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= (tα) 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 = (vj) andw= (wj),j = 1, a:
v=w ⇔ vj=wj, j= 1, a; v < w ⇔ vj< wj, j= 1, a;
v <=w ⇔ vj≤wj, j= 1, a (product order relation).
v≤w ⇔ v <=wandv6=w.
Using the product order relation on Rp, the hyperparallelepiped Ωt0,t1 ⊂R p, with
the diagonal opposite pointst0= (t10, . . . , tp0) andt1= (t11, . . . , tp1), can be written as
the interval [t0, t1].
∗
Balkan Journal of Geometry and Its Applications, Vol.14, No.2, 2009, pp. 75-88. c
Now, we introduce theC∞
-class Lagrange 1-forms densities
fα= (fαℓ) :J1(T, M)→Rr, kα= (kαℓ) :J1(T, M)→Rr, ℓ= 1, r, α= 1, p.
Suppose thatDβfαℓ =Dαfβℓ, and Dβkℓα=Dαkℓβ,α, β= 1, p, α6=β, ℓ= 1, r, where
Dβis the total derivative (closeness conditions, complete integrability conditions) and
Z
γt0,t1
kαℓ(t, x(t), xγ(t))dtα>0,
wherexγ(t) =
∂x
∂tγ(t),γ= 1, p, are partial velocities andγt0,t1 is a piecewiseC
1-class
curve joining the pointst0 and t1. The closed Lagrange 1-forms densitiesfαℓ andkℓα
will be used to define certain quotients of curvilinear integral functionals. Also we accept that the Lagrange matrix density
g= (gba) :J1(T, M)→Rms, 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= (hb a) :J
1(T, M)→Rqs
, 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
fα1(t, x(t), xγ(t))dtα
Z
γt0,t1
kα1(t, x(t), xγ(t))dtα
, . . . ,
Z
γt0,t1
fαr(t, x(t), xγ(t))dtα
Z
γt0,t1
krα(t, x(t), xγ(t))dtα
,
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
Fℓ(x(·)) =
Z
γt0,t1
fαℓ(t, x(t), xγ(t))dtα, Kℓ(x(·)) =
Z
γt0,t1
kℓα(t, x(t), xγ(t))dtα,
(M F P)
min
x(·)
µ
F1(x(·))
K1(x(·)), . . . ,
Fr(x(·))
Kr(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∞+
p
X
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 inteinte-grals, 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))dtα=
µ Z
γt0,t1
fα1(t, x(t), xγ(t))dtα, . . . ,
Z
γt0,t1
fαr(t, x(t), xγ(t))dtα
¶
and we formulate a simplifiedP DI&P DE-constrained minimum problem
(M P)
min
x(·) 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))dtβ. Consider the
followingP DI&P DE-constrained variational problem
(SP)
min
x(·) 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)) = (µaαb(t)), ν = (να(t)) = (ναba (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.
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 Pℓ(x◦(·)),
ℓ= 1, r, where
Pℓ(x
◦
)
min
x(·) F
ℓ(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,
Fj(x(·))≤Fj(x◦
(·)), j = 1, r, j6=ℓ.
Proof. In order to prove the direct implication, we suppose that the func-tion x◦
(·)∈ F(Ωt0,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
Fj(y(·))≤Fj(x◦
(·)), j= 1, r, j6=k; Fk(y(·))< Fk(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 programPℓ(x◦(·)),ℓ= 1, r.
Conversely, let us consider that the function x◦
(·)∈ F(Ωt0,t1) is an optimal
so-lution of all problemsPℓ(x◦(·)), ℓ = 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 Fj(y(·))≤Fj(x◦
(·)), j = 1, r, and there is k ∈ {1, . . . , r} such that Fk(y(·)) < Fk(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 problemPk(x◦(·)). Therefore, the functionx◦(·)∈ F(Ωt0,t1) is an efficient solution
Lemma 2.5. Let ℓ be fixed between 1 and r. If the function x◦
(·)∈ F(Ωt0,t1) is a [normal] optimal solution of the scalar problemPℓ(x◦(·)), then there exist the real vec-tors(λjℓ),j= 1, r, and the matrix functionsµℓ,νℓ, such that the following conditions
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 Pℓ(x◦(·)),ℓ= 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◦
∈Rr and the smooth matrix functions µ◦
(t) = (µ◦
α(t)),
ν◦
(t) = (ν◦
α(t)), which satisfy the following conditions
(F V)ℓ
the following relations are obtained
(F V) 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
For a fixed indexℓ, we consider the following pair of extremizing problems
(F P R)ℓ
min
x(·)
Fℓ(x(·))
Kℓ(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,
Fj(x(·))−Rj
◦(x ◦
(·))Kj(x(·))<
= 0, j= 1, r, j6=ℓ.
