Generalized Duality for a Nondifferentiable Control Problem

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1 Aeca Joal of Appled Matheatcs ad Statstcs, 4, Vol., No. 4, 93- Avalable ole at Scece ad Edcato Pblsh DO:.69/aas--4-3 Geealzed Dalty fo a Nodffeetable Cotol Poble. Hsa,*, Vkas K. Ja, Abdl Raoof Shah Depatet of Matheatcs, Jaypee Uvesty of Eee ad echoloy, Ga, da Depatet of Statstcs, Uvesty of Kash, Saa, da *Coespod atho: hsa@yahoo.co, avkas3@yahoo.co Receved Mach 3, 4; Revsed May, 4; Accepted Je 3, 4 Abstact A eealzed dal to a cotol poble cota sppot ctos s folated ad vaos dalty theoes ae establshed de eealzed covexty hypotheses. hs dal odel epesets the cobato of Wolfe ad Mod-We type dal odels to the cotol poble ad hece t s descbed as a eealzed dal. Soe specal cases ae obtaed. A close elatoshp of dalty eslts wth those of the olea poa pobles volv sppot ctos s dcated. Keywods: cotol poble, sppot cto, optalty coos, eealzed dalty, covese dalty, olea poa Cte hs Atcle:. Hsa, Vkas K. Ja, ad Abdl Raoof Shah, Geealzed Dalty fo a Nodffeetable Cotol Poble. Aeca Joal of Appled Matheatcs ad Statstcs, vol., o. 4 (4: 93-. do:.69/aas todcto [3], Hsa et al. cosdeed the follow cotol poble cota sppot ctos: ( ( (CP: Mze f t, x, + S t K ( x, sbect to xb xa =, =, ( x t = h tx,,, t ( tx,, + S x t C, =,,..., t (3 whee ( x: R s a dffeetable state vecto cto wth ts devatve x ad : R s a sooth cotol vecto cto, ( R deotes a -desoal Ecldea space ad = ab, s a eal teval. [ ] ( f : R R R, : R R R, =,,... ad h: R R R ae cotosly dffeetable. K S xt C, =,,...,. ae the (v S( xt ad sppot ctos of the copact sets K ad C ( =,,..., espectvely. Deote the patal devatves of f whee by ft, f x ad f y, f f f f ft =, =,,...,, t x x x f f f =,,...,, whee spescpts deote the vecto copoets. Slaly we have ht,h x, h ad t, x,. Desate by X. he space of cotosly dffeetable state ctos x: R sch that x( a = ad xb = ad s eqpped wth the o x = x D x, + ad U, the space of pecewse cotos cotol vecto ctos : R hav the fo o.. he dffeetal eqato ( wth tal coos expessed b as x( t = x( a + h( s, x( s, ( s ds, t ay be wtte a as H= H( x,, whee : (, H X U C R, C(, R be the space of cotos cto fo to R defed as H( x, ( t = h( t,x ( t, ( t. the devato of the optalty coos, soe costat qalfcato to ake the eqalty costats locally solvable s eeded. Fo ths the Fechet devatve of Dx H( x, = Q( x,, (say wth espect to x aely Q = Q ( x, = [ D H ( x,, H ( x, ] ae (,, eqed to be sectve. Hsa et al. [3] establshed the follow Ftz type ecessay coos fo the cotol poble (CP: x

