3 Necessary efficiency conditions for the problem (MFP)

P DI&P DE -constrained optimization problems with curvilinear functional quotients as objective vectors

Ariana Pitea, Constantin Udri¸ste and S¸tefan Mititelu

Abstract. In this work we introduce and perform a study on the multi- time multi-objective fractional variational problem of minimizing a vector of quotients of path independent curvilinear integral functionals (M F P) subject to certain partial differential equations (P DE) and/or partial dif- ferential inequations (P DI), using a geometrical language. The paper is organized as follows: §1 formulates a P DI&P DE-constrained optimiza- tion problem.§2 states and proves necessary conditions for the optimality of the problem (M P) of minimizing a vector of path independent curvi- linear integral functionals constrained by P DIs and P DEs. §3 analyzes necessary efficiency conditions for the problem (M F P), and §4 studies different types of dualities.

M.S.C. 2000: 49J40, 49K20, 58E17, 65K10, 53C65.

Key words:P DI&P DEconstraints, multi-objective fractional variational problem, Pareto optimality, quasiinvexity, duality.

1 P DI&P DE-constrained optimization problem

Let (T, h) and (M, g) be Riemannian manifolds of dimensionspandn, respectively.

The local coordinates onT andM will be writtent= (t^α) andx= (xⁱ), respectively.

LetJ¹(T, M) be the first order jet bundle associated to T andM.

To develop our theory, we recall the following relations between two vectorsv = (v^j) andw= (w^j),j = 1, a:

v=w ⇔ v^j=w^j, j= 1, a; v < w ⇔ v^j< w^j, j= 1, a;

v <=w ⇔ v^j≤w^j, j= 1, a (product order relation).

v≤w ⇔ v <=wandv6=w.

Using the product order relation on R^p, the hyperparallelepiped Ωt0,t1 ⊂R^p, with the diagonal opposite pointst0= (t¹₀, . . . , t^p₀) andt1= (t¹₁, . . . , t^p₁), can be written as the interval [t0, t1].

Balkan Journal of Geometry and Its Applications, Vol.14, No.2, 2009, pp. 65-78.∗

c

°Balkan Society of Geometers, Geometry Balkan Press 2009.

Now, we introduce theC^∞-class Lagrange 1-forms densities

fα= (f_α^{`</sup>) :J<sup>1</sup>(T, M)→R<sup>r</sup>, kα= (k<sub>α</sub><sup>`}) :J¹(T, M)→R^r, `= 1, r, α= 1, p.

Suppose thatDβf_α^{`</sup> =Dαf<sub>β</sub><sup>`}, and Dβk^{`</sup><sub>α</sub>=Dαk<sup>`}_β,α, β= 1, p, α6=β, `= 1, r, where Dβis the total derivative (closeness conditions, complete integrability conditions) and

Z

γt0,t1

k_α^`(t, x(t), xγ(t))dt^α>0,

wherexγ(t) = ∂x

∂t^γ(t),γ= 1, p, are partial velocities andγt0,t1 is a piecewiseC¹-class curve joining the pointst0 and t1. The closed Lagrange 1-forms densitiesf_α^{`</sup> andk<sup>`}_α will be used to define certain quotients of curvilinear integral functionals. Also we accept that the Lagrange matrix density

g= (g^b_a) :J¹(T, M)→R^ms, a= 1, s, b= 1, m, m < n, ofC^∞-class defines the partial differential inequations (P DI) (of evolution) (1.1) g(t, x(t), xγ(t))<= 0, t∈Ωt0,t1,

and the Lagrange matrix density

h= (h^b_a) :J¹(T, M)→R^qs, a= 1, s, b= 1, q, q < n, defines the partial differential equation (P DE) (of evolution)

(1.2) h(t, x(t), xγ(t)) = 0, t∈Ωt0,t1.

The purpose of this work is to study the multitime multi-objective fractional vari- ational problem of minimizing a vector of quotients of path independent curvilinear functionals





 Z

γt0,t1

f_α¹(t, x(t), xγ(t))dt^α Z

γt0,t1

k_α¹(t, x(t), xγ(t))dt^α , . . . ,

Z

γt0,t1

f_α^r(t, x(t), xγ(t))dt^α Z

γt0,t1

k^r_α(t, x(t), xγ(t))dt^α





,

knowing that the functionx(t) satisfies the boundary conditionsx(t0) =x0,x(t1) = x1, or x(t)|∂Ωt0,t1 = given, the partial differential inequations of evolution (1.1), and the partial differential equation of evolution (1.2). Such a problem is called P DI&P DE-constrained optimization problem(see also [2]-[8], [10], [11], [19], [20]).

