Mixed Type Second-order Duality for a Nondifferentiable Continuous Programming Problem

Transcription

1 heoretical Mathematics & Applications, vol.3, no.1, 13, SSN: (print), (online) Scienpress Ltd, 13 Mied ype Second-order Dality for a Nondifferentiable Continos Programming Problem. Hsain 1 and Santosh K. Shrivastav Abstract A mied type second-order dal is formlated for a class of continos programming problems in which the integrand of the objective fnctional contains sqare root of positive semi-definite qadratic form; hence it is nondifferentiable. Under second-order psedoinveity and second-order qasi-inveity, varios dality theorems are proved for this pair of dal nondifferentiable continos programming problems. A pair of dal continos programming problems with natral bondary vales is constrcted and it is briefly indicated that the dality reslts for this pair of problems can be validated analogosly to those for the earlier models. Lastly, it is pointed ot that or dality reslts can be regarded as dynamic generalizations of those for a nondifferentiable nonlinear programming problem, already treated in the literatre. 1 Department of Mathematics, Jaypee University of Engineering and echnology, Gna, M.P, ndia: ihsain11@yahoo.com Department of Mathematics, Jaypee University of Engineering and echnology, Gna, M.P, ndia: santosh_147@rediffmail.com Article nfo: Received : November 1, 1. Revised : Janary 1, 13 Pblished online : April 15, 13

2 14 Mied ype Second-order Dality Mathematics Sbject Classification: 9C3; 9C11; 9C; 9C6 Keywords: Continos programming; Second-order inveity; Second-order psedoinveity; Second-order qasi-inveity; Mied type Second-order dality; Nondifferentiable nonlinear programming 1 ntrodction A nmber of researchers have stdied second-order dality in mathematical programming. A second order dal to a Nonlinear programming problem was first formlated by Mangasarian [1].Sbseqently Mond [] established varios dality theorem nder a condition which is called second-order conveity, which is mch simpler than that sed by Mangasarian [1]. Mond and weir [3] reformlated the second-order and higher-order dals to validate dality reslts. t is fond that second-order dal to a mathematical programming problem offers a tighter bond and hence enjoys comptational advantage over a first-order dal. n the spirit of Mangasarian [1], Chen [4] formlated second-order dal for a constrained variational problem and established varios dality reslts nder an involved inveity-like assmption. Later, Hsain et al. [5], stdied Mond-Weir type second-order dality for the problem of [4], by introdcing continos-time version of second-order inveity and generalized second-order inveity, validated sal dality reslts. Sbseqently Hsain and Masoodi [6] presented Wolf type dality while Hsain and Srivastava [7] formlated Mond-weir type dal for a class of continos programming containing sqare root of a qadratic form to rela to the assmption of second-order psedoinveity and second-order qasi-inveity. n this paper, in order to combine dal formlations of [6] and [7], a mi type dal to the non-differentiable continos programming problem of Chandra et al. [8] is constrcted and a nmber of dality reslts are proved nder

3 . Hsain and S.K. Shrivastav 15 appropriate generalized second-order inveity established. A relation between or dality reslts and those of a nondifferentiable nonlinear programming problem is pointed ot throgh natral bondary vale variational problems. Related Pre-reqisites n n Let = [a, b] be a real interval, φ : R R R and n n m ψ : R R R be twice continosly differentiable fnctions. n order to ( ) consider tt ( ) t ( ) n φ,,, where : R is differentiable with derivative, denoted by and, respectively, that is, φ φ the first order of φ with respect to ( t) and ( t) φ, φ,, φ, φ, φ φ,, φ 1 n 1 n φ = =., Denote by φ the Hessian matri ofφ and ψ the m n Jacobian matrices respectively, that is matri φ φ =, i, j 1,,... n i j =, ψ the m n Jacobian ψ ψ ψ 1 n ψ ψ ψ 1 n ψ = m m m ψ ψ ψ 1 n m n he symbols φ, φ, φ and ψ have analogos representations. n Designate by X the space of piecewise smooth fnctions : R, with the.