2 94 Aeca Joal of Appled Matheatcs ad Statstcs Poposto. (Ftz Joh Necessay Coos: f x, s a optal solto of (CP ad the Fechet devatve Q s sectve, the thee exsts Laae ltples α R ad pecewse sooth : R, : R, z: R ad ω : R sch that fo all t, tx t tx t = (,, + x (,, + α ω + µ ( t hx ( tx,, = µ ( t, t = α (,, + (,, tx t tx + µ ( t h ( tx,, =, t ( x ω t tx,, + x t t =, t ( µ tzt = S xt K xtω t = S xt C, =,,..., ω z t K, t C, =,,..., ( α, t, t ( α, t, t, t As [5], Hsa et al. [3] poted ot f the optal solto fo (CP s oal, the the Ftz Joh type optal coos edce to the follow Kash-Kh- cke optal coos: x, s a optal solto ad s Poposto. f oal ad Q s sectve, thee exst pecewse sooth : R wth = (,,...,, : R, z: R ad ω : R, =,,..., sch that tx t tx t = (,, + x (,, + ω + x = ( t h ( tx,, ( t µ µ (,, + (,, tx t tx + µ ( t h ( tx,, =, t (4 (5 ( t x ( tx,, + x ( t ω ( t =, t (6 = ( t zt = S xt K (7 xt ω t = S xt C, =,,..., (8 ( t, t,,,...,. = (9 ω z t K, t C, =,,...,. ( Us the Kash-Kh-cke type ecessay optalty coos, Hsa et al. [3] costcted the follow Wolf type dal cotol poble to (CP ad poved vaos dalty eslts: f( tx,, + µ ( t zt ( (,, (WCD: Maxze tx ( t + = xt ω ( t + + µ ( t ( h( tx,, x( t sbect to α ( b x a =, = tx t tx t = + µ ( t h x µ ( t =, t (,, + (,, + ω tx,, + t tx,, + µ ( t h =, t,,,,..., t t = ω z t K, t C, =,,..., he poble (WCD s a dal to (CP ass f + t (., µ ( t ( h x ( t = x fo all z( t R ad that s psedo covex (, ( t R,,,...,. ω = Hsa et al. [4] the weakeed the eealzed covexty fo dalty by costct a Mod-We type dal to (CP ve below. (M-WCD: Maxze { f ( t, x, + ( t z ( t } sbect to = = xb xa tx t tx x t t = + µ ( t hx ( tx,, = µ ( t, t (,, + x (,, + ω t ( tx,, + z+ ( tx,, + µ ( t h ( tx,, =, t { ( t, x, + x( t ω } = µ ( t ( h( t, x, x ( t, t z( t K,,,..., ω t C = whee ( K ad C, =,,..., ae copact sets R, ad ( : R, ω: R ad µ : R ae pecewse sooth ctos ad

3 Aeca Joal of Appled Matheatcs ad Statstcs 95 Hsa et al. [4] poved dalty theoes fo the assptos of poble (CP ad (M-WCD de the f t,, + z t z t K psedocovexty of { } fo ad qas covexty of (,, + ( ω = ad µ ( (,, ( t t t t h t x fo all ω t C =,,,...,. We evew soe well kow facts abot a sppot cto fo easy efeece. Let Γ be a copact covex set R. he the sppot cto of Γ deoted by S xt Γ s defed as ( { } ( υ υ S xt Γ = ax xt t: t Γ, t A sppot cto, be covex ad eveywhee fte, has a sbdffeetal the sese of covex aalyss, that s, thee exsts z sch that S yt Γ S xt Γ + zt yt xt fo all ( ( ( x. he sbdffeetal of S( xt Γ s ve by S( xt Γ = { zt Γ : zt