On Multiobjective Duality For Variational Problems

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1 The Open Operational Research Jornal, 202, 6, -8 On Mltiobjective Dality For Variational Problems. Hsain *,, Bilal Ahmad 2 and Z. Jabeen 3 Open Access Department of Mathematics, Jaypee University of Engineering and Technology, Gna, MP, ndia 2 Department of Statistics, University of Kashmir, Srinagar, Kashmir, ndia 3 Department of Mathematics, National nstitte of Technology, Srinagar, Kashmir, ndia Abstract: n this paper two types of dals are considered for a class of variational problems involving higher order derivatives. The dality reslts are derived withot any se of optimality conditions. One set of reslts is based on Mond- Weir type dal that has the same objective fnctional as the primal problem bt different constraints. The second set of reslts is based on a dal of an axiliary primal with single objective fnction. Under varios convexity and generalized convexity assmptions, dality relationships between primal and its varios dals are established. Problems with natral bondary vales are considered and the analogs of or reslts in nonlinear programming are also indicated. AMS-Mathematics Sbject Classification: Primary 90C30, Secondary 90C, 90C20, 90C26. Keywords: Variational Problem, Mltiobjective dality, Convexity, Generalised convexity.. NTRODUCTON Calcls of Variations offers a powerfl techniqe for the soltion of varios important problems appearing in dynamics of rigid bodies, optimization of orbits, theory of vibrations and many areas of science and engineering. The sbject of calcls of variation primarily concerns with finding optimal vale of a definite integral involving a certain fnction sbject to fixed point bondary conditions. Mond and Hanson [5] were the first to represent the problem of calcls of variation as a mathematical programming problem in infinite dimensional space. Since that time many researches contribted to this sbject extensively. For somewhat comprehensive list of references, one may conslt Hsain and Jabeen [] and Hsain and Rmana [2]. The treatment in [] has been for the real valed objective fnction while in [2] for vector valed fnction. n this research, we consider a vector valed fnction for the primal problem and its minimality in the Pareto sense. Both eqality and ineqality constraints are considered in the formlations. n establishing dality reslts we consider two types of dal problems to the primal problem. The first one has vector valed objective whereas the second set of reslts are based on the dality relations between an axiliary problems and its associated dal as defined in Mond and Hanson [7]. Dality theorems, nlike in case of classical mathematical programming, are not based on optimality criteria bt on certain types of convexity and generalized convexity reqirements. Finally mltiobjective variational problems with natral bondary vales rather than fixed end *Address correspondence to this athor at the Department of Mathematics, Jaypee University of Engineering and Technology, Gna, MP, ndia Tel: Fax: ihsain@yahoo.com points are mentioned and the analogs of or reslts in nonlinear programming are pointed ot. 2. PRE-REQUSTES n the treatment of the following problem (VP, by minimality we mean Pareto minimality. Now consider the following mltiobjective variational problem involving higher order derivatives. (VP Minimize f ( t, x, x, x,..., f p ( t, x, x, x dt x( a= 0 = x( b ( x( a= 0 = x( b (2 gt, ( x, x, x < 0, t (3 ht, ( x, x, x = 0, t (4 where (i for = [ a,b] R, f : R n R n R n R, g : R n R n R n R m and h : R n R n R n R k are continosly differentiable fnctions, and (ii X designates the space of piecewise smooth fnction x : R n having its first and second order derivatives x and x respectively eqipped with the norm. x = x + Dx + D 2 x, where the differentiation operator D is given by = Dx x t (= t a ( sds /2 202 Bentham Open