4 16 Mied ype Second-order Dality norm = + D, where the differentiation operator D is given by hs d = D ecept at discontinities. dt t ( ) ( ) D t s ds = =, Now we incorporate the following definitions which are reqired in the sbseqent analysis. a Definition.1 (Second-order nve) f there eist a vector fnction ( t,, ) η = η R n n n n where η : R R R and with η = at t = a and t = b, t dt where sch that for a scalar fnction φ ( t,, ), the fnctional φ (,, ) n n φ : R R Rsatisfies 1 { ( ) ( ) ( t,, ) dt ( t,, ) t G t dt { ηφ ( ) + ( η) φ ( ) + η β( )} φ φ β β then φ (,, ) t,, D t,, G t dt, t dt is second-order inve with respect toη where G = φ Dφ + D φ D φ n, and β C(, R ) 3 continos vector fnctions., the space of n -dimensional t dt Definition. (Second- order Psedoinve) f the fnctional φ (,, ) satisfies { ηφ ( η) φ η β( )} + D + G t dt 1 φ( t,, ) dt φ( t,, ) β( t) G β( t) dt,

5 . Hsain and S.K. Shrivastav 17 t dt is said to be second-order psedoinve with respect to η. hen φ (,, ) t dt Definition.3 (Second- order Qasi-inve) f the fnctional φ (,, ) satisfies 1 { ( ) ( ) φ( t,, ) dt φ( t,, ) β t G β t dt then φ (,, ) { ηφ ( η) φ η ( ) β( )} + D + G t t dt, t dt is said to be second-order qasi-inve with respect toη. Remark.1 f φ does not depend eplicitly on t, then the above definitions redce to those given in [] for static cases. he following giving Schwartz ineqality is also reqired for the validation of or dality reslts. Lemma.1 (Schwartz ineqality) t states that ( ) ( ( ) ( ) ( )) ( ) ( ) ( ) ( ) ( ) ( ) 1 / 1 / t B t z t t B t t z t B t z t, t (1) ( ) with eqality in (1) if and only if B( t) ( t) q( t) z( t) = for some q( t) R. Proposition.1 f (CP) attains an optimal soltion at = X, then there eist Lagrange mltiplier τ R and piecewise smooth : m y R +, not both zero, and n also piecewise smooth ωˆ : R, satisfying for all t. ( ) ˆ ( ) ( ) τ f t,, + B(t ) ω+ y(t ) g t,, D τ f (t,, ) + y(t ) g =,t ( ) ( ) y t g t,, =,t

6 18 Mied ype Second-order Dality ( ) 1 / ( ) ( ) ( ) ( ) ( ) ( ) t B t ω t = t B t t,t ω(t) B(t) ω(t) 1,t he Fritz John necessary optimality conditions given in Proposition7..1 become the Karsh-Khn-cker type optimality conditions if τ= 1. For τ= 1, it sffices that the following slater s conditions [8] holds. ( ) + ( ) + ( ) < g t,, g t,, v(t ) g t,, v(t ),For some v X and all t. Consider the following class of non-differentiable continos programming problem stdied in [8]. CP Minimize ( ) ( ) 1/ ( ): Where Sbject (i) [ a,b] f t,, + (t) B(t)(t) dt ( ) =,( b) =β ( ) ( ) ( 3) a g t,,,t = is a real interval (ii) f : R R R,g: R R R n n n n m are twice continosly differentiable fnction with respect to its argment t () and t (). (ii) : R n is for times differentiable with respect to t and these derivatives are defined by,, and.... (iii) B (t) is a positive semi-definite n n matri with B (.) continos on. he poplarity of mathematical programming problems as (CP) seems to stem from the fact that, even thogh the objective fnctions and /or constraint fnctions are differentiable, a simple formlation of the dal may be given. Non-differentiable mathematical programming deals with mch more general kinds of fnctions by sing generalized sb-differentials and qasi differentials.