xt = S( xt Γ }. Let NΓ ( xt be oal coe at a pot xt Γ. he y( t NΓ ( x f ad oly f S( y( t Γ = x( t y( t o eqvaletly, xt s the sbdffeetal of s at y( t. ths pape, we popose a eealzed dal to (CP ad pove vaos dalty theoes de appopate eealzed covexty asspto. Fo o dalty eslts, specal cases ae dedced ad t s show that o eslts deved ths eseach ca be cosdeed as dyac eealzato of those of olea poa pobles hav sppot ctos.. Geealzed Dalty Let { } { } M =,,...,, N =,,...,, α M, α =,,... wth α β = φ, α β, ad α = M, ad Jα N, α = α =,,... wth Jα Jβ = φ, α β, ad Jα = N. α = We popose the follow eealzed dal to the poble (CP ad pove vaos dalty theoe de appopate eealzed covexty coo: f( tx,, ( t z( t + ( tx,, (GCD: Maxze ( t + xt ω ( t + + µ ( t ( h ( tx,, x sbect to xb xa =, = ( tx t tx t = + µ ( t hx ( tx,, + µ ( t =, t (,, + x (,, + ω (,, + + (,, ( t h ( tx,,, t tx z t t tx + µ = ( (3 t t, x, + t ω t, α =,,..., (4 α µ t h t, x, x t, α =,,..., (5 Jα,,,..., t = (6 ω z t K, t C, =,,..., (7 heoe (Weak dalty: let ( x, be feasble fo (CP ad ( x,,, z, µω,,..., ω wth ( =,..., ad (,..., µ µ µ feasble ( x,, µ, x,,, z, ω,..., ω, = feasble fo (GCD. f fo all f ( t,.,. + (. z( t + ( t ( ( t,.,. + (. ω ( t + µ ( t ( h ( t,.,. x (. J ( t,.,. s psedocovex, ad ( t, ad α + (. ω ( t ( µ ( t ( h ( t,.,. x (., α =,,..., ae Jα qascovex, the ( CP Sp ( GCD f. Poof: Sce ( x, s feasble fo (CP ad (,,,,,,..., x z ( (,, + ω α ( µω ω s feasble fo (GCD, we have t t x x t t t x, + x t ω t, α =,,..., α ad (,, µ t h t x x t Jα µ ( t ( h ( t, x, x ( t, α =,,...,. Jα

4 96 Aeca Joal of Appled Matheatcs ad Statstcs ( tx,, By the qascovexty of ( t α + ( t ω ( t ad µ ( t h ( t, x, x ( t,,,...,, Jα above eqalty espectvely yelds, ad ( x x x ( tx,, + ω ( t t + ( ( tx,, α ( x J α + ( µ ( t h ( tx,, Hece ( x x µ ( t h ( tx,, µ ( t ( x x x ( tx,, + ω ( t t + ( ( tx,, M ad ( x N J + ( µ ( t h ( tx,, ( x x µ ( t h ( tx,, + µ ( t α = the. Cob the above eqaltes ad the s eqalty costats ( ad (3, we have x ( tx,, ( x x ( tx,, + ( t + ω ( t ( µ ( t hx ( t, x, µ ( t + +. J ( tx,, ( t ( tx,, + ( + + µ ( t h ( tx,, J hs, becase of psedocovexty of f t z t t + µ J (,.,. + (. + at ( x,,we have ( t h ( t,.,. x (. f tx t zt t + µ ( t ( h ( tx,, x ( t J + (,, + + ( t,.,. (. ω ( t ( tx,, + x t ω ( t f tx t z t t + ( J (,, + + µ ( t h ( tx,, x ( t ( tx,, ω ( t + x t Us ( t z( t S( ( t K,. ad x t ω t S x t C =,,...,, toethe wth feasblty of ( x, fo (CP the above eqalty, we have { (,, + ( } f t x S t K f tx t z t t + ( J yeld (,, + + µ ( t h ( tx,, x ( t ( CP Sp ( GCD f. ( tx,, ω ( t + x t heoe (Sto Dalty: f ( x, s a optal solto of (CP ad s oal, the thee exst pecewse sooth z: R, µ : R, : R, =,,..., ad ω : R, =,,..., sch that (,,,,...,,,...,, xz ω ω µ s feasble fo (GCD, ad the coespod vales of (CP ad (GCD ae eqal. f the hypotheses of heoe hold, the,,,,...,,,..., xz ω ω, µ s a optal solto of (GCD. Poof: Sce (, x s a optal solto of (CP ad s oal, the fo Poposto, thee exst pecewse sooth z: R, : R, =,,...,, ω : R, =,,..., ad µ : R sch that coos (4-( hold. So,,,,...,,,..., xz ω ω, µ s feasble fo (GCD ad vew of coos (4, (5, (6, (9 ad (, the eqalty of the obectve ctoals follows. f f ( t,.,. + (. z+ ( t ( ( t,.,. + (. ω ( t s + µ ( t ( h ( t,.,. + x (. psedocovex, ad (,.,. + (. ω t t t ad µ ( t h ( t,.. x ( t ae qascovex fo all z( t K ad ω ( t C, =,,..,, the fo heoe (,,,,...,,,...,, optal solto of (GCD. xz ω ω µ st be a

5 Aeca Joal of Appled Matheatcs ad Statstcs 97 heoe 3 (Stct covese dalty: Let the poble x, that satsfes the (CP have a optal solto oalty coo ad ˆ ˆ ( ˆ, ˆ,,...,, ˆ, ˆ, ˆ,..., ˆ x zµω ω be optal solto of (GCD f f t z t t + µ ( t ( h ( t,.,. + x (. (,.,. + (. ˆ + + ( t,.,. stctly psedocovex, ad ( t ad µ (,.,. + (. (. ω ( t + ( t,.,. (. ω ( t s t ( h t x ae qas covex fo all z K, ω C,,,..., x, = x ˆ, ˆ, =, the.e., ( x ˆ, ˆ s a optal solto of (CP. Poof: We shall asse that ( x, ( x ˆ, ˆ ad exhbt a cotadcto, sce ( x, s a optal solto of (CP, t follows fo heoe thee exst : R, =,,...,, : R, =,,..., ad R ( ω µ :, =,,...,, sch that (,,,,...,,,...,,,..., xz µ µ ω ω s a optal solto of (GCD. Hece (,, + + (,, + ω f tx z tx t + µ ( t ( h ( tx,, x J ˆ, ˆ, ˆ ˆ ˆ (, ˆ, ˆ ˆ ω f tx z tx t =. + ˆ µ ( t ( h ( tx, ˆ, ˆ xˆ J toethe wth the feasblty of ( x, ( ˆ ˆ ˆ ˆ, ˆ,,...,, ˆ, ˆ,,..., ˆ x z x( t ( t S( x( t C, we have Also fo (CP ad µω ω fo (GCD. ω = fo ω C, =,,...,, ˆ (( ( t, x, + x ( t ω ( t α ˆ (( (, ˆ, ˆ ˆ ˆ + ω Jα Jα t x x t t ˆ µ t h t, x, x t ( ˆ ˆ µ t h t, xˆ, ˆ xˆ ω, α =,,..., Jα hese, becase of qascovexty hypothess ad e the plcato ad the s eqalty costats of (GCD, we have (, ˆ, ˆ ˆ x tx tx, ˆ, ˆ + ( ˆ ˆ x x + ω ( t J ( ˆ (, ˆ, µ hx txˆ ˆ µ ( t + + J (, ˆ, ˆ ˆ tx + ( tx, ˆ, ˆ + ( ˆ ( ˆ µ h ( tx, ˆ, + ˆ J hs, vew of the stct psedocovexty of f t z t t + µ ( t ( h ( t,.,. + x (. ples (,.,. + (. ˆ + + (,, ˆ ˆ (,, ( t,.,. (. ω ( t ( ω, f tx + z+ tx + x + ˆ µ ( t ( h ( tx,, + x J (, ˆ, ˆ ˆ ( (, ˆ, ˆ ˆ ˆ f tx + tx + x ω + ˆ µ ( t ( h ( tx, ˆ, ˆ + xˆ J f( tx,, + x z+ ( ( tx,, + x ω =. + µ ( t ( h ( tx,, + x J hs vew of ( ad (3, yelds. whch ves ( f ( t, x, + ( t zˆ ( t > ( + f t, x, t z t, > t zˆ t z t. vew of zˆ ( t S( ( t K (, ths ples t z t S t K ( > ( S t K S t K, whch s absd. Hece ( x = ( x ˆ ˆ 3. Covese Dalty,,. ad