2 2 The Open Operational Research Jornal, 202, Volme 6 Hsain et al. Ths D d except at discontinities. dt We denote the set of feasible soltions of the problem (VP by K P, i.e., K P = x X x( a= 0 = x ( b, gt, ( x, x, x < 0, t x( a= 0 = x( b, ht, ( x, x, x = 0, t The following convention for ineqalities for vectors in R n given in Mangasarian [3] will be sed throghot the development of the theory: f x, y R n,then x >y x i >y i i =,..., n x y x i >y i, and x y x > y x i > y i,i =,..., n. Definition 2.: A feasible soltion of the problem (VP i.e., x K P is said to be Pareto minimm if there exists no ˆx K p sch that dt f t, ˆx, x, ˆ xˆ,..., f p t, ˆx, x, ˆ xˆ dt f ( t, x, x, x,..., f p ( t, x, x, x dt Pareto maximality can be defined in the same way except that the ineqality in the above definition is reversed. n the sbseqent analysis the following reslt plays a significant role. PROPOSTON 2.: Sppose > 0, R p sch that Let x(k t P is an optimal soltion of the problem, : Min P xt (K P T f ( t, x, x, x Then x( t is an optimal soltion of (MP in the Pareto sense. Proof: Assme x(is t not a Pareto optimal of (MP. Then there exists an ˆx(K t P sch that f i f j ( t, ˆx, x, ˆ xˆ < f i ( t, x, x, x, i =, 2,..., p. ( t, ˆx, x, ˆ xˆ < f j ( t, x, x, x, i j. Hence T f ( t, ˆx, x, ˆ xˆ < T f ( t, x, x, x. This contradicts the assmption that T f ( t, x, x, x over K P. x minimizes n the sbseqent sections some dality reslts by introdcing two types of dals to (VP will be established. 3. MOND-WER TYPE MULTOBJECTVE DUALTY Consider the following Mond-Weir [4] dal to (VP (M-WD: Maximize f ( t,,, dt,..., f p ( t,,, dt ( a= 0 = ( b, (5 ( a= 0 = ( b, (6 T f ( t,,, + yt ( T g ( t,,, + zt ( T h ( t,,, D T f ( t,,, + yt ( T g ( t,,, + zt ( T h t,,, ( T f +D 2 ( t,,, + yt zt t,,, ( T g ( T h ( t,,, + = 0, t (7 ( yt ( T gt,, (, + zt ( T ht,, (, > 0, (8 > 0, R p, yt (> 0, t (9 Let K D be the set of the feasible soltions of (M-WD. Theorem 3.: Sppose (A : x(k t P (A 2 : (,, yt (, zt (K D (A 3 : T f ( t,.,.,. is psedo-convex (A 4 : { yt ( T gt,.,.,. + zt ( T ht,.,.,. }dt Then T f ( t, x, x, x > T f ( t,,,. is qasiconvex. Proof: Since yt (> 0, t, gt, ( x, x, x < 0, t and ht, ( x, x, x = 0, t, we have ( yt ( T gt, ( x, x, x + zt ( T ht, ( x, x, x < 0 (0 Combining this ineqality with (8 We have, yt ( T gt, ( x, x, x + zt ( T ht, ( x, x, x < ( yt ( T gt,, (, + zt ( T ht,, (, By the hypothesis (A 4, this yields

3 On Mltiobjective Dality For Variational Problems The Open Operational Research Jornal, 202, Volme > ( x T ( yt ( T g ( t,,, + zt ( T h ( t,,, + ( x T ( yt ( T g ( t,,, + zt ( T h ( t,,, T yt ( T g + x T yt = x ( ( t,,, + zt ( T h ( t,,, dt ( ( T g ( t,,, + zt ( T h ( t,,, + ( x T yt ( T g ( t,,, + zt ( T h t,,, ( ( x T D( y( t T g ( t,,, + zt ( T h ( t,,, + ( x T yt ( T g ( t,,, + zt ( T h t,,, ( ( x T D( y( t T g ( t,,, + zt ( T h ( t,,, (By integration by parts T yt ( dt = x ( ( T g ( t,,, + zt ( T h ( t,,, D y( t T g ( t,,, + zt ( T h t,,, T D y( t T g x + x ( t,,, + zt ( T h t,,, ( T D 2 yt ( T g ( ( t,,, + zt ( T h ( t,,, dt (By integration by parts Using (7, we have, 0 ( x T { T f ( t,,, D T f t,,, +D 2 T f ( t,,, }dt 0 {( x T T f ( t,,, + ( x T T f ( t,,, }dt T T f ( t,,, x T D T f x { ( t,,, dt + x ( x T T f ( t,,, + x T D T f + x ( t,,, }dt 0 T D T f ( t,,, T T f ( t,,, Ths, by integration by parts sing the bondary conditions, we have, { ( x T T f ( t,,, + x T T f + x ( t,,, }dt 0 T T f ( t,,, This, becase of the hypothesis (A 3 implies, T f ( t, x, x, x T f ( t,,,. Theorem 3.2: Assme (B : x(k t P (B 2 : (,(, t y(, t z( t K D (B 3 : ( yt ( T gt,.,.,. + zt ( T ht,.,.,. is qasi-convex (B 4 : T f ( t,.,.,. is psedoconvex (B 5 : T f ( t, x, x, x = T f ( t,,, Then x(is t an optimal soltion of (VP and (,(, t y(, t z( t is an optimal soltion of the problem (M-WD. Proof: Assme that x is not Pareto-optimal of (VP. Then there exists an ˆx(K t P sch that f i ( t, ˆx, x, ˆ xˆ f i ( t, x, x, x, for all i and j p f j ( t, ˆx, x, ˆ xˆ < f j ( t, x, x, x for some j, Since > 0, this implies, T f ( t, ˆx, x, ˆ xˆ < T f ( t, x, x, x By the hypothesis (B 5, this ineqality implies, T f ( t, ˆx, x, ˆ xˆ < T f ( t,,,. This contradicts the conclsion of Theorem 3. ths establishing the Pareto optimality of x(for t (VP. Similarly we can show that (,(, t y(, t z( t is Pareto optimal for (M-WD. We state the following theorem withot proof as it is similar to Theorem 3.4 of [6]. Theorem 3.3: Assme, (C : x(k t P (,(, t y(, t z( t K D (C 2 : T f ( t, x, x, x = T f ( t,,, (C 3 : ( yt ( T gt,.,.,. + zt ( T ht,.,.,. is convex (C 4 : T f ( t,.,.,. is qasiconvex. Then x(= t (, t t. 4. WOLFE TYPE MULTOBJECTVE DUALTY To establish dality reslts similar to the preceding ones bt nder different convexity and generalized convexity

4 4 The Open Operational Research Jornal, 202, Volme 6 Hsain et al. assmptions, we formlate the following Wolfe type dal to the problem (P stated in the Preposition 2.. We assme that is known and > 0. ( WCD : Maximize: ( T f ( t, x, x, x + yt ( T gt, ( x, x, x + zt ( T ht, ( x, x, x : x(a = 0 = x(b, x(a = 0 = x(b ( ( T f ( t,,, + yt ( T g ( t,,, + zt ( T h ( t,,, D( T f ( t,,, + yt ( T g ( t,,, + zt ( T h t,,, +D ( 2 T f ( t,,, + yt ( T g ( t,,, + zt = 0, t ( T h ( t,,, (2 yt (0, t (3 n the following L P represents the set of feasible soltions of (P and L D the set of feasible soltions of ( WCD. Theorem 4.: Assme (H x(l t P ( (, t y(, t z( t L D (H 2 T f ( t,.,.,. and ( yt ( T gt,.,.,. + zt ( T ht,.,.,. are convex. Then, T f ( T f ( t,,, + yt ( T gt,, (, t, x, x, x dt. +z( t T ht,, (, Proof: By the convexity of T f ( t,.,.,. T f ( t, x, x, x T f ( t,,, + x T T f t,,, + x + x T T f, we have T T f ( t,,, ( t,,, dt (4 From the dal constraint (2, we have, ( x T T f ( t,,, + yt ( T g ( t,,, + zt ( T h t,,, ( ( D T f ( t,,, + yt ( T g ( t,,, + zt ( T h t,,, ( ( T g ( t,,, + zt ( T h ( t,,, dt = 0 +D 2 T f ( t,,, + yt This, by integrating by parts and sing the bondary conditions as earlier, implies ( ( x T T f ( t,,, + yt ( T g ( t,,, + zt ( T h t,,, + ( x T ( T f ( t,,, + yt ( T g ( t,,, + zt ( T h ( t,,, T T f + x ( ( t,,, + yt ( T g ( t,,, + zt ( T h ( t,,, dt = 0 Using this, in (4 we have T f ( t, x, x, x T f ( t,,, ( ( x T yt ( T g ( t,,, + zt ( T h t,,, + ( x T ( yt ( T g ( t,,, + zt ( T h ( t,,, T yt ( T g + x ( ( t,,, + zt ( T h ( t,,, dt By the hypothesis (H 2, this implies T f ( t, x, x, x dt T f ( t,,, + ( yt ( T gt,, (, + zt ( T ht,, (, ( yt ( T gt, ( x, x, x + zt ( T ht, ( x, x, x, Since x L P, this implies T f ( t, x, x, x T f t,,, +y( t T gt,,, +z( t T ht,,, dt. This proves the theorem. The following theorem gives a sitation in which a Pareto optimal soltion of (VP exists. Theorem 4.2. Sppose (F : x(l t P, ( (, t y(, t z( t L D (F 2 : T f ( t,.,.,. and ( yt ( T gt,.,.,. + zt ( T ht,.,.,. are convex, (F 3 : T f ( t,,, + yt ( T gt,, (, T f ( t, x, x, x = +z( t T ht,,, dt Then x t ( and y(, t z t ( (,( t are optimal soltions of (P and (WCD. Hence x(is t a Pareto optimal soltion of (VP. The last part of the conclsion follows from Proposition 2.. Proof: Sppose x(does t not minimize (P then there exist, x (L t P sch that T f ( t, x (, t x (, t x ( t < T f ( t, x, x, x = ( T f ( t,,, + yt ( T gt,, (, + zt ( T ht,, (, This contradicts the conclsion of Theorem 4.. Hence x t (minimizes (P.