7 . Hsain and S.K. Shrivastav 19 Hsain et al. [5] formlated the following Wolf type second-order dal and established dality reslts for the pair of problems (CP) and (W-CD) nder the { } second-order psedoinveity of ( ) + + ( ) respect to η. f t,.,. (.) B( t )w( t ) y( t ) g t,.,. dt with ( WCD ): 1 f t,, + (t) B(t)w(t) + y(t) g t,, β(t) L β(t) dt Maimize ( ) ( ) Sbject to ( ) = = ( ) a b, ( ) ( ) ( ) f t,, + (t) B(t) + y(t) g t,, D f + y(t) g + L β(t) =, t, y( t ),t, w(t) B(t)w(t) 1,t. where ( ) ( ) ( ( ) ( )) ( ) ( ) ( ( ) ( )) ( ) ( ) ( ) L= f t,, + y t g t,, D[ f t,, + y t g t,, ] ( ) ( ) ( ) 3 + D[ f t,, + y t g t,, ] D[ f t,, + y t g t,, ] Recently, Hsain and Srivastava [7] investigated Mond-Weir type dality by ( ) constrcting the following dal to (CP) nder ( ( ) ( )) + ( ) ( ) for all n w( t ) R f t, t, t w B t w t dt, is second-order psedo-inve and y ( t) ( ) second-order qasi-inve with respect to the same η. (M WCD) : g t,.., dt is Maimize 1 f(t,,) + (t) B(t)w(t) β(t) F β(t) dt

8 13 Mied ype Second-order Dality Sbject to where ( ) ( ) = = ( b) ( ) + + ( ) ( ( ) ( ) ) ( ) a f t,, B(t )w(t ) y(t ) g t,, D f t,, + y(t ) g t,, + F + K β(t ) = 1 y( t) g ( t,, ) β( t) K β( t) dt, t ( ) y t,t ( ) ( ) ( ) wt Btwt 1,t 3 F t,,,, = f Df + D f D f, t and 3 ( ) ( ) ( ) ( ) ( ) K t,,,, = y t g Dy t g + D y t g D y t g, t n this paper, we propose a mied type second-order dal (MiCD) to (CP) and prove varios dality theorems nder the assmptions of second-order psedoinveity and second-order qasi-inveity. We formlate a pair of nondifferentiable continos programming problem with natral bondary vales rather than fied end points. Finally, it is also pointed ot that or dality reslts derived in this research can be regarded as the generalization of those of the nonlinear programming problem. 3 Mied ype second-order dality n this section we formlate the following mied type second-order dal Mied (CD) to (CP): Mied (CD): 1 Maimize f ( t,,) + ( t) B(t ) ω( t) + y ( t) g ( t,, ) β( t) H β( t) dt i

9 . Hsain and S.K. Shrivastav 131 Sbject to a ( ) =, b ( ) = (4) f (, t, ) + Btwt () )) + yt () g (, t, ) i ( ) D f ( t,, ) + yt ( ) g ( t,, ) + Hβ ( t) =, t (5) 1 y ( t) g ( t,, ) β( t) G β( t) dt. = 1,,3... r (6) Where w() t B() t w() t 1, and y() t, t (7) r i M = 1,,...m, = 1,,,...r with = M and β = φ if β. = ( ) { } ( ) ( ) ii H = f (, t, ) y() tg(, t, ) D( f (, t, ) i ( y tg t ) D f t ( y tg t ) () (,, ) ) + ( (,, ) () (,, ) ) i i 3 i ( ) D( f (, t, ) y() tg (, t, ) ) ( iii) H= f t ( yt g t ) D f t ( yt g t ) + D ( f (, t, ) ( yt () g (, t, ) ) ) 3 D( f (, t, ) ( yt () g (, t, ) ) ) and (,, ) () (,, ) ( (,, ) () (,, ) ) ( iv) G = ( y() tg (, t, ) ) D ( y() tg (, t, ) ) and i i 3 + D ( y( tg ) ( t,, )) D ( y( tg ) ( t,, )), = 1,,..., r. i i