6 98 Aeca Joal of Appled Matheatcs ad Statstcs ths secto, we shall pove the covese dalty de the asspto f, ad h ae twce cotosly dffeetable. he poble (GCD ay be wtte the follow fo: ψ txz,,,,,...,, µ,..., µ, ω,..., ω Maxze sbect to θ = = xb xa txt,, t, zt,,...,, =, t ω,..., ω, µ,..., µ ( θ txt,, t, zt,,...,, µ,..., µ =, t ˆ t t, x, + x t ω t, α =,,..., α Jα whee θ = θ ˆ µ t h t, x, + x t, α =,,..., ( t, t,,,...,. = ω z t K, t C, t, =,,..., (. x x x = = f + t t + ω t + µ t h + µ t. = = + z+ + h θ θ µ wth = ( txt t x = x( txt t hx = hx( txt,, t, etc.., x(., (., (., µ (.,,,,,,, Cosde θ as def a µ (., z (., ω (.,..., ω (. app ( Q : X U V Z W B ad (., x(., (., (., (., z(., (.,..., (. θ µ ω ω as def a app Q : X U Λ V Z W B whee ( B ad B ae Baach spaces, ( Λ, V, Z ad W ae spaces of pecewse sooth ctos,, z µ ad (,..., ω ω ω =. ode to apply the eslts of [], soe estctos ae θ. = ad eqed o the eqalty costats θ (. =. t sffces f the Fechet devatves ( (,,,, x µ z ( ( x,,, µ, z Q = Q Q Q Q Q ad Q = Q Q Q Q Q have weak closed ae. the follow theoe, we f f tx,,, tx,, h= h tx,,. wte = = ad heoe 4 (Covese Dalty: Asse that (C : f, ad h ae twce cotosly dffeetable. (C : Q ad Q have weak closed ae., (C 3 : ( t,.,. + (. ω (,.,. (. f t + z + ( t h ( t,.,. s + µ ( t J x ( t psedocovex. (C 4 : ( t ( ( t,.,. (. ω. ad α µ ( t ( h ( t,.,. x ( t ae qascovex, Jα (C 5 : σ ( t A σ ( t σ ( t =, t σ t s, whee a appopate vecto cto, ad x + ( t xx x + ( t x + µ ( t hxx + µ ( t h x A = ( t x ( t + + µ ( t hx µ ( t h + + (C 6 : µ ( a = µ ( b ad ( t( x + ω, α (C 7 : ae lealy ( µ ( t hx + µ α =,,..., Jα depedet. x, s a optal solto of (CP ad the he optal vales of (CP ad (GCD ae eqal. x,,,...,, z, µ,..., µ, ω,..., ω s Poof: Sce a optal solto of (GCD, by Poposto thee exst Laaes ltples τ R, pecewse sooth β : R, : R, η : R, τα, α =,,...,, τα, α =,,..., sch that ( xx xx xx ( x x µ x + β t f + t + µ t h + t f + t + t h + τ α ( µ ( t hx + µ ( t =, t α = Jα τα R ad τ + ( t( x + ω + ( µ ( t hx + µ J + τ α ( t( x + ω α = τ + z+ ( t + µ ( t h J + β t f + t + µ t h ( x x x (8