(SP R)ℓ
min
x(·) F
ℓ(x(·))−Rℓ
◦(x ◦
(·))Kℓ(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,
Fj(x(·))−Rj◦(x ◦
(·))Kj(x(·))<
= 0, j = 1, r, j6=ℓ, With the statements of Jagannathan [3], we have
Lemma 3.1. The functionx◦
(·)∈ F(Ωt0,t1) is optimal in(F P R)ℓ if and only if it is optimal in(SP R)ℓ,ℓ= 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)ℓ,ℓ= 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)ℓ,ℓ= 1, r, according
to Lemma 2.4. Also, for any ℓ = 1, r, if the function x◦
(·) ∈ F(Ωt0,t1) is optimal
for problem (F P R)ℓ, then it is optimal for problem (SP R)ℓ, ℓ = 1, r (according to
Lemma 3.1).
The sufficiency. Let us suppose that the function x◦
(·) ∈ F(Ωt0,t1) is
effi-cient for the problem (SP R)ℓ, for allℓ= 1, r. Then it is optimal for problem (F P R)ℓ,
ℓ= 1, r, according to Lemma 3.1. Also, for anyℓ= 1, r, the functionx◦
(·)∈ F(Ωt0,t1)
is optimal for problem (F P R)ℓ, 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)ℓ,ℓ= 1, r.
Consider λ = (λ1, . . . , λr) ∈ Rr and the matrix functions µ: Ωt0,t1 →R msp,
ν: Ωt0,t1 →R
qspsuch that the auxiliary Lagrange 1-form
L= (Lα),
Lα(t, x(t), xγ(t), λ, µ(t), ν(t)) =λj
¡
fαj(t, x(t), xγ(t))−Rj◦(x ◦
(·))kj
α(t, x(t), xγ(t))
¢
+< µα(t), g(t, x(t), xγ(t))>
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◦
,Λ2◦
∈
Rr and the smooth functions 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))dtα. 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◦
(·)], the following implication holds
(M F D)
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
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) Λ1◦
, b)-quasiinvex at the point y(·)
and−Kℓ(x(·))is(ρ′′ℓ
, b)-quasiinvex at the pointy(·)with respect toη andθ;
c) the functional
Z
Then, the inequalityπ(x◦
(·))≤δ(y(·), yγ(·),Λ1◦,Λ2◦, µ(·), ν(·))is false.
The proof will be given in a further paper (see also, [9]).
Theorem 4.3 (Direct duality). Letx◦
the vectorsΛ1◦
(·)) is an efficient solution of the dual
(M F D)andπ(x◦
Proof. We take into account that the pointx◦
(·) is a normal efficient solution of the problem (M F P). Therefore, according to Theorem 3.3, there are Λ1◦
,Λ2◦
∈Rr and the smooth functions µ◦
: Ωt0,t1 → R msp,ν◦
: Ωt0,t1 →R
qsp satisfying the relations
(M F V)C. Also,< να◦(t), h(t, x
Therefore, the statements are proved.
Theorem 4.4 (Converse duality). Let(x◦
(·), x◦ ficient solution of the dual problem (M F D). Suppose the following conditions are fulfilled:
d)the functional
Z with respect toη andθ, respectively;
f)ρ′ℓΛ1◦
(·). According to Theorem 3.3, there are the vectors ¯Λ1◦
satisfying the conditions (M F V)C. It follows
<µ¯α(t), g(t,x(·),¯ x¯γ(·))>+<ν¯α(t), h(t,x(·),¯ x¯γ(·))>= 0, α= 1, p the maximal efficiency of the point (x◦
(·), x◦
(·)) is contradicted. Hence, ¯x(·) =x◦
Remark 4.5. To make a computer aided study ofP DI&P DE-constrained optimiza-tion problems we can perform symbolic computaoptimiza-tions via Maple software package
(see also [1], [16]).
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Authors’ addresses:
Ariana Pitea and Constantin Udri¸ste,
University Politehnica of Bucharest, Faculty of Applied Sciences, 313 Splaiul Independent¸ei, Bucharest, RO-060042, Romania. E-mail: [email protected], [email protected]
S
¸tefan Mititelu, Technical University of Civil Engineering, Department of Mathematics and Informatics,