10 13 Mied ype Second-order Dality ( v) G= ( y() tg (, t, ) ) D ( y() tg (, t, ) ) i M i M 3 + D ( y() tg (, t, )) D ( y() tg(, t, )), i M i M heorem 3.1 (Weak Dality) Let be feasible for (CP) and (,y,w,β ) feasible for Mied (CD). f for all feasible (,, ywβ,, ), is second-order psedoinve and f ( t,.,.) + y ( t) g ( t,.,.) + (.) B( t) w( t) dt i i y ( t) g ( t,.,.) dt, = 1,,... r is second-order qasi-inve with respect to the same η, then infimm(cp) spremm Mied(CD) Proof: By the feasibility of and (,y,w,β ) for (CP) and Mied (CD) respectively, we have 1 ( y ( t) g ( t,, ) ) dt ( y ( t) g ( t,, ) β( t) G β( t) ) dt, i = 1,,..., r. By second-order qasi-inveity of this ineqality yields i i i ( ) ( ) i y t g t,, dt, = 1,,..., r, ( ) ( ) ( ) ( ) ( ) ( ) i i [( η y t g t,, ) + Dη ( y t g t,, ) + η G β t] dt,

11 . Hsain and S.K. Shrivastav 133 = 1,,..., r. his implies i M ( ) ( ) [( η y t g t,, ) ( η) ( ) ( ) η β( ) + D ( y t g t,, ) + G t ] dt i M = η y ( t) g ( t,,) D y ( t) g ( t,,) + G β( t) dt i M i M t = b + η y ( t) g ( t,,) i t = a Using η =, at t = aand t = b, we obtain, (by integrating by parts) ( ) ( ) ( ) ( ) β( ) i M i M η [( y t g t,, ) D( y t g t,, ) + G t] dt Using (5), we have ( ) ( ) ( ) ( ) ( ) i η [ f t,, + B t wt + y t g t,, ( ) ( ) ( ) β ( ) D( f t,, + y t g t,, ) + H t] dt i ( ) ( ) ( ) ( ) ( ) i = [ η ( f t,, + B t wt + y t g t,, ) ( η) ( ) ( ) η β( ) + D ( f + y t g t,, ) + H t ] dt i t = b η ( f + y( t) g( t,, )) i t = a From this, as earlier η = at t = a and t = b, we get, (by integrating by parts)

12 134 Mied ype Second-order Dality ( ) + ( ) ( ) + ( ) ( ) i [ η ( f t,, B t wt y t g t,, ) ( η) ( ) ( ) ( ) η β( ) + D ( f t,, + y t g t,, ) + H t] dt i which by second-order psedo-inveity of implies ( ) ( ) ( ) ( ) ( ) ( f t,, Bt wt y t g t,, ) dt + + i ( ) + ( ) ( ) ( ) + ( ) ( ) ( f t,, t Bt wt y t g t,, ) dt i 1 [( f( t,, ) + ( t) B( t) wt ( ) + y( t) g ( t,, ) ) β( t) H β( t) ] dt hs from y( t) and ( ) ( f ( t,, ) + ( t) B( t) w( t) ) dt i gt,,, t. i he above gives, 1 [( f( t,, ) + ( t) B( t) wt ( ) + y( t) g ( t,, ) ) β( t) H β( t) ] dt wt Bt wt t, the Schwartz ineqality (1), this ineqality Since ( ) ( ) ( ) 1, implies ( ) ( ) + ( ) ( ) ( ) 1/ ( f t,, t B t t ) dt 1 [( f( t,, ) + ( t) B( t) wt ( ) + y( t) g ( t,, ) ) β( t) H β( t) ] dt i yielding infimm(cp) spremm Mied (CD)

13 . Hsain and S.K. Shrivastav 135 heorem 3. (Strong Dality) f is a optimal soltion of (CP) and normal [8],then there eist piecewise smooth y: R m and w: ( ) n R sch that,y,w, ( t) β = is feasible for Mied (CD), and the corresponding vales of (CP) and Mi (CD) are eqal. f, for all feasible (,,y,w,β ), f ( t,, ) + y ( t) g ( t,, ) + ( t) B( t) w( t) dt i is second-order psedo-inve and ( ) ( ) ( y t g t,, ) dt, = 1,,..., r is i second-order qasi-inve, then (,y,w, β ( t) ) is an optimal soltion of Mi (CD). Proof: Since is an optimal soltion of (CP) and normal [8], then by Proposition.1, there eist piecewise smooth y: R that (,, ) + ( ) ( ) + ( ) (,,, ) ( ) ( ) ( ) f t B t w t y t g t ( ) D f t,, + y t g t,,, =, t yt ( ) gt (,, ) =, t 1/ ( ( ) ( ) ( )) ( ) ( ) ( ) t B t t = t B t w t, t wt ( ) Btwt ( ) ( ) 1, t y( t), t his implies that (,y,w, ( t) ) m and w: R n sch β = is feasible for Mi (CD) and the corresponding vale of (CP) and Mi (CD) are eqal. f f( t,, ) + y g( t,, ) ++ t ( ) Btwt ( ) ( ) dt i o is second-order psedo-inve, and