7 Aeca Joal of Appled Matheatcs ad Statstcs 99 + t + ( t + τ α t x µ ( t h + α = α (9 + τ α µ ( t h =, t α = Jα α ( + x + ( t ( x + τ ω β ω + t + t =,, t η ( + x + ( t ( x + τ ω β ω + ( t + η ( t =, α, α =,,..,, t x, ( ( τ h x + β t h + t h = β t t ( α ( h x + ( t hx + ( t h = ( t τ β β t, Jα, α =,,..., ( t x ( t ( t N, C, (3 τ + β ω (4 α β t + τ t x N ω, J,,..., C α α = (5 α + t NK z (6 α (7 α τ t + x ω = τ α µ t h + x = (8 Jα η = (9 ( τη, t, τ,..., τ, τ,..., τ, t (3 ( τ, β t, t, ητ,,..., τ, τ,..., τ, t (3 Mltply ( by ( t, α, α =,,.., ad s (9, we have ( + α t α τ ω ( x + β t t + ω α ( + t t x ω + = α Fo (7, we have β( t ( t( x + ω α = α + ( t ( t x whch ca be wtte as t, t α = ( t x α ( β ( t( x + ω (3 Mltply (3 by µ ( t, Jα, α =,,...,, we have ( ( t β ( t µ ( t hx Jα + γ ( t µ ( t h Jα α t h x + Jα τ µ = µ β t. Jα Fo (8, we have β ( t µ ( t h x γ ( t µ ( t h + Jα Jα t = b = µ ( t β ( t µ ( t β( t t = a Jα Jα (by teat by pats whch o s the hypothess (C 5, ve the elato ca be wtte the atx fo as ( β γ ( t h + ( t µ x µ J t, t α =. µ ( t h Jα (33 Us the eqato costats of (GCD, (8 ad (9 espectvely. We have ad τ ( t( x + ω α = α + + α = Jα τ ( µ ( t hx µ ( t ( xx xx xx ( x x x + β t f + t + µ t h + γ t f + t + µ t h =, t τ α = α + α = Jα t τ µ t h ( x x x ( + t f + t + t h β µ + γ t f + t + µ t h =, t (34 (35

8 Aeca Joal of Appled Matheatcs ad Statstcs Cob (34 ad (35, we have α = τ τ + α = ( t α ( t( x + ω ( µ ( t hx + µ ( t Jα Jα µ x + ( t xx x + ( t x + µ ( t hxx + µ ( t h x β ( t + =. f ( t x t x t γ + + µ ( t hx µ ( t h + + t h Ppeltply by ( ( t ( t τ β, γ α = + α = τ β, γ + ( β( t, γ ( t β, γ, ths ves t t ( t α ( t( x + ω ( µ ( t hx + µ ( t J t t α µ ( t h Jα x x + ( t xx + ( t x + µ ( t hxx + µ ( t hx β ( t =. γ ( t + ( t x + ( t + µ ( t hx + µ ( t h hs, o s (3 ad (33 ves x x + ( t xx + ( t x + µ ( t hxx + µ ( t hx β ( t t, t =. γ ( t + ( t x + ( t µ ( t hx µ ( t h + + ( β γ vew of the hypothess (C 4, ths yelds Us (36, we have γ β t = = t, t (36 τ ( t( x + ω α = α + τ ( µ ( t hx + µ ( t =. α = Jα hs becase of (C 6, yelds τα τ =, τα τ =,. α =,,..., f τ =, the τ α = = τ α, α =,,...,. ad fo ( η t = t. ad (, we have, ττ Coseqetly, we have,,..., τ, τ,..., τ, =, β( t, γ ( t, η( t t es a cotadcto to (3. Hece τ >, ply τα >, τα >, α =,,...,. Fo ( toethe wth ( ad ( toethe wth (3, we have ad x + x ω, t (37 ( x x ω, + = (38 h x =, t (39 ( t( h x, t, J (4 µ = he elatos (4-(6, we have x t ω t = S x t C, =,,..., (4 ( t z t = S K (4 Fo (37, (39 ad (4, we have tx,, + S x t C, =,,..., htx,, x =. ply that ( x, s feasble fo (CP. Fo (38, (4 ad (4, we have ( f ( t, x, + S ( ( t K f( tx,, + ( t zt = + ( t ( ( t, x, + x ω ( t + µ ( t ( h ( tx,, x J vew of the hypotheses (C 3 ad (C 4, by heoe, x, fo (CP follows. the optalty of 4. Specal Cases f = M ad J = N, the (GCD becoes (WCD whch s Wolfe type dal to (CD de the psedocovexty of