Introducing the notations F^`(x(·)) =

Z

γt0,t1

f_α^{`</sup>(t, x(t), xγ(t))dt<sup>α</sup>, K<sup>`}(x(·)) = Z

γt0,t1

k^`_α(t, x(t), xγ(t))dt^α, the above-mentioned extremizing problem can be written

(M F P)















 minx(·)

µF¹(x(·))

K¹(x(·)), . . . ,F^r(x(·)) K^r(x(·))

¶

subject to

x(t0) =x0, x(t1) =x1, g(t, x(t), xγ(t))<

= 0, t∈Ωt0,t1, h(t, x(t), xγ(t)) = 0, t∈Ωt0,t1.

LetC^∞(Ωt0,t1, M) be the space of all functionsx: Ωt0,t1 →M of C^∞-class, with the norm

kxk=kxk∞+ Xp

α=1

kxαk∞. The set

F(Ωt0,t1) ={x∈C^∞(Ωt0,t1, M)|x(t0) =x0, x(t1) =x1, g(t, x(t), xγ(t))<= 0, h(t, x(t), xγ(t)) = 0, t∈Ωt0,t1}

is called theset of all feasible solutionsof the problem (M F P).

Partial differential inequations/equations mathematically represent a multitude of natural phenomena, and in turn, applications in science and engineering ubiquitously give rise to problems formulated asP DI&P DE-constrained optimization. The areas of research who strongly motivate theP DI&P DE-constrained optimization include:

shape optimization in fluid mechanics and medicine, material inversion - in geophysics, data assimilation in regional weather prediction modelling, structural optimization, and optimal control of processes.P DI&P DE-constrained optimization problems are generally infinite dimensional in nature, large and complex. As a result, this class of optimization problems present significant reasoning and computational challenges, many of which have been studied in recent years in Germany, USA, Romania, etc.

As computing power grows and optimization techniques become more advanced, one wonders whether there are enough commonalities among P DI&P DE-constrained optimization problems from different fields to develop ratiocinations and algorithms for more than a single application. This question has been the topic of many papers, conferences and recent scientific grants.

The basic optimization problems of path independent curvilinear integrals with P DE constraints or with isoperimetric constraints, expressed by the multiple inte- grals or path independent curvilinear integrals, were stated for the first time in our works [12]-[18]. The papers [15], [17], [18] focuss onmultitime maximum principlein multitime optimal control problems.

2 Necessary conditions of optimality

In order to obtain necessary conditions for the optimality of the problem (M F P), we start with a vector of path independent curvilinear functionals,

F(x(·)) = Z

γt0,t1

fα(t, x(t), xγ(t))dt^α= µ Z

γt0,t1

f_α¹(t, x(t), xγ(t))dt^α, . . . , Z

γt0,t1

f_α^r(t, x(t), xγ(t))dt^α

¶

= (F¹(x(·)), . . . , F^r(x(·))),

and we formulate a simplifiedP DI&P DE-constrained minimum problem

(M P)















minx(·) F(x(·)) subject to

x(t₀) =x₀, x(t₁) =x₁, g(t, x(t), xγ(t))<

= 0, t∈Ωt0,t1, h(t, x(t), xγ(t)) = 0, t∈Ωt0,t1.

We are interested in finding necessary conditions for the optimality, respectively efficiency conditions, for the problem (M P) in the domainF(Ωt0,t1).

Definition 2.1. A feasible solution x^◦(·)∈ F(Ωt0,t1) is calledefficient pointfor the program (M P) if and only if for any feasible solutionx(·)∈ F(Ωt0,t1), the inequality F(x(·))<

=F(x^◦(·)) implies the equalityF(x(·)) =F(x^◦(·)).

To analyze the previous problem, we start with the case of a single functional.