14 136 Mied ype Second-order Dality i ( ) ( ) y t g t,, dt, = 1,,..., r is second-order qasi-inve with respect to the sameη, then from weak dality ( ) heorem3.1,y,w, β ( t) is an optimal soltion of Mi (CD). heorem 3.3 (Converse Dality) Let ( ywβ,,, ) be an optimal soltion at which ( A1 ) for all = 1,,3... r, either ( a ) β( t) G + ( y g ) β( t) dt >, and β ( t) ( y g ) dt, or i i ( b ) ( ) ( β t G + y g ) β( t) dt <, and β ( t) ( y g ), i dt i ( A ) the vectors H,G, = 1,,...,r and j = 1,,...n are linearly independent, j j where H j is the jth row of the matri H and G j is the jth row of the matri G 3 and ( A ) the vectors i i ( y( tg ) ) D ( y( tg ) ), = 1,,3,..., r are linearly independent. f, for all (,, ywβ,, ), is second-order psedoinve and f ( t,...) + (.) B( t) w( t) + y ( t) g ( t,.,.) dt i i y ( t) g ( t,.,.) dt, = 1,,..., r, is second-order qasi-inve, then is an optimal soltion of (CP). Proof: Since ( yβ,,, ) is an optimal soltion of Mi (CD), (), by Proposition.1, there τ R, τ R, = 1,,..., r, and piecewise smooth m m θ : R, r: R and p: R n sch that

15 . Hsain and S.K. Shrivastav = τ[( f + y() tg + Btwt () ()) Df ( + y() tg ) β() t Hβ() t i i D( β() t H β() t ) D ( β() t H β() t ) + D ( β() t H β() t ) 1 4 D t H t + θ () t [( f + yt () g ) D f + yt () g D f + ( yt () g) ( ) ( ) ( ) ( β() β() ) ] ( ) ( ) D( D f + yt g ) + D ( D f + yt g ) + Hβ t ( () ( ( () ) ( ()) (8) 3 4 D( Hβ( t)) + D ( Hβ) D ( Hβ) + D ( Hβ) ] r 1 1 τ{ ( y() tg) Dy ( () tg) ) ( β() tg β()) t + D( β() tg β()) t = 1 i D( β() tg β()) t + D( β( t) G β( t)) D ( β( t) G β( t))}, t 3 4 i 1 i i ( g () t g ()) t () t ( g + g ()) t + r () t =, i (9) i 1 ( g ( t) g ( t)) ( t)( g + g ( t)) + r ( t) =, i, = 1,,..., r (1) τ β β θ β τ β β θ β ( ) τ Btt () () θ() tbt () pt () Btwt () () =, t (11) ( τ β θ ) ( τ β θ ) r ( t) H + ( t) G = (1) = 1 1 τ { y ( t) g β( t) G β( t)} dt =, = 1,,3,..., r (13) ( ) i ( ) pt () wt () Btwt () () 1 =, t (14) rt ( ) yt ( ) =, t (15) ( τ τ1 τ τr ) ( τ τ τ τ θ ),,,..., rt ( ), pt ( ), t (16),,,..., rt ( ), ( t), pt ( ), t (17) 1 r Becase of assmption ( A ), (1) implies τ β( t) θ( t) =, =,1,,... r (18) i Mltiplying (1) by y ( t), t, = 1,,..., r, and smming over i, we have, 1 τ{ y() t g β()( t yg) β()} t + θ(){( t yg) + ( yg) β()} t + y() tr() t =

