SUFFICIENT EFFICIENCY CONDITIONS FOR A MINIMIZING ...

U.P.B. Sci. Bull., Series A, Vol. 72, Iss. 2, 2010

ISSN 1223-7027

SUFFICIENT EFFICIENCY CONDITIONS FOR A MINIMIZING FRACTIONAL PROGRAM Ariana Pitea1 , Constantin Udris¸te2

Consider˘ am problema de minimizare a programului (MFP) ˆın care obiectivul este un vector de cˆ aturi de funct¸ionale integrale curbilinii cu restrict¸ii inecuat¸ii cu derivate part¸iale (IDP) ¸si/sau ecuat¸ii cu derivate part¸iale (EDP). Scopul acestei lucr˘ ari este de a introduce ¸si studia condit¸ii suficiente de eficient¸a ˘ a unei solut¸ii realizabile a problemei (MFP). Rezultatele prezentate ˆın §2 sunt originale, ele finalizˆ and rezultate recente, al c˘ aror studiu este init¸iat ˆın [7] ¸si [8]. We consider the minimizing fractional program (MFP), where the objective is a vector of functionals quotients of paths integrals and the constraints are partial differential inequations (PDI) and partial differential equations (PDE). The aim of this work is to introduce and study sufficient conditions for the efficiency of a feasible solution of the problem (MFP). The results discussed in §2 are new and finalize a recent research initiated in [7] and [8]. Keywords: Efficient solution, PDE, PDI, normal efficient solution. MSC2000: primary 49J40, 49K20; secondary 58E17, 65K10, 53C65.

1. Minimizing fractional programs Before presenting our results, we need the following background which is necessary for introducing notations and for the completeness of the exposition. For more details, we address the reader to [7], [8]. Let (T, h) and (M, g) be Riemannian manifolds of dimensions p and n, respectively. Denote by t = (tα ), α = 1, p, and x = (xi ), i = 1, n, the local coordinates on T and M , respectively. Consider J 1 (T, M ) be the first order jet bundle associated to T and M . Using the product order relation on Rp , [5], the hyperparallelepiped Ωt0 ,t1 , in p R , with the diagonal opposite points t0 = (t10 , . . . , tp0 ) and t1 = (t11 , . . . , tp1 ), can be written as being the interval [t0 , t1 ]. Suppose γt0 ,t1 is a piecewise C 1 -class curve joining the points t0 and t1 . The closed Lagrange 1-forms densities of C ∞ -class fα = (fα` ) : J 1 (T, M ) → Rr ,

kα = (kα` ) : J 1 (T, M ) → Rr ,

1

` = 1, r,

α = 1, p

Lecturer, Department of Mathematics and Informatics I, Faculty of Applied Sciences, University ”Politehnica” of Bucharest, Romania, E-mail: [email protected] 2 Professor, Department of Mathematics and Informatics I, Faculty of Applied Sciences, University ”Politehnica” of Bucharest, Romania, E-mail: [email protected] 13

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Ariana Pitea, Constantin Udris¸te

determine the following path independent curvilinear functionals (actions) Z Z ` α ` ` F (x(·)) = fα (t, x(t), xγ (t)) dt , K (x(·)) = kα` (t, x(t), xγ (t)) dtα , γt0 ,t1

γt0 ,t1

∂x (t), γ = 1, p, are partial velocities. ∂tγ The closeness conditions (complete integrability conditions) are

where xγ (t) =

Dβ fα` = Dα fβ` ,

Dβ kα` = Dα kβ` ,

α, β = 1, p, α 6= β, ` = 1, r,

where Dβ is the total derivative. Suppose K ` (x(·)) > 0, for all ` = 1, r, and accept that the Lagrange matrix densities g = (gab ) : J 1 (T, M ) → Rms , a = 1, s, b = 1, m, m < n, h = (hba ) : J 1 (T, M ) → Rqs ,

a = 1, s,

b = 1, q, q < n,

of C ∞ -class define the partial differential inequations (PDI) (of evolution) g(t, x(t), xγ (t)) < = 0, t ∈ Ωt0 ,t1 , and the partial differential equations (PDE) (of evolution) h(t, x(t), xγ (t)) = 0,

t ∈ Ωt0 ,t1 .