5 On Mltiobjective Dality For Variational Problems The Open Operational Research Jornal, 202, Volme 6 5 We can similarly prove that ( y(, t z(, t ( t maximizes ( WCD. Theorem 4.3 Assme (G : x(l t P, ( y(, t z(, t ( t L D (G 2 : T f ( T f ( t,,, + yt ( T gt,, (, t, x, x, x = +z( t T ht,, (, dt (G 3 : ( T f ( t,.,.,. + yt ( T gt,.,.,. + zt ( T ht,.,.,. is convex Then yt ( T gt, ( x, x, x = 0, t Proof: By hypotheses (G 2 and (G 3, we have T f ( t, x, x, x = T f ( t,,, + yt ( T gt,,, +z( t T ht,,, T f ( t, x, x, x + y( t T gt, x, x, x +z( t T ht, x, x, x dt T f ( x T ( t,,, + y( t T g ( t,,, +z( t T h ( t,,, + ( x T f T ( t,,, + y( t T g ( t,,, +z( t T h t,,, T f T ( t,,, + y( t T g ( t,,, +z( t T h ( t,,, + x = T f ( t, x, x, x + y( t T gt, ( x, x, x +z( t T ht, x, x, x dt dt T f ( x T ( t,,, + y( t T g ( t,,, +z( t T h ( t,,, T f ( x T ( t,,, + y( t T g ( t,,, +z( t T h t,,, ( x T D T f ( t,,, + y( t T g ( t,,, +z( t T h ( t,,, dt + ( x T f T ( t,,, + y( t T g ( t,,, +z t t,,, ( T h ( x T D T f ( t,,, + y( t T g ( t,,, +z( t T h ( t,,, T f T ( t,,, + y( t T g ( t,,, +z( t T h ( t,,, + x dt dt dt (ntegrating by parts = T f ( t, x, x, x + y( t T gt, ( x, x, x +z( t T ht, x, x, x dt + x + y( t T g ( t,,, T f T t,,, +z( t T h ( t,,, T D T f t,,, +z( t T h ( t,,, x x + y( t T g ( t,,, dt ( t,,, + y( t T g ( t,,, +z t t,,, T D T f ( T h +. x T f T D 2 ( t,,, + y( t T g ( t,,, +z( t T h ( t,,, (Using bondary conditions and ntegrating by parts = T f ( t, x, x, x + y( t T gt, ( x, x, x +z( t T ht, x, x, x dt T f + ( x T ( t,,, + y( t T g ( t,,, +z( t T h ( t,,, D T f ( t,,, + y( t T g ( t,,, +z( t T h t,,, T f +D 2 ( t,,, + y t +z t t,,, ( T g ( t,,, ( T h dt dt = ( T f ( t, x, x, x + y( t T gt, ( x, x, x + z( t T ht, ( x, x, x (Using (3 This implies ( y( t T gt, ( x, x, x + z( t T ht, ( x, x, x 0 (5 Bt since y( t 0,gt, ( x, x, x 0 and ht, ( x, x, x = 0,t yield, ( y( t T gt, ( x, x, x + z( t T ht, ( x, x, x 0 (6 Combining (5 and (6, we have ( y( t T gt, ( x, x, x + z( t T ht, ( x, x, x = 0 This, becase of ht, ( x, x, x = 0,t, gives y( t T gt, ( x, x, x = 0.

