmad t al. Jounal of Inqualitis and Alications 203, 203:33 Pag 2 of 4 htt:// Thom 5.2 (Stong duality) Lt * b a LU otimal solution to (IVP) at which Kuhn- Tuck constaints qualification a satisfid. Thn th ist ξ *L >0,ξ *U >0and μ * 0 such that ( *, ξ *L, ξ *U, μ * ) is fasibl fo (MWD) and th two objctivs hav th sam valu. Futh, if th hyothsis of wak duality Thom 5. holds fo all fasibl solutions (y *, ξ *L, ξ *U, μ * ), thn ( *, ξ *L, ξ *U, μ * ) is a LU otimal solution to (MWD). Poof Sinc * is a LU otimal solution to (IVP) and th Kuhn-Tuck constaints qualification a satisfid at *, thn by Thom 2., thistmultilisξ *L >0,ξ *U >0, μ * j 0, j =,2,...,m,suchthat ξ *L F L( *) + ξ *U F U( *) + μ * j g j( * ) =0, * ) =0, which yilds that ( *, ξ *L, ξ *U, μ * ) is a fasibl solution fo (MWD) and cosonding objctiv valus a sam. Futh, if ( *, ξ *L, ξ *U, μ * ) is not a LU otimal solution of (MWD), thn th ists a fasibl solution (y *, ξ *L, ξ *U, μ * )fo(mwd)suchthat F ( *) < LU F ( y *), which contadicts wak duality Thom 5..Hnc( *, ξ *L, ξ *U, μ * ) is a LU otimal solution to (MWD). Thom 5.3 (Stict convs duality) Lt X 0 b an on subst of R n. Lt F and g j, j =,2,...,m, b diffntiabl on X 0. Suos that * and (y *, ξ *L, ξ *U, μ * ) a th fasibl solutions to (IVP) and (MWD), sctivly. Assum that ξ *L F L + ξ *U F U is stictly (, ) ρ (η, θ)-sudo-inv and m μ* j g j is (, ) ρ 2 (η, θ)-quasi-inv with sct to η, θ at y * with (ρ + ρ 2 ) 0 and ξ *L F L( *) + ξ *U F U( *) ξ *L F L( y *) + ξ *U F U( y *). (5) Thn * = y *. Poof Now w assum that * y * and hibit a contadiction. Fom th assumtion that (y *, ξ *L, ξ *U, μ * ) is a fasibl solution fo (MWD), w gt ξ *L F L( y *) + ξ *U F U( y *) + μ * j g j( y * ) =0. (6) Sinc μ * j 0, j =,2,...,m, fom th fasibility of * and (y *, ξ *L, ξ *U, μ * )to(ivp)and (MWD), sctivly, w obtain * ) y * ).

13 Ahmad t al. Jounal of Inqualitis and Alications 203, 203:33 Pag 3 of 4 htt:// As > 0, using th oty of an onntial function, w gt [ m * ) m y *)} ] 0, which togth with th assumtion that m μ* j g j is (, ) ρ 2 (η, θ)-quasi-inv at y * givs μ * j g j( y * )( η(*,y *) ) ( + ρ 2 θ *, y *) 2 0. Thfo, fom (6) and th hyothsis that (ρ + ρ 2 ) 0, th abov inquality yilds ( η( *,y *) )[ ξ *L F L( y *) + ξ *U F U( y *)] ( + ρ θ *, y *) 2 0, which togth with th assumtion that ξ *L F L +ξ *U F U is stictly (, ) ρ (η, θ)-sudoinv at y * givs [ (ξ *L F L ( * )+ξ *U F U ( * )) (ξ *L F L (y * )+ξ *U F U (y *))} ] >0. Sinc > 0, using th oty of an onntial function, w gt ξ *L F L( *) + ξ *U F U( *) > ξ *L F L( y *) + ξ *U F U( y *), which contadicts (5). This comltsth oof. 6 Conclusion In this a, w hav divd sufficint otimality conditions fo a class of intval-valud otimization oblms und gnalizd inv functions. Wak, stong and stict convs duality thoms a discussd fo two tys of th dual modls. It will b intsting to obtain th otimality and duality thom fo a class of intval-valud ogamming und gnalizd invity assumtions in which th involvd functions a non-smooth. Moov, it will also b intsting to s whth th scond-od duality sults fo a class of intval-valud ogamming oblm hold o not. This will oint th futu sach of th authos. Comting intsts Th authos dcla that thy hav no comting intsts. Authos contibutions All authos caid out th oof. All authos concivd of th study, and aticiatd in its dsign and coodination. All authos ad and aovd th final manuscit. Autho dtails Datmnt of Mathmatics and Statistics, King Fahd Univsity of Ptolum and Minals, Dhahan, 326, Saudi Aabia. 2 Pmannt addss: Datmnt of Mathmatics, Aligah Muslim Univsity, Aligah, , India. 3 Datmnt of Alid Mathmatics, Indian School of Mins, Jhakhand, Dhanbad , India. Acknowldgmnts Izha Ahmad thanks th King Fahd Univsity of Ptolum and Minals, Dhahan-326, Saudi Aabia fo th suot und th Intnal Pojct No. IN037. Th authos wish to thank th fs fo thi sval valuabl suggstions which hav considably imovd th sntation of this aticl. Rcivd: 27 Mach 203 Acctd: 3 Jun 203 Publishd: 3 July 203

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