On the set C ∞ (Ωt0 ,t1 , M ) of all functions x : Ωt0 ,t1 → M of C ∞ -class, we set the norm p X kxk = kxk∞ + kxα k∞ . α=1

Denote by n F(Ωt0 ,t1 ) = x ∈ C ∞ (Ωt0 ,t1 , M ) | x(t0 ) = x0 , x(t1 ) = x1 , or x(t)|∂Ωt0 ,t1 = given, o g(t, x(t), xγ (t)) < 0, h(t, x(t), x (t)) = 0, t ∈ Ω γ t ,t 0 1 = the set of all feasible solutions of the problem (MFP). The aim of this work is to introduce and study sufficient efficiency conditions for the variational problem  ¶ µ 1 F r (x(·)) F (x(·))   min ,..., r , K 1 (x(·)) K (x(·)) x(·) (MFP)   subject to x(·) ∈ F(Ωt0 ,t1 ). The authors of this paper and S¸tefan Mititelu have introduced and studied such variational problems. More exactly, they have given necessary conditions for the efficiency of a feasible solution of the problem (MFP) and studied some types of dualities [7], [8]. Definition 1.1. A feasible solution x◦ (·) ∈ F(Ωt0 ,t1 ) is called efficient for the program (MFP) if and only if for any feasible solution x(·) ∈ F(Ωt0 ,t1 ), the inequality F (x(·)) < F (x◦ (·)) F (x(·)) F (x◦ (·)) implies the equality = . K(x(·)) = K(x◦ (·)) K(x(·)) K(x◦ (·))

Sufficient efficiency conditions for a minimizing fractional program

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Definition 1.2. Let x◦ be an optimal solution of the problem (MFP). Suppose there are in Rr the vectors Λ1◦ and Λ2◦ having all components nonnegative but al least one positive and the smooth matrix functions µ◦α and να◦ such that ∂fα ∂kα (t, x◦ (t), x◦γ (t)) > − < Λ2◦ , (t, x◦ (t), x◦γ (t)) > ∂x ∂x ∂h ∂g + < µ◦α (t), (t, x◦ (t), x◦γ (t)) > + < να◦ (t), (t, x◦ (t), x◦γ (t)) > ∂x ∂x µ ∂fα ∂kα (t, x◦ (t), x◦γ (t)) > − < Λ2◦ , (t, x◦ (t), x◦γ (t)) > −Dγ < Λ1◦ , ∂xγ ∂xγ ∂g + < µ◦α (t, x◦ (t), x◦γ (t)), (t, x◦ (t), x◦γ (t)) > ∂xγ ¶ ∂h ◦ ◦ ◦ + < να (t), (t, x (t), xγ (t)) > = 0, ∂xγ t ∈ Ωt0 ,t1 , α = 1, p (Euler − Lagrange PDEs).

< Λ1◦ ,

Then x◦ (·) is called normal optimal solution of problem (MFP). Let ρ be a real number and b : C ∞ (Ωt0 ,t1 , M )×C ∞ (Ωt0 ,t1 , M ) → [0, ∞) a functional. To any closed 1-form a = (aα ) we associate the path independent curvilinear functional Z A(x(·)) = γt0 ,t1

aα (t, x(t), xγ (t)) dtα .

The following definition of the quasiinvexity [5], [7], [8], helps us to state the results included in our main section. Definition 1.3. The functional A is called [strictly] (ρ, b)-quasiinvex at the point x◦ (·) if there is a vector function η : J 1 (Ωt0 ,t1 , M ) × J 1 (Ωt0 ,t1 , M ) → Rn , such that η(t, x◦ (t), x◦γ (t), x◦ (t), x◦γ (t)) = 0, 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 µ · Z ◦ ◦ < (A(x(·)) = A(x (·))) ⇒ b(x((·), x (·)) < η(t, x(t), xγ (t), x◦ (t), x◦γ (t)), γt0 ,t1