16 138 Mied ype Second-order Dality 1 τ{ y() tg β() t ( yg) β()} t i i i i Using (18) in (19), we have + θ( t){ ( yg) + ( yg) β( t)} =, i = 1,,..., r (19) 1 τ{ y() tg β() t ( yg) β()} t i i τβ( t){ ( yg) + ( yg) β( t)} =, i, = 1,,..., r. () i i τ{ β() t ( y g ) dt + β() t ( y g ) β() t dt} i i 1 + τ [ ( yg) β( t) ( ( yg)) ] β( t) dt=, = 1,,..., r. (1) i i, gives 1 τ[ β()( t ( y g )) dt + β() t ( ( y g )) β() t dt] i i τ + β( t) G β( t) dt = () τ τ () t ( y g ) dt β + ()( t ( y g ) G ) () t dt (3) i β + β = i f for all =,1,... r, τ =, then (18) implies θ ( t) =, t From (1), we have rt ( ) =, t and pt () = from (11) and (14). τ, τ1, τ,... τr, rt ( ), θ ( t), pt ( ) =, t hs ( ) his gives a contradiction. Hence there eists an {,1,,...r} τ >. f β ( t), t, ths (18) gives ( τ τ ) β( t) =, = 1,,..., r. sch that

17 . Hsain and S.K. Shrivastav 139 his implies thatτ = τ >, from (), we have β() t ( ( y g ) ) dt + β() t ( G + ( y g ) ) β() t dt = i i his contradicts ( A1 ). Hence β ( t) =, t. Using (4) and β ( t) =, t,(8) gives r τ τ yg D yg = 1 i i ( ){ ( ) ( )} =, (4) Which by the linear independence of yields τ i i = τ, for all {,1,,... r} Now (9) and (1) gives g rt ( ), t i, i i { ( yg ) D ( yg )}, = 1,,3,..., r i τ + = gives ( ) g., i (5) i τ g + r( t) =, t i, implies g (.), i, = 1,,3,... r. (6) Gives g (.) implies is a feasible soltion of (CP). i i Mltiplying (5) by y ( t), i and to y ( t), i, = 1,,.., r From (11), we have Hence ( ) p t B( t) ( t) = B t t τ ( ) ω ( ) 1 1 ( ) ( ) ( ) t () Btwt () () = t () Bt () t () wt () Btwt () () 7 f pt ( ) >, t, (14) implies wt () Btwt () () = 1 and (5) implies f pt ( ) =, t, then (7) gives ( ) 1 t () Btwt () () = t () Btt () (), t

18 14 Mied ype Second-order Dality hs in either Case, we have ( ) 1 t () Btwt () () = t () Btt () (), t B( t) ( t) =, t, 1 his gives (,, ) f t + ( t () Btt () ()) dt 1 = f( t,, ) + t ( ) Bt ( ) wt ( ) + yg( t,, ) β( t) Hβ( t) dt i implies is optimal to (CP). 4 Special Cases f = M, then Mi (CD) becomes (WD) and from heorem , it follows that it is a second-order dal to (CP), if { ( ) + ( ) ( ) + ( ) ( ) ( )} f t,, y t g t,, t B t w t dt is a second-order psedoinve. f is empty set and = M for some { 1,,..., r }, to (M-WD) which is a second-order dal to (CP). f { ( ) + ( ) ( ) ( )} f t,, t B t w t dt is second-order psedoinve and y( t) ( ) qasi-inve. then Mi (CD) redces g t,, dt, is second-order f B( t ) =,t, the dal problem Mied (CD) will redce to the Mied type dal treated by Hsain and Bilal [9].

19 . Hsain and S.K. Shrivastav Natral Bondary Vales n this section, we formlate a pair of nondifferentiable mied type dal variational problems with natral bondary vales rather than fied end points. (CP ): Minimize ( ) 1 f ( t,, ) + ( t) B( t) ( t) dt Sbject to gt (,, ), t (MiCD ): Maimize 1 ( f ( t, ( t), ( t) ) + ( t) B( t) w( t) + y ( t) g ( t,, ) β() t H β()) t dt Sbject to ( f ( t,, ) + B( t) wt ( ) + y( t) g ( t,, )) ( ( ) ( ) ( )) β ( ) i D f t,, + y t g t,, + H t =, t 1 ( y () t g (, t, ) β() t G β()) t dt, = 1,,3,, r i ( ) ( ) ( ) ( ) wt Btwt 1, yt, t y( t) g ( t,, ) =, i and ( ) ( ) ( ) f t,, + y t g t,, =, at t= aand t= b. i