Let s = (sα) : J¹(Ωt0,t1, M) → R^p be a closed Lagrange 1-form density of C^∞- class which produces the action S(x(·)) =

Z

γt0,t1

sβ(t, x(t), xγ(t))dt^β. Consider the followingP DI&P DE-constrained variational problem

(SP)











minx(·) S(x(·)) subject to g(t, x(t), xγ(t))<

= 0, t∈Ωt0,t1, h(t, x(t), xγ(t)) = 0, t∈Ωt0,t1. We define the auxiliary Lagrange density 1-formL= (Lα) as

Lα(t, x(t), xγ(t), λ, µ(t), ν(t)) =λsα(t, x(t), xγ(t))+< µα(t), g(t, x(t), xγ(t))>

+< να(t), h(t, x(t), xγ(t))>, α= 1, p,

where λ is real number and µ(t) = (µα(t)) = (µ^a_αb(t)), ν = (να(t)) = (ν_αb^a (t)) are Lagrange multipliers subject to the condition that the 1-form L = (Lα) is closed.

Extending the results in [15], [18], the necessary conditions for the optimality of a feasible solutionx^◦(·)∈ F(Ωt0,t1) in the problem (SP) are











∂Lα

∂x (t, x^◦(t), x^◦_γ(t))−Dγ∂Lα

∂xγ(t, x^◦(t), x^◦_γ(t)) = 0, α= 1, p (Euler-Lagrange PDE)

< µα(t), g(t, x^◦(t), x^◦_γ(t))>= 0, t∈Ωt0,t1, α= 1, p, µα(t)>

= 0, t∈Ωt0,t1, α= 1, p.

Definition 2.2. Ifλ6= 0, the optimal feasible solutionx^◦(·) of the problem (SP) is callednormal.

Without loss of generality, if x^◦(·) is an optimal normal solution of the problem (SP), we can assume thatλ= 1.

The following Theorem describes the previous necessary optimality conditions in the language of [4], [15], [18], [19].

Theorem 2.3. Lets= (sα)be a closed1-form ofC^∞-class. Ifx^◦(·)∈ F(Ωt0,t1)is a normal optimal solution of the problem (SP), then there exist the multipliersλ,µ(t), ν(t)satisfying the following conditions:

(V C)





































λ∂sα

∂x (t, x^◦(t), x^◦_γ(t))+< µα(t),∂g

∂x(t, x^◦(t), x^◦_γ(t))>

+< να(t),∂h

∂x(t, x^◦(t), x^◦_γ(t))>−Dγ

µ λ∂sα

∂xγ(t, x^◦(t), x^◦_γ(t)) +< µα(t), ∂g

∂xγ(t, x^◦(t), x^◦_γ(t))>+< να(t), ∂h

∂xγ(t, x^◦(t), x^◦_γ(t))>

¶

= 0, t∈Ωt0,t1, α= 1, p (Euler-Lagrange PDEs)

< µα(t), g(t, x^◦(t), x^◦_γ(t))>= 0, t∈Ωt0,t1, α= 1, p, µα(t)>

= 0, t∈Ωt0,t1, α= 1, p, [λ= 1].

Now we turn back to the vector problem (M P). To develop further our theory, we need the result in the following Lemma (for the single-time case, see [3]).

Lemma 2.4. The function x^◦(·)∈ F(Ωt0,t1) is an efficient solution of the problem (MP) if and only if x^◦(·) is an optimal solution of each scalar problem P`(x^◦(·)),

`= 1, r, where

P`(x^◦)



















minx(·) F^`(x(·)) subject to

x(t₀) =x₀, x(t₁) =x₁, g(t, x(t), xγ(t))<

= 0, t∈Ωt₀,t₁, h(t, x(t), xγ(t)) = 0, t∈Ωt0,t1, F^j(x(·))≤F^j(x^◦(·)), j = 1, r, j6=`.

Proof. In order to prove the direct implication, we suppose that the func- tion x^◦(·)∈ F(Ω_t0,t1) is an efficient solution of the problem (M P) and there is k∈ {1, . . . , r} such that x^◦(·) ∈ F(Ωt0,t1) is not an optimal solution of the scalar problemPk(x^◦(·)). Then there exists a functiony(·)∈ F(Ωt0,t1) such that

F^j(y(·))≤F^j(x^◦(·)), j= 1, r, j6=k; F^k(y(·))< F^k(x^◦(·)).