∂aα (t, x◦ (t), x◦γ (t)) > + < Dγ η(t, x(t), xγ (t), x◦ (t), x◦γ (t)), ∂x ¸ ¶ ∂aα ◦ ◦ α ◦ ◦ 2 < (t, x (t), xγ (t)) > dt [ − < Λ2◦ , (t, x◦ (t), x◦γ (t)) > < Λ1◦ ,   ∂x ∂x     ∂h ∂g   + < µ◦α (t), (t, x◦ (t), x◦γ (t)) > + < να◦ (t), (t, x◦ (t), x◦γ (t)) >    ∂x ∂x  µ    ∂kα ∂fα   (t, x◦ (t), x◦γ (t)) > − < Λ2◦ , (t, x◦ (t), x◦γ (t)) > −Dγ < Λ1◦ ,   ∂x ∂x  γ γ     ∂g + < µ◦α (t, x◦ (t), x◦γ (t)), (t, x◦ (t), x◦γ (t)) > (MFP)◦ ∂x γ   ¶    ∂h  ◦ ◦ ◦  + < να (t), (t, x (t), xγ (t)) > = 0,   ∂xγ     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,   1◦ Λ ≥ 0, < e, Λ1◦ >= 1, e = (1, . . . , 1) ∈ Rr .

2. Efficiency sufficient conditions We shall establish efficiency sufficient conditions for the problem (MFP). Theorem 2.1. Let us consider the vectors Λ1◦ , Λ2◦ from Rr and the functions x◦ (·), µ◦ (·), ν ◦ (·) which satisfy the conditions (MFP)◦ . Suppose that the following properties hold: a) the functional Z £ 1◦ 2◦ < Λ , F (x(·)) > − < Λ , K(x(·)) > = < Λ1◦ , fα (t, x(t), xγ (t)) > γt0 ,t1

¤ − < Λ2◦ , kα (t, x(t), xγ (t)) > dtα is (ρ1 , b)-quasiinvex at the point x◦ (·) with respect to η and θ; Z b) the functional < µ◦α (t), g(t, x(t), xγ (t)) > dtα is (ρ2 , b) -quasiinvex at γt0 ,t1

the point x◦ (·) with respect to η and θ; Z c) the functional < να◦ (t), h(t, x(t), xγ (t)) > dtα is (ρ3 , b)-quasiinvex at γt0 ,t1

the point x◦ (·) with respect to η and θ; d) one of the integrals of a)-c) is strictly (ρ1 , b), (ρ2 , b) or (ρ3 , b)-quasiinvex at the point x◦ (·) with respect to η and θ; e) ρ1 + ρ2 + ρ3 ≥ 0; ` ◦ 2◦ ` ◦ f) Λ1◦ ` F (x (·)) − Λ` K (x (·)) = 0, for each ` = 1, r. Then the point x◦ (·) is an efficient solution of problem (MFP).


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Proof. Let us suppose that the point x◦ (·) is not an efficient solution for problem (MFP). Then, there is a feasible solution x(·) for problem (MFP), such that F ` (x(·)) F ` (x◦ (·)) ≤ , K ` (x(·)) K ` (x◦ (·))

` = 1, r,

the case x(·) = x◦ (·) being excluded. That is ` 2◦ ` 1◦ ` ◦ 2◦ ` ◦ Λ1◦ ` F (x(·)) − Λ` K (x(·)) ≤ Λ` F (x (·)) − Λ` K (x (·)),

` = 1, r.

