Advances in Automatic Control, Modelling & Simulation
On Efficiency Conditions for Multiobjective Variational Problems Involving Higher Order Derivatives∗ SAVIN TREANTA University Politehnica of Bucharest Faculty of Applied Sciences Department of Mathematics-Informatics Splaiul Independentei 313 060042, BUCHAREST, ROMANIA savin
[email protected]
CONSTANTIN UDRISTE University Politehnica of Bucharest Faculty of Applied Sciences Department of Mathematics-Informatics Splaiul Independentei 313 060042, BUCHAREST, ROMANIA
[email protected]
Abstract: This paper aims to formulate and prove necessary and sufficient conditions of efficiency for a class of multiobjective variational problems involving higher order derivatives. Consider a multiobjective optimization problem of minimizing a vector of simple integral functionals subject to certain higher order differential equations and/or inequations. We establish sufficient efficiency conditions for a feasible solution using the notion of quasiinvexity. Key–Words: efficient solution, quasi-invexity, multiobjective variational problem.
1
Our framework and problem describtion
for the completeness of the exposition, we set the following notations. Let consider the real interval I := [t0 , t1 ] ⊆ R and
In this work we extend and further develop some optimization results connected to the efficiency of a feasible solution for a class of multiobjective nonfractional programming problems. We introduce and perform a study on the multiobjective variational problem of minimizing a vector of simple integral functionals (MVP) constrained by higher order differential equations and inequations. This paper is strongly motivated by its applications in natural phenomena and in wide areas of research for new technology as well, where there are needed derivatives of order higher than one or two (engineering, chemistry, games theory, etc.) The passing from the first order derivatives to the higher order derivatives is not a facile task because it requests specific techniques, a new quasiinvexity and an appropriate mathematical framework. In time, several authors have been interested in the study of vector programming problems which involve a generalized convexity (see [4]-[6], [8]). Thus, many of them extend this notion and develop a multitime optimization theory, using a geometrical language (see [7]). For other different ideas but connected to this subject, we address the readers to the works [1]-[3] and [9]. Before presenting our results,
f = (fα ) : I × Rn(k+1) → Rp , α = 1, p, (f1 (t, x(t), x(1) (t), ..., x(k) (t)), ..., fp (t, x(t), x(1) (t), ..., x(k) (t))), dk a C k+1 -class function, where x(k) (t) := x(t), dtk with k ≥ 1 a fixed natural number. Also, let be given g = (g1 , ..., gm ) : I × Rn(k+1) → Rm , with m < n, and h = (h1 , ..., hr ) : I ×Rn(k+1) → Rr , with r < n, two C k+1 -class functions. Assume that the previous C k+1 -class Lagrangians, fα (t, x(t), x(1) (t), ..., x(k) (t)),
generate the simple integral functionals ∫
t1
Fα (x(·)) :=
fα (t, x(t), x(1) (t), ..., x(k) (t))dt,
t0
α = 1, p. ∞
Let C ([t0 , t1 ], Rn ) be the space of all functions x : [t0 , t1 ] → Rn of C ∞ -class, with the norm
∗ 15th WSEAS International Conference on Automatic Control, Modelling & Simulation (ACMOS-13), Brasov, Romania, June 1-3, 2013.
ISBN: 978-1-61804-189-0
α = 1, p,
∥x∥ := ∥x∥∞ +
k ∑ β=1
157
∥x(β) ∥∞ .
Advances in Automatic Control, Modelling & Simulation
As usual, two vectors, u = (u1 , ..., us ) , v = (v1 , ..., vs ), in Rs can be related via u = v ⇔ u i = vi ,
scalar λ and the piecewise smooth functions, p(t) and q(t), satisfying
u ≤ v ⇔ ui ≤ vi
∂L (χ 0 (t), p(t), q(t), λ) ∂x x
u < v ⇔ ui < vi , u ≼ v ⇔ u ≤ v, u ̸= v, i = 1, s.
−
Also, we underline that the argument of our Lagrangians is a graph
d ∂L (χ 0 (t), p(t), q(t), λ) dt ∂x(1) x
+... + (−1)k
χx (t) := (t, x(t), x(1) (t), ..., x(k) (t)).
dk ∂L (χ 0 (t), p(t), q(t), λ) = 0 dtk ∂x(k) x
(higher order Euler-Lagrange ODEs) Using these ingredients, we formulate the multiobjective variational problem (MVP) (constrained optimization problem), (∫
min F (x(·)) = x(·)
∫
t1
)
t1
f1 (χx (t))dt, ..., t0
p(t) ≥ 0,
p(t)g(χx0 (t)) = 0,
fp (χx (t))dt t0
subject to x (·) ∈ F (I), where the set (domain) F (I) of all feasible solutions is x ∈ C ∞ (I, Rn ) , x(tε ) = xε , x(β) (tε ) = xβε g(χx (t)) ≤ 0, h(χx (t)) = 0, t ∈ I, β = 1, k − 1, with ε ∈ {0, 1}. In this paper we are looking for necessary and sufficient efficiency conditions for the foregoing multiobjective variational problem (MVP). Next section introduces the necessary mathematical tools which will be used for proving our main results.