20 14 Mied ype Second-order Dality 6 Nondifferentiable Nonlinear Programming Problems f all the fnctions of the problem (CP ) and Mi (CD ) are independent of t as b a= 1. hen these problem will redces to the following dal problem to (CP ) formlated by Zhang and Mond [1] : (CP1): Minimize f ( ) + ( B) 1/ Sbject to g( ), Maimize 1 f + y g + Bw β H β Mi (CD 1 ): ( ) ( ) i i Sbject to f + Bw+ y g + y g + Hβ = where ( ) ( ) ( ) i 1 yg( ) ˆ i β Gβ w Bw 1, y Hˆ = f yg and G= yg, = 1,,..., r. ( ) n heorem 3.3, the symbols ˆ i i ( ) ( ) H j i i j,h,g and G will respectively become as H = f ( ) yg ( ) = f( ) yg ( ) H j = f yg i, G = yg ( ) = yg ( ) and Gj = yg ( ) i i i For the problems (CP ) and Mi (CD ) will be stated as the following theorem (heorem 6.1) established by Yang and Zhang [11].heorem 6.1 is heorem of [11] and the rectified version of [1]. j j

21 . Hsain and S.K. Shrivastav 143 heorem 6.1 Let (, y,w,β ) be an optimal soltion of Mi (CD ) at which ( ) 1 A for all = 1,,..,r. either or β (a) the n n Hessian matri β i y g () (b) the n nhessian matri i y g (), y g () is positive definite, and i y g () is negative definite and i ( A ) he vectors f( ) yg ( ), yg ( ) i j, i = 1,,..,r. j = 1,,...,n are linearly independent, where it is the j th row of the j matri H and G. A the vectors { yg( ), = 1,,.., r} are linearly independent. ( ) 3 i f, for all feasible (,,y,w,p ), ( ) + ( ) + ( ) psedoconve and ( ) = 1 i y g.,,,..,r f. yg.. Bw i respect to the sameη, then is an optimal soltion to (CP ). is second-order is second-order qasiconcave with References [1] O.L. Mangasarian, Second and higher order dality in nonlinear programming, Jornal of Mathematical Analysis and Applications, 51, (1979), 65-6.

22 144 Mied ype Second-order Dality [] B. Mond, Second order dality in nonlinear programming, Opsearch, 11, (1974), [3] B. Mond and. Weir, Generalized Conveity and High Order-Dality, Jornal of Mathematical Analysis and Applications, 46(1), 1974), [4] X. Chen, Second-Order Dality for Variational Problems, Jornal of Mathematical Analysis and Applications, 86, (3), [5]. Hsain, A. Ahmed and Mashoob Masoodi, Second-Order Dality for Variational Problems, Eropean Jornal of Pre and Applied Mathematics, (), (9), [6] qbal Hsain and Mashoob Masoodi, Second-Order Dality for a Class of Nondifferentiable Continos Programming Problems, Eropean Jornal of Pre and Applied Mathematics, 5(3), (1), [7]. Hsain and Santosh K. Shrivastav, On Second-Order Dality in Nondifferentiable Continos Programming, American Jornal of Operations Research, (3), (1), [8] S. Chandra, B.D. Craven and. Hsain, A class of nondifferentiable continos programming problems, J. Math. Anal. Appl., 17, (1985), [9]. Hsain and Bilal Ahmad, Mied ype Second-order Dality for Variational Problems, Jornal of nformatics and Mathematical Sciences, 5, (13), [1] J. Zhang and B. Mond, Dality for a non-differentiable programming problem, Blletin of the Astralian Mathematical society, 55, (1997), -44. [11] Xin Min Yang and Ping Zhang, On Second-Order Converse Dality for a Nondifferentiable Programming Problem, Blletin Astralian Mathematical society, 7(), (5), 65-7.