Making the sum after ` = 1, r, we get < Λ1◦ , F (x(·)) > − < Λ2◦ , K(x(·)) > ≤ < Λ1◦ , F (x◦ (·)) > − < Λ2◦ , K(x◦ (·)) > . According to condition a), it follows · Z ∂fα ◦ b(x(·), x (·)) < η(t, x(t), xγ (t), x◦ (t), x◦γ (t)), < Λ1◦ , (t, x◦ (t), x◦γ (t)) > ∂x γt0 ,t1 ∂kα (t, x◦ (t), x◦γ (t)) >> + < Dγ η(t, x(t), xγ (t), x◦ (t), x◦γ (t)), ∂x ¸ ∂kα ∂fα (t, x(t), xγ (t)) > − < Λ2◦ , (t, x◦ (t), x◦γ (t)) >> dtα < Λ1◦ , ∂xγ ∂xγ

− < Λ2◦ ,

< − ρ1 b(x(·), x◦ (·))kθ(x(·), x◦ (·))k2 . = Z

Applying property b), the inequality
dt < =

γt0 ,t1

< µ◦α (t), g(t, x◦ (t), x◦γ (t)) > dtα

µ ∂g (t, x◦ (t), x◦γ (t)) >> < η(t, x(t), xγ (t), x◦ (t), x◦γ (t)), < µ◦α (t), ∂x γt0 ,t1 ¶ ∂g ◦ ◦ ◦ ◦ ◦ + < Dγ η(t, x(t), xγ (t), x (t), xγ (t)), < µα (t), (t, x (t), xγ (t)) >> dtα ∂xγ < − ρ2 b(x(·), x◦ (·))kθ(x(·), x◦ (·))k2 . = (2) Taking into account condition c), the equality Z Z < να◦ (t), h(t, x(t), xγ (t)) > dtα = < να◦ (t), h(t, x◦ (t), x◦γ (t)) > dtα Z

b(x(·), x◦ (·))

γt0 ,t1

implies

γt0 ,t1

µ ∂h < η(t, x(t), xγ (t), x◦ (t), x◦γ (t)), < να◦ (t), (t, x◦ (t), x◦γ (t)) >> ∂x γt0 ,t1 ¶ ∂h ◦ ◦ ◦ ◦ ◦ + < Dγ η(t, x(t), xγ (t), x (t), xγ (t)), < να (t), (t, x (t), xγ (t)) >> dtα ∂xγ < − ρ3 b(x(·), x◦ (·))kθ(x(·), x◦ (·))k2 . = (3)

b(x(·), x◦ (·))

Z

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Summing side by side relations (1), (2), (3) and using condition d), it follows Z ∂fα ◦ b(x(·), x (·)) < η(t, x(t), xγ (t), x◦ (t), x◦γ (t)), < Λ1◦ , (t, x◦ (t), x◦γ (t)) > ∂x γt0 ,t1 ∂kα ∂g (t, x◦ (t), x◦γ (t)) > + < µ◦α (t), (t, x◦ (t), x◦γ (t)) > ∂x ∂x Z ∂h ◦ ◦ α ◦ ◦ < Dγ η(t, x(t), + < να (t), (t, x (t), xγ (t)) >> dt + b(x(·), x (·)) ∂x γt0 ,t1 − < Λ2◦ ,

∂fα ∂kα (t, x◦ (t), x◦γ (t)) > − < Λ2◦ , (t, x◦ (t), x◦γ (t)) > ∂xγ ∂xγ ∂g ∂h + < µ◦α (t), (t, x◦ (t), x◦γ (t)) > + < να◦ (t), (t, x◦ (t), x◦γ (t)) >> dtα ∂xγ ∂xγ xγ (t), x◦ (t), x◦γ (t)), < Λ1◦ ,

< − (ρ1 + ρ2 + ρ3 ) b(x(·), x◦ (·))kθ(x(·), x◦ (·))k2 . This inequality implies that b(x(·), x◦ (·)) > 0, therefore we obtain Z ∂fα (t, x◦ (t), x◦γ (t)) > < η(t, x(t), xγ (t), x◦ (t), x◦γ (t)), < Λ1◦ , ∂x γt0 ,t1 − < Λ2◦ ,