9 Aeca Joal of Appled Matheatcs ad Statstcs f t z t t = + µ ( t ( ht (,.,. + x (. (,.,. + (. + + ( t,.,. (. ω ( t. f = φ ad α M, α,,...,, the (GCD becoes (M-WCD s a Mod-We type dal to (CP f ( f ( t,.,. + (. z ( t s psedocovex, ad ( t,.,. ( t ht (,.,. ad = + (. ω ( t µ t ae + xt qascovex. D t, =,,..., be postve se = (fo soe { } Let B( t ad defte atces ad cotos o. he St K = xt Btxt, t whee {, } = (,, =,,..., C = { D t ω t ω t D t ω t t } ( K = B t z t z t B t z t t ad S xt C xt D txt t whee,. Replac the sppot cto by ts coespod sqae oot of a qadatc fo, we have { } (CP : Mze (,, + ( sbect to f t x S t K = = xb xa tx,, + xt D txt, t, =,,..., h tx,, = x t, t f t x t B t z t (GCD : Maxze + + sbect to (,, + ( tx,, ( t + x( t D ( t ω ( t µ ( t ( h ( tx,, x ( t J = = xb xa tx B t z t t = (,, µ ( t hx = µ ( t, t ( tx,, x + D ( t ω ( t tx B t z t t tx (,, + + (,, + µ ( t h ( tx,, =, t z t B t z t t,,,,,..., ω t D t ω t t = t t, x, + D t x t, α =,,..., α µ t h t,., x t, α =,,..., α ( t, t,,,..,, = hese dal odels ae ot explctly epoted the lteate. Howeve, the dalty elatoshp betwee (CD ad (GCD ca be establshed aaloosly to that of the poble of peced secto. 5. Nolea Poa Poble f all the ctos volved the folato of (CP ad (GCD ae depedet of t, these poble edce to the follow olea poa pobles wth sppot ctos whch do ot appea the lteate. f x, + S K (NP: Mze sbect to x, + S x C, =,,..., h( x, = (, + ( ( x, x ω x z (GND: Maxze µ h ( x, J sbect to f (, ( (, x x + x x + ω + µ hx( x, = = ( x. + z+ ( x. + µ h ( x, =, z K ad ω C, =,,..., ( ( x x, + ω, α =,,..., α µ h ( x,, α =,,..., J α ( t,, t, =,,..., o h( x, ad eplac (,, (, s( K by f ( x, ( x ad ( f x x ad s x K espectvely, we et the follow pobles stded by Hsa ad Jabee []: Pal (P : Mze f ( x + s( x K sbect to x + s x C, =,,..., h( x =

10 Aeca Joal of Appled Matheatcs ad Statstcs Dal (GD: Maxze sbect to α f x + x z ( x ω µ h ( x J ( x ω x + z+ + = = Jα ( +, =,,..., x ω α t h x =,,..., µ α Refeeces z K, ω C, =,,...,. [] S. Chada, B.D. Cave, ad. Hsa, A class of odffeetal Cotol pobles, J. Opt. heoy Appl. 56 (988, []. Hsa ad Z. Jabee, Mxed dalty fo a Poa cota sppot ctos, J. Appl. Math & copt Vol. 5 (4, No - pp. -5. [3]. Hsa, A. Ahad ad Abdl Raoof Shah, O a cotol poble wth sppot ctos, sbtted fo pblcato. [4]. Hsa, Abdl Raoof Shah ad Rsh K. Padey, Dalty fo a cotol Poble wth sppot cto, sbtted fo pblcato. [5] B. Mod ad M. Haso, Dalty fo cotol poble, SAM J. Cotol 6 (968, 4-.