These relations contradict the efficiency of the function x^◦(·) ∈ F(Ωt0,t1) for the problem (M P). Consequently, the pointx^◦(·)∈ F(Ωt0,t1) is an optimal solution for each programP`(x<sup>◦</sup>(·)),`= 1, r.

Conversely, let us consider that the function x^◦(·)∈ F(Ωt0,t1) is an optimal so- lution of all problemsP`(x<sup>◦</sup>(·)), ` = 1, r. Suppose that x^◦(·) ∈ F(Ωt0,t1) is not an efficient solution of the problem (M P). Then there exists a functiony(·)∈ F(Ωt0,t1) such that F^j(y(·))≤F^j(x^◦(·)), j = 1, r, and there is k ∈ {1, . . . , r} such that F^k(y(·)) < F^k(x^◦(·)). This is a contradiction to the assumption that the function x^◦(·)∈ F(Ωt0,t1) minimizes the functionalF^k(x(·)) on the set of all feasible solutions of problemP_k(x^◦(·)). Therefore, the functionx^◦(·)∈ F(Ω_t0,t1) is an efficient solution of the problem (M P)

Lemma 2.5. Let ` be fixed between 1 and r. If the function x<sup>◦</sup>(·)∈ F(Ωt0,t1) is a [normal] optimal solution of the scalar problemP`(x^◦(·)), then there exist the real vec- tors(λj`),j= 1, r, and the matrix functionsµ`,ν`, such that the following conditions are satisfied

λj`∂f_α^j

∂x (t, x^◦(t), x^◦_γ(t))+< µ`α(t),∂g

∂x(t, x^◦(t), x^◦_γ(t))>

+< ν`α(t),∂h

∂x(t, x^◦(t), x^◦_γ(t))>−Dγ

µ λj`∂f_α^j

∂xγ(t, x^◦(t), x^◦_γ(t)) +< µ`α(t), ∂g

∂xγ

(t, x^◦(t), x^◦_γ(t))>+< ν`α(t), ∂h

∂xγ

(t, x^◦_γ(t), x^◦_γ(t))>

¶

= 0, t∈Ωt0,t1, α= 1, p (Euler-Lagrange PDEs)

< µ`α(t), g(t, x<sup>◦</sup>(t), x<sup>◦</sup><sub>γ</sub>(t))>= 0, t∈Ωt0,t1, α= 1, p, µ`α(t)>

= 0, t∈Ωt0,t1, α= 1, p, λj`≥0 [λ``= 1].

For a proof, see [9].

Definition 2.6. The function x^◦(·)∈ F(Ωt0,t1) is callednormal efficient solution of the problem (MP) if it is normal optimal solution for at least one of the problems P`(x<sup>◦</sup>(·)),`= 1, r.

It follows the main result of this section.

Theorem 2.7. Ifx^◦(·)∈ F(Ωt0,t1)is a normal efficient solution of the problem (MP), then there exist a vector λ^◦ ∈R^r and the smooth matrix functions µ^◦(t) = (µ^◦_α(t)), ν^◦(t) = (ν_α^◦(t)), which satisfy the following conditions

(M V)









































< λ^◦,∂fα

∂x(t, x^◦(t), x^◦_γ(t))>+< µ^◦_α(t),∂g

∂x(t, x^◦(t), x^◦_γ(t))>

+< ν^◦_α(t),∂h

∂x(t, x^◦(t), x^◦_γ(t))>−Dγ µ

< λ^◦,∂fα

∂xγ(t, x^◦(t), x^◦_γ(t))>

+< µ^◦_α(t), ∂g

∂xγ(t, x^◦(t), x^◦_γ(t))>+< ν_α^◦(t), ∂h

∂xγ(t, x^◦(t), x^◦_γ(t))>

¶

= 0, t∈Ωt0,t1, α= 1, p (Euler-Lagrange PDEs)

< µ^◦_α(t), g(t, x^◦(t), x^◦_γ(t))>= 0, t∈Ωt0,t1, α= 1, p, µ^◦_α(t)>= 0, t∈Ωt0,t1, α= 1, p,

λ^◦≥0,

< e, λ^◦>= 1, e= (1, . . . ,1)∈R^r.