6 6 The Open Operational Research Jornal, 202, Volme 6 Hsain et al. This, together with y( t T gt, ( x, x, x <0, t, implies y( t T gt, ( x, x, x = 0, t Theorem 4.4. Sppose (R : ( y(, t z(, t ( t L D and (L t P (R 2 : y( t T gt,, (, = 0,t (R 3 : T f ( t,.,.,. and ( y( t T gt,.,.,. + z( t T ht,.,.,. are convex Then (is t an optimal soltion of (P and hence of (VP. Proof: f (is t the only feasible soltion of (P, the conclsion is self evident. So, assme that x(is t another feasible soltion of (P. Then by the hypotheses (R and (R 3, we have T f ( t, x, x, x T f ( t,,, ( ( x T T f t,,, T T f ( t,,, + x dt T T f ( t,,, + x Now integrating by parts, we have, = T f ( t,,, x T T f ( t,,, x + x T T f ( t,,, T T f t,,, = T f ( t,,, + x T D T f ( t,,, + x dt (Using bondary conditions ( = T f ( t,,, + x dt ( dt ( x T D T f t,,, ( x T D( T f ( t,,, dt T D T f ( t,,, T T f ( t,,, D( T f ( t,,, + D 2 T f t,,, = ( T f ( t,,, ( dt ( x T y( t T g ( t,,, + z( t T h ( t,,, ( ( T g ( t,,, + z( t T h ( t,,, D y( t T g ( t,,, + z( t T h t,,, +D 2 y t dt Ths, by integrating by parts and sing bondary conditions, as earlier, we get = T f ( t,,, ( ( x T y( t T g ( t,,, + z( t T h t,,, + ( x T ( y( t T g ( t,,, + z( t T h ( t,,, + ( x T y( t T g ( t,,, + z( t T h ( t,,, dt ( T f ( t,,, + y( t T gt,, (, + z( t T ht,, (, dt y( t T gt, ( x, x, x + z( t T ht, ( x, x, x T f ( t,,, (Using hypothesis (A, (A 2 and x L p. This implies that minimizes dt T f ( t, x, x, x over L p. Remark: n Theorem 4.4, we assme that a part of feasible soltion of ( WCD is a feasible soltion of (P. t is a natral qestion if there is any set of appropriate conditions nder which this assmption is tre. The following theorem gives one sch set of conditions. Theorem 4.5. Assme (Q : x(l t p and ( y(, t z(, t ( t L D (Q 2 : gt,.,.,. and ht,.,.,. are differentiable convex fnctions (Q 3 : ( gt, ( (, t (, t ( t + ht, ( (, t (, t ( t = 0 (Q 4 : x + x + x T g t,,, T g (Q 5 : x + x ( t,,, 0, t + x T h t,,, T h ( t,,, 0, t T g ( t,,, T h ( t,,, Then L p. Proof: By the convexity of gt,.,.,. and ht,.,.,., we have gt, ( x, x, x g( t,,, + ( x T g ( t,,, (7 + ( x T g ( t,,, + x ( t,,, 0 T g ht, ( x, x, x h( t,,, + x + ( x T h ( t,,, + x T h ( t,,, T h t,,, 0 (8

7 On Mltiobjective Dality For Variational Problems The Open Operational Research Jornal, 202, Volme 6 7 Using (3 and (4 together with the hypotheses (Q, (Q 2 and (Q 3, we have gt,, (, 0, t (9 and ht,, (, 0, t (20 By (5 and (6, we have gt, ( (, t (, t ( t + ht, ( (, t (, t ( t 0,t (2 The hypothesis (Q 2 with (7 implies gt, ( (, t (, t ( t + ht, ( (, t (, t ( t = 0,t (22 Bt gt,, (, < 0, t. Hence by (22 we have ht,, (, 0, t. (23 The ineqalities (20 and (2 imply ht,, (, = 0, t. (24 The relations (9 and (24 imply that (L t p 5. VARATONAL PROBLEMS WTH NATURAL BOUNDARY VALUES t is possible to constrct variational problems with natral bondary vales rather than the problem with fixed end point considered in the preceding sections. The problems of Section 2 can be formlated as follows: (VP N :Maximize f ' (t, x, x, x,..., f ' (t, x, x, x g(t, x, x, x 