The previous result represents a generalization of Valentine’s necessary conditions (see [10]). Definition 2.2 The optimal solution x0 (·) of problem (SVP) is called normal optimal solution if λ ̸= 0. Without a loss of generality, we can assume that λ = 1. In this work we are interested in finding necessary and sufficient conditions of efficiency for the optimality of the vector problem (MVP) in the domain F (I). In this respect, we give the following two definitions. Definition 2.3 A feasible solution x0 (·) ∈ F (I) is named efficient solution (or Pareto minimum) in (MVP) if there exists no other(feasible ) solution x(·) ∈ F (I) such that F (x(·)) ≼ F x0 (·) . Consider ρ a real number and b : [C ∞ ([t0 , t1 ], Rn )]k+1 → [0, ∞) a functional. Denote
2
(
Preliminaries
∫
(
)
η t, x, x(1) , ..., x(k−1) , x0(k) := ηtxx0 .
}
t1
min I (x(·)) = x(·)
)
b x, x0 , x0(1) , ..., x0(k−1) := bxx0
Let start with the case of a single functional, considering the following scalar variational problem (SVP), {
(∀)t ∈ I.
Also, let a : I × Rn(k+1) → R be a real function that determines the following simple integral functional
X (χx (t)) dt t0
∫
subject to x (·) ∈ F (I).
t1
A(x(·)) =
a (χx (t)) dt. t0
Consider the auxiliary Lagrange function L,
Definition 2.4 The functional A(·) is [strictly] (ρ, b)-quasiinvex at x0 if there exist the vector functions η = (η1 , ..., ηn ), with the property
L(χx (t), p(t), q(t), λ) := λX (χx (t)) +pa (t)ga (χx (t)) + q ζ (t)hζ (χx (t)),
dζ ηtx0 x0 = 0, ζ ∈ {0, 1, ..., k − 1}, (∀) t ∈ I, dtζ
(see summation over the repeated indices!) that allows us to establish necessary conditions of optimality for (SVP). Theorem 2.1 Consider that x0 , a feasible solution of the problem (SVP), is an optimal solution and X, g, h are C k+1 -class functions. Then, there exist a
ISBN: 978-1-61804-189-0
and θ : [C ∞ ([t0 , t1 ], Rn )]k+1 → Rn such that, for any x [x ̸= x0 ], we have (
158
)
A(x) ≤ A(x0 )
Advances in Automatic Control, Modelling & Simulation
∫
=⇒ (bxx0 ∫
+bxx0
t1 t0
∫
+... + bxx0
t1
ηtxx0
t0
and there exists k ∈ {1, ..., p} such that
∂a (χ 0 (t))dt ∂x x
∫
dηtxx0 ∂a (χ 0 (t))dt dt ∂x(1) x
t0
t0
∫
t0
p ∑
To develop our theory, we establish the following Lemma 3.1 The feasible solution x0 (·) ∈ F (I) is an efficient solution of the problem (MVP) if and only if x0 (·) ∈ F (I) is an optimal solution of each scalar problem Pl (x0 ), l = 1, p,
x(·)
∫
t1
t0
t1
∂g ∂h d pl (t) (1) (χx0 (t)) + ql (t) (1) (χx0 (t)) − dt ∂x ∂x
fj (χx (t)) dt ≤
t0
fj (χy (t)) dt ≤
∫
t1 t0
fj (χx0 (t)) dt
+(−1)k +(−1)k
∫
t0
t1
t0
fj (χy (t)) dt ≤
∫
x(·)
t1
ISBN: 978-1-61804-189-0
{
∂g (χ 0 (t)) ∂x(k) x }
ql (t)
∂h (χ 0 (t)) = 0 ∂x(k) x
t1
t0
t0
t0
fl (χx0 (t)) dt =
fl (χx (t)) dt, l = 1, p. Define the C k+1 -
∫
t1
Gj (x(·)) := t0
[
]
fj (χx (t)) − Rj0 + ϕj (χx (t)) dt = 0.
Therefore, the scalar problem Pl (x0 ), l ∈ {1, ..., p} fixed, is changed into ∫
t1
max x(·)
fj (χx0 (t)) dt,
t1
class functions, ϕj : I × Rn(k+1) → R, ϕj (χx (t)) ≥ 0, j = 1, p, j ̸= l, as follows
fk (χx0 (t)) dt.
t1 t0
}
pl (t)
Proof. Consider Rl0 :=
This contradicts the efficiency of the function x0 (·) ∈ F (I) in (MVP). Consequently, we proved the direct implication. ” ⇐= ” Let x0 (·) ∈ F (I) be an optimal solution of each scalar problem Pl (x0 ), l = 1, p. Assume that x0 (·) ∈ F (I) is not an efficient solution of the problem (MVP). Consequently, there exists a function y (·) ∈ F (I) such that ∫
dk dtk
{
∫
fj (χx0 (t)) dt min
fk (χy (t)) dt