Proof. If the functionx^◦(·)∈ F(Ωt0,t1) is a [normal] efficient solution of the prob- lem (M P), according to Lemma 2.4, the pointx^◦(·)∈ F(Ωt0,t1) is a normal efficient solution of each scalar problemP`(x<sup>◦</sup>(·)), ` = 1, r. According to Lemma 2.5, there exist the matrix λ<sub>j`</sub>, j, `= 1, r, and the functions µ_{`α</sub>, ν<sub>`α}, satisfying the following conditions

(F V)`







































λj`

∂f_α^j

∂x (t, x^◦(t), x^◦_γ(t))+< µ`α(t),∂g

∂x(t, x^◦(t), x^◦_γ(t))>

+< ν`α(t),∂h

∂x(t, x^◦(t), x^◦_γ(t))>−Dγ

µ λj`∂f_α^j

∂x_γ(t, x^◦(t), x^◦_γ(t)) +< µ`α(t), ∂g

∂xγ

(t, x^◦(t), x^◦_γ(t))>+< ν`α, ∂h

∂xγ

(t, x^◦(t), x^◦_γ(t))>

¶

= 0, t∈Ωt0,t1, α= 1, p (Euler-Lagrange PDEs)

< µ`α(t), g(t, x<sup>◦</sup>(t), x<sup>◦</sup><sub>γ</sub>(t))>= 0, t∈Ωt0,t1, α= 1, p, µ`α(t)>= 0, t∈Ωt0,t1, α= 1, p,

λj`≥0 [λ``= 1].

Making the sum of all relations (F V)` from`= 1 to`=rand denoting Λj=

Xr

`=1

λj`, Mα(t) = Xr

`=1

µ`α(t), Nα(t) = Xr

`=1

ν`α(t), the following relations are obtained

(F V)





































Λj∂f_α^j

∂x (t, x^◦(t), x^◦_γ(t))+< Mα(t),∂g

∂x(t, x^◦(t), x^◦_γ(t))>

+< Nα(t),∂h

∂x(t, x^◦(t), x^◦_γ(t))>−Dγ

µ Λj∂f_α^j

∂xγ(t, x^◦(t), x^◦_γ(t)) +< Mα(t), ∂g

∂xγ(t, x^◦(t), x^◦_γ(t))>+< Nα(t), ∂h

∂xγ(t, x^◦(t), x^◦_γ(t))>

¶

= 0, t∈Ωt0,t1, α= 1, p (Euler-Lagrange PDEs)

< Mα(t), g(t, x^◦(t), x^◦_γ(t))>= 0, t∈Ωt0,t1, α= 1, p, Mα(t)>= 0, t∈Ωt0,t1, α= 1, p,

λj≥0, j= 1, r.

We divide the relations (F V) byS= Xr

j=1

Λj ≥1 and we denote

Λ^◦_j =Λj

S , M_α^◦(t) =Mα(t)

S , N_α^◦(t) = Nα(t)

S .

Thus we obtain the relations from the statement

3 Necessary efficiency conditions for the problem (MFP)

Consider x^◦(·)∈ F(Ωt0,t1) being a feasible solution of the problem (M F P) and for each indexj between 1 andr, let us introduce the real number

R_◦^j(x^◦(·)) = F^j(x^◦(·)) K^j(x^◦(·)).

For a fixed index`, we consider the following pair of extremizing problems

(F P R)`



















 minx(·)

F^{`</sup>(x(·)) K<sup>`}(x(·)) subject to

x(t0) =x0, x(t1) =x1

g(t, x(t), xγ(t))<= 0, t∈Ωt0,t1, h(t, x(t), xγ(t)) = 0, t∈Ωt0,t1, F^j(x(·))−R◦^j(x^◦(·))K^j(x(·))<

= 0, j= 1, r, j6=`.

(SP R)`



















minx(·) F^{`</sup>(x(·))−R<sup>`}_◦(x^◦(·))K^`(x(·)) subject to

x(t0) =x0, x(t1) =x1

g(t, x(t), xγ(t))<

= 0, t∈Ωt0,t1, h(t, x(t), xγ(t)) = 0, t∈Ωt0,t1,

F^j(x(·))−R^j◦(x^◦(·))K^j(x(·))<= 0, j = 1, r, j6=`, With the statements of Jagannathan [3], we have

Lemma 3.1. The functionx^◦(·)∈ F(Ωt0,t1) is optimal in(F P R)` if and only if it is optimal in(SP R)`,`= 1, r.