0, t h(t, x, x, x = 0, t ( M WD N : Maximize f (t,,, (VP 0 : dt,..., f (t,,, The conditions ( and ( are poplarly known as natral bondary conditions in calcls of variation. Theorems of the section 3 for ( VP N and ( M WD N can easily be proved in view of earlier analysis in this research. The problems of the Section 4 can be written with natral bondary vales as follows: For given 0 < R p. (P N : Minimize T f (t, x, x, xd t (WCD N : Maximize g(t, x, x, x 0, t. h(t, x, x, x = 0, t. T f (t,,, + y(t T g(t,,, +z(t T h(t,,, dt ( T f (t,,, + y(tg (t,,, +z(t T h (t,,, D( T f (t,,, + y(t T g (t,,, +z(t T h (t,,, + D 2 ( T f (t,,, + y(t T g (t,,, +z(t T h (t,,, = 0,t, T f (t,,, + y(t T g (t,,, + z(t T h (t,,, = 0, at t = a,t = b T f (t,,, + y(t T g (t,,, +z(t T h (t,,, = 0, at t = a,t = b > 0, y(t>0,t. 6. MULTOBJECTVE NONLNEAR PROGRAM- MNG PROBLEMS f all the fnctions in the problems ( VP N, (WCD N, and (P N are independent of t, then these problems redce to the following problems: Minimize ( f (, f 2 (,..., f p ( g(x 0, h(x = 0. ( T f (t,,, + y(tg (t,,, +z(t T h (t,,, D T f (t,,, + y(t T g (t,,, + z(t T h (t,,, + D 2 ( T f (t,,, + y(t T g (t,,, +z(t T h (t,,, = 0, t, ( yt ( T gt,, (, + zt ( T ht,, (, > 0 > 0, y(t > 0, t. T f (t,,, + y(t T g (t,,, +z(t T h (t,,, = 0, at t = a,t = b ( T f (t,,, + y(t T g (t,,, +z(t T h (t,,, = 0, at t = a,t = b ( (M WP 0 :Maximize ( f (, f 2 (,..., f p ( T f ( + y T g ( + z T h ( = 0, y T g( + z T h(>0. For given > 0, R p, we have (VP 0 : Minimize T f ( g( 0, h( = 0. (WCD 0 : Maximize T f ( + y T g( + z T h( T f ( + y T g ( + z T h ( = 0, y T g( + z T h(>0. y>0.

8 8 The Open Operational Research Jornal, 202, Volme 6 Hsain et al. Theorems for the pair of Mond-Weir [4] type dal problems (VP 0 and (M WP 0 and Theorems for the pair of Wolf type dal problems (VP 0 and (WCD 0 are simple to be validated, albeit validations of these theorems are not explicitly mentioned in the literatre. ACKNOWLEDGEMENT Declared none. CONFLCT OF NTEREST Declared none. REFERENCES [] Hsain, Jabeen Z. On variational problems involving higher order derivatives. J Math Compt (-2: [2] Hsain, Mattoo, Rmana G. Mltiobjective dality in variational problems with higher order derivatives. Commn Netw 200 2(: [3] Mangasarian OL. Nonlinear programming. New York: McGraw Hill 969. [4] Mond B, Weir T. Generalized concavity and dality. n: Schiable S, Ziemba WT, Eds. Generalized concavity in optimization and economics. New York: Academic Press 98. [5] Mond B, Hanson MA. Dality for variational problems. J Math Anal Appl 967 8: [6] Singh C. Dality theory in mltiobjective differentiable programming. J nf Optim Sci 988 9: [7] Wolfe P. A dality theorem for nonlinear programming. Q Appl Math 96 9: Received: September 22, 20 Revised: March 2, 202 Accepted: March 22, 202 Hsain et al. Licensee Bentham Open. This is an open access article licensed nder the terms of the Creative Commons Attribtion Non-Commercial License ( by-nc/3.0/ which permits nrestricted, non-commercial se, distribtion and reprodction in any medim, provided the work is properly cited.