∂kα ∂g (t, x◦ (t), x◦γ (t)) > + < µ◦α (t), (t, x◦ (t), x◦γ (t)) > ∂x ∂x

+ < να◦ (t), Z + γt0 ,t1

∂h (t, x◦ (t), x◦γ (t)) >> dtα ∂x

< Dγ η(t, x(t), xγ (t), x◦ (t), x◦γ (t)), < Λ1◦ ,

− < Λ2◦ ,

∂fα (t, x◦ (t), x◦γ (t)) > ∂xγ

∂kα ∂g (t, x◦ (t), x◦γ (t)) > + < µ◦α (t), (t, x◦ (t), x◦γ (t)) > ∂xγ ∂xγ

+ < να◦ (t),

∂h (t, x◦ (t), x◦γ (t)) >> dtα ∂xγ

< − (ρ1 + ρ2 + ρ3 ) kθ(x(·), x◦ (·))k2 . According to [13], §9, we have the following Lemma 2.1. A total divergence is equal to a total derivative. Integrating by parts the second integral and using Lemma 2.1, the previous inequality leads us to a contradiction, that is 0 < − (ρ1 + ρ2 + ρ3 ) kθ(x(·), x◦ (·))k2 . Therefore, the point x◦ (·) is an efficient solution for problem (MFP).

¤

Replacing the integrals from hypotheses b), c), of Theorem 2.1 by the integral Z £ ¤ < µ◦α (t), g(t, x(t), xγ (t)) > + < να◦ (t), h(t, x(t), xγ (t)) > dtα , γt0 ,t1


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the following statement is obtained. Corollary 2.1. Let x◦ (·) be a feasible solution of problem (MFP), µ◦ (·), ν ◦ (·) be functions and Λ1◦ , Λ2◦ vectors from Rr such that the relations (MFP)◦ are satisfied. Suppose that the following conditions are fulfilled: a) the functional Z £ 1◦ 2◦ < Λ1◦ , fα (t, x(t), xγ (t)) > < Λ , F (x(·)) > − < Λ , K(x(·)) > = γt0 ,t1

¤ − < Λ2◦ , kα (t, x(t), xγ (t)) > dtα is (ρ1 , b)-quasiinvex at the point x◦ (·) with respect to η and θ; b) the functional Z (< µ◦α (t), g(t, x(t), xγ (t)) > + < να◦ (t), h(t, x(t), xγ (t)) >)dtα γt0 ,t1

is (ρ2 , b)-quasiinvex at the point x◦ (·) with respect to η and θ; c) one of the integrals from a) or b) is strictly-quasiinvex at the point x◦ (·); d) ρ1 + ρ2 ≥ 0; ` ◦ 2◦ ` ◦ e) Λ1◦ ` F (x (·)) − Λ` K (x (·)) = 0, for each ` = 1, r. Then, the point x◦ (·) is an efficient solution of problem (MFP). For other developments of optimization problems of path independent curvilinear integrals with PDE constraints or with isoperimetric constraints as multiple integrals or path independent curvilinear integrals, see [2] ÷ [6] and [9] ÷ [16]. For a computer aided study of PDE and/or PDI optimization problems using Maple, see [1] and [14]. 3. Conclusions We considered the minimizing fractional program (MFP), where the objective is a vector of functionals quotients of paths integrals and the constraints are partial differential inequations (PDI) and equations (PDE). In this work, we introduced and studied sufficient conditions for the efficiency of a feasible solution of problem (MFP). The present study completes previous results obtained with S¸tefan Mititelu and included in papers [7] and [8]. REFERENCES [1] Maria Teresa Calapso and C. Udri¸ste, Isothermic surfaces as solutions of Calapso PDE, Balkan J. Geom. Appl., 13(2008), No. 1, 20-26. [2] S ¸ t. Mititelu, Extensions in invexity theory, J. Adv. Math. Stud., 1(2008), No. 1-2, 63-70. ¸ t. Mititelu, Optimality and duality for invex multi-time control problems with mixed con[3] S straints, J. Adv. Math. Stud., 2(2009), No. 1, 25-34. [4] S ¸ t. Mititelu, Invex sets and preinvex functions, J. Adv. Math. Stud., 2(2009), No. 2, 41-52. [5] Ariana Pitea, Integral Geometry and PDE Constrained Optimization Problems, Ph. D Thesis, ”Politehnica” University of Bucharest, 2008.

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