Using Lemma 2.4 and Lemma 3.1, we can formulate

Theorem 3.2. The functionx^◦(·)∈ F(Ωt0,t1)is an efficient solution for the problem (MFP) if and only if it is an optimal solution for each problem(SP R)`,`= 1, r.

Proof. We shall prove this statement using the double implication.

The necessity. Let us suppose that the function x^◦(·) ∈ F(Ωt0,t1) is efficient for the problem (M F P). Then it is optimal for problem (F P R)`,`= 1, r, according to Lemma 2.4. Also, for any ` = 1, r, if the function x<sup>◦</sup>(·) ∈ F(Ωt0,t1) is optimal for problem (F P R)`, then it is optimal for problem (SP R)`, ` = 1, r (according to Lemma 3.1).

The sufficiency. Let us suppose that the function x^◦(·) ∈ F(Ω_t0,t1) is effi- cient for the problem (SP R)`, for all`= 1, r. Then it is optimal for problem (F P R)`,

`= 1, r, according to Lemma 3.1. Also, for any`= 1, r, the functionx^◦(·)∈ F(Ωt0,t1) is optimal for problem (F P R)`, therefore it is optimal for problem (M P F) (according to Lemma 2.4)

Remark. The function x^◦(·) ∈ F(Ωt0,t1) is a normal efficient solution of the problem (M F P) if it is a normal optimal solution for at least one of the scalar problems (F P R)`,`= 1, r.

Consider λ = (λ1, . . . , λr) ∈ R^r and the matrix functions µ: Ωt0,t1 →R^msp, ν: Ωt0,t1 →R^qspsuch that the auxiliary Lagrange 1-formL= (Lα),

Lα(t, x(t), xγ(t), λ, µ(t), ν(t)) =λj

¡f_α^j(t, x(t), xγ(t))−R^j_◦(x^◦(·))k_α^j(t, x(t), xγ(t))¢ +< µα(t), g(t, x(t), xγ(t))>

+< να(t), h(t, x(t), xγ(t))>, α= 1, p,

be closed. Having in mind the background introduced above, we can state the main results of this section. First of all, we shall introduce our necessary efficiency condi- tions.

Theorem 3.3 (Necessary efficiency conditions). Let the function x^◦(·) ∈ F(Ωt0,t1)be a normal efficient solution of problem(M F P). Then there existΛ^1◦,Λ^2◦∈ R^r and the smooth functions M^◦: Ωt0,t1 →R^msp, N^◦: Ωt0,t1 →R^spq, such that we have

(M F V)























































Λ^1◦_j ∂f_α^j

∂x (t, x^◦(t), x^◦_γ(t))−Λ^2◦_j ∂k^j_α

∂x (t, x^◦(t), x^◦_γ(t)) +< M_α^◦(t),∂g

∂x(t, x^◦(t), x^◦_γ(t))>+< N_α^◦(t),∂h

∂x(t, x^◦(t), x^◦_γ(t))>

−Dγ

½ Λ^1◦_j ∂f_α^j

∂xγ(t, x^◦(t), x^◦_γ(t))−Λ^2◦_j ∂k_α^j

∂xγ(t, x^◦(t), x^◦_γ(t)) +< M_α^◦(t, x^◦(t), x^◦_γ(t)), ∂g

∂xγ(t, x^◦(t), x^◦_γ(t))>

+< N_α^◦(t), ∂h

∂xγ(t, x^◦(t), x^◦_γ(t))>

¾

= 0,

t∈Ωt0,t1, α= 1, p (Euler-Lagrange PDEs)

< M_α^◦(t), g(t, x^◦(t), x^◦_γ(t))>= 0, t∈Ωt0,t1, α= 1, p, M_α^◦(t)>

= 0, t∈Ωt0,t1, α= 1, p,

Λ^1◦≥0, < e,Λ^1◦>= 1, e= (1, . . . ,1)∈R^r.

Proof. There areλ¹_{j`</sub>,λ<sup>2</sup><sub>j`},j= 1, r, and the functionsµ`α(t),ν`α(t), such that

(F V)`















































λ¹_j`∂f_α^j

∂x (t, x^◦(t), x^◦_γ(t))−λ²_j`∂k^j_α

∂x (t, x^◦(t), x^◦_γ(t))

¸

+< µ`α(t),∂g

∂x(t, x^◦(t), x^◦_γ(t))>+< ν`α(t),∂h

∂x(t, x^◦(t), x^◦_γ(t))>

−Dγ

½ λ¹_j`∂f_α^j

∂xγ(t, x^◦(t), x^◦_γ(t))−λ²_j`∂k^j_α

∂xγ(t, x^◦(t), x^◦_γ(t)) +< µ`α(t), ∂g

∂xγ

(t, x^◦(t), x^◦_γ(t))>+< ν`α(t), ∂h

∂xγ

(t, x^◦(t), x^◦_γ(t))>

¾

= 0, t∈Ωt0,t1, α= 1, p (Euler-Lagrange PDEs)

< µ`α(t), g(t, x<sup>◦</sup>(t), x<sup>◦</sup><sub>γ</sub>(t))>= 0, t∈Ωt0,t1, α= 1, p, µ`α(t)>

= 0, t∈Ωt0,t1, α= 1, p, λj`≥0 [λ``= 1].

We make the sum of all relations (F V)` after`= 1, rand denoting Λ¯¹_j =

Xr

`=1

λ¹_j`, Λ¯²_j = Xr

`=1

λ²_j`, M¯α(t) = Xr

`=1

µ`α(t), N¯α(t) = Xr

`=1

ν`α(t) we obtain

(F V)















































Λ¯¹_j∂f_α^j

∂x(t, x^◦(t), x^◦_γ(t))−Λ¯²_j∂k^j_α

∂x (t, x^◦(t), x^◦_γ(t)) +<M¯α(t),∂g

∂x(t, x^◦(t), x^◦_γ(t))>+<N¯α(t),∂h

∂x(t, x^◦(t), x^◦_γ(t))>

−Dγ

½ Λ¯¹_j

·∂f_α^j

∂xγ(t, x^◦(t), x^◦_γ(t))−Λ¯²_j∂k_α^j

∂xγ(t, x^◦(t), x^◦_γ(t)) +<M¯α(t), ∂g

∂xγ(t, x^◦(t), x^◦_γ(t))>+<N¯α(t), ∂h

∂xγ(t, x^◦(t), x^◦_γ(t))>

¾

= 0, t∈Ωt0,t1, α= 1, p (Euler-Lagrange PDEs)

<M¯α(t), g(t, x^◦(t), x^◦_γ(t))>= 0, t∈Ωt0,t1, α= 1, p, M¯α(t)>

= 0, t∈Ωt0,t1, α= 1, p, Λ¯¹_j ≥0.

Dividing the relations (F V) byS= Xr

j=1

Λ¯¹_j ≥1 and denoting

Λ^1◦_j =Λ¯¹_j

S , Λ^2◦_j = Λ¯²_j

S , M_α^◦(t) =M¯α(t)

S , N_α^◦(t) = N¯α(t)

S ,

the relations (F V) take the form (M F V)

4 A dual program theory

Letρ be a real number andb:C^∞(Ωt0,t1, M)×C^∞(Ωt0,t1, M)→[0,∞) a func- tional. Leta = (aα) be a closed Lagrange 1-form. We associate the path indepen- dent curvilinear functional A(x(·)) =

Z

γt0,t1

aα(t, x(t), xγ(t))dt^α. The definition of the quasiinvexity (see also [2], [7], [10], [11], [20]) helps us to state the results included in this section.

Definition 4.1. The functional A is called [strictly] (ρ, b)-quasiinvex at the point x^◦(·) if there is a vector function η: J¹(Ωt0,t1, M)×J¹(Ωt0,t1, M) → Rⁿ, vanish- ing at the point (t, x^◦(t), x^◦_γ(t), x^◦(t), x^◦_γ(t)), and the functional θ:C^∞(Ωt0,t1, M)× C^∞(Ωt0,t1, M)→Rⁿ,such that for any x(·) [x(·)6=x^◦(·)], the following implication holds

(A(x(·))<=A(x^◦(·)))→ Ã

b(x(·), x^◦(·)) Z

γt0,t1

{< η(t, x(t), xγ(t), x^◦(t), x^◦_γ(t)),

∂aα

∂x (t, x^◦(t), x^◦_γ(t))>+< Dγη(t, x(t), xγ(t), x^◦(t), x^◦_γ(t)),

∂aα

∂xγ(t, x^◦(t), x^◦_γ(t))>}dt^α[<]<= −ρb(x(·), x^◦(·))kθ(x(·), x^◦(·))k²

¶ . We associate a multi-objective variational dual problem to the problem (M F P), preserving the same set of feasible solutions:

(M F D)





























































 maxy(·)

µF¹(y(·))

K¹(y(·)), . . . ,F^r(y(·)) K^r(y(·))

¶

subject to

y(t0) =x0, y(t1) =x1

Λ^1◦<sub>`</sub> ∂f<sub>α</sub><sup>`</sup>

∂y (t, y(t), yγ(t))−Λ^2◦<sub>`</sub> ∂k<sub>α</sub><sup>`</sup>

∂y (t, y(t), yγ(t)) +< µα(t),∂g

∂y(t, y(t), yγ(t))>+< να(t),∂h

∂y(t, y(t), yγ(t))>

−Dγ

½ Λ^1◦<sub>`</sub> ∂f<sub>α</sub><sup>`</sup>

∂yγ(t, y(t), yγ(t))−Λ^2◦<sub>`</sub> ∂k<sub>α</sub><sup>`</sup>

∂yγ(t, y(t), yγ(t)) +< µα(t), ∂g

∂yγ(t, y(t), yγ(t))>+< να(t), ∂h

∂yγ(t, y(t), yγ(t))>

¾

= 0, t∈Ωt0,t1, α= 1, p

< µα(t), g(t, y(t), yγ(t))>+< να(t), h(t, y(t), yγ(t))> >= 0, α= 1, p, t∈Ωt0,t1

Λ^1◦≥0, < e,Λ^1◦>= 1, e= (1, . . . ,1)∈R^r.

To formulate our original results, we use the minimizing functional vectorπ(x(·)) of the problem (M F D) at the pointx(·)∈ F(Ωt0,t1) and the maximizing functional vector

δ(y(·), yγ(·),Λ^1◦,Λ^2◦, µ(·), ν(·)) of the dual problem (M F D) at

(y(·), yγ(·),Λ^1◦,Λ^2◦, µ(·), ν(·))∈∆, where ∆ is the domain of the problem (M F D).

Theorem 4.2 (Weak duality). Let x^◦(·) be a feasible solution of the problem (M F P)andy(·)be a normal efficient solution of the dual problem(M F D). Assume that the following conditions are fulfilled:

a) Λ^1◦_{`</sub> >0,Λ<sup>2◦</sup><sub>`} >0,`= 1, r, Λ<sup>1◦</sup><sub>`</sub> F<sup>`</sup>(y(·))−Λ<sup>2◦</sup><sub>`</sub> K^`(y(·)) = 0;

b) for any` = 1, r, the functionalF<sup>`</sup>(x(·))is (ρ^{0`</sup>, b)-quasiinvex at the point y(·) and−K<sup>`}(x(·))is(ρ^00`, b)-quasiinvex at the pointy(·)with respect toη andθ;

c) the functional Z

γt0,t1

[< µα(t), g(t, x(t), xγ(t))>+< να(t), h(t, x(t), xγ(t))>]dt^α is(ρ⁰⁰⁰, b)- quasiinvex aty(·)with respect toη andθ;

d)one of the functionals of b),c)is strictly (ρ^0`, b)-quasiinvex;

e) ρ<sup>0`</sup>Λ<sup>1◦</sup><sub>`</sub> +ρ<sup>00`</sup>Λ<sup>2◦</sup><sub>`</sub> +ρ⁰⁰⁰ >

= 0.

Then, the inequalityπ(x^◦(·))≤δ(y(·), yγ(·),Λ^1◦,Λ^2◦, µ(·), ν(·))is false.

The proof will be given in a further paper (see also, [9]).

Theorem 4.3 (Direct duality). Letx^◦(·)∈ F(Ω_t0,t1)be a normal efficient solution of(M F P)and suppose that the hypotheses of Theorem4.2are satisfied. Then there are