A necessary and sufficient condition on the equivalence between local and global optimal solutions in variational control problems

Nonlinear Analysis 191 (2020) 111640

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Nonlinear Analysis www.elsevier.com/locate/na

A necessary and sufficient condition on the equivalence between local and global optimal solutions in variational control problems Savin Treanţă a ,∗, Manuel Arana-Jiménez b , Tadeusz Antczak c a

Department of Applied Mathematics, Faculty of Applied Sciences, University Politehnica of Bucharest, Bucharest, Romania b Department of Statistics and Operational Research, Faculty of SSCC and Communication, University of Cádiz, Cádiz, Spain c Faculty of Mathematics and Computer Science, University of Lódź, Lódź, Poland

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Article history: Received 5 March 2019 Accepted 14 September 2019 Communicated by Enrico Valdinoci MSC: 49J40 65K10 90C26 90C30 49K20

abstract In this paper, optimality conditions are investigated for a class of PDE&PDIconstrained variational control problems. Thus, an efficient condition for a local optimal solution of the considered PDE&PDI-constrained variational control problem to be its global optimal solution is derived. The theoretical development is supported by a suitable example of nonconvex optimization problem. © 2019 Elsevier Ltd. All rights reserved.

Keywords: Global optimal solution Efficient condition Control PDE&PDI-constrained variational control problem

1. Introduction It is very well-known that a minimizer of an optimization problem is unique if its objective function is strictly convex. Polyak [10] extended the definition of a unique minimizer to the notion of a unique sharp minimizer. An excellent survey in this area and on weak sharp solutions associated with variational-type inequalities can be found in Hiriart-Urruty and Lemar´echal [5]. Over time, a considerable interest has been given in achieving the necessary and sufficient conditions so that a real-valued continuous function has the property that any local extreme is also global. In this regard, we mention the works of Zang and Avriel [16], ∗ Corresponding author. E-mail addresses: [email protected] (S. Treanţă), [email protected] (M. Arana-Jiménez), [email protected] (T. Antczak).

https://doi.org/10.1016/j.na.2019.111640 0362-546X/© 2019 Elsevier Ltd. All rights reserved.

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S. Treanţă, M. Arana-Jiménez and T. Antczak / Nonlinear Analysis 191 (2020) 111640

Zang et al. [17,18], Horst [6], Martin [8], Ivanov [7], Antczak and Arana-Jim´enez [1], Arana-Jim´enez and Antczak [2], Treant¸˘ a and Arana-Jim´enez [14,15] and Treant¸˘a [12]. In this paper, by using several variational techniques developed in Giannessi [4], Clarke [3], AranaJim´enez and Antczak [2], Treant¸˘ a [11,13] and Mititelu and Treant¸˘a [9], we develop a new mathematical framework on the equivalence between local and global optimal solutions in multidimensional controlled variational problems. More precisely, under only continuity assumptions of the functionals involved in the considered control problem, an efficient condition is formulated and proved such that any local optimal solution associated with the considered first-order PDE&PDI-constrained variational control problem is also its global optimal solution. To the best of authors knowledge, there are no works available dealing with this topic for (multidimensional) variational control problems with mixed constraints involving first-order PDE. Also, since controlled variational problems occur often in real-life applications, all these things motivate the present study. The paper is structured as follows. Section 2 includes some notations, working hypotheses and problem formulation. Section 3 contains the main results of this paper. Thereby, under only continuity assumptions of the involved functionals, an efficient condition is established so that any local optimal solution associated with the considered PDE&PDI-constrained variational control problem is also its global optimal solution. In Section 4, an example is provided to illustrate the results. Finally, Section 5 concludes the paper. 2. Notations and problem description In order to introduce the extremum problem for which the optimality is studied in the paper, we start with the following notations and working hypotheses: ▶ Θ ⊂ Rm is a compact domain in Rm and the point Θ ∋ θ = (θν ), ν = 1, m; ▶ let X˜ be the space of all functions x : Θ → Rn and let X ⊂ X˜ be the space of piecewise smooth state functions x : Θ → Rn with the norm ∥ x ∥=∥ x ∥∞ +

m ∑

∥ xν ∥ ∞ ,

∀x ∈ X ,

(2.1)

ν=1 ∂x where xν denotes ∂θ ν; ▶ also, denote by U˜ the space of all functions u : Θ → Rk and by U ⊂ U˜ the space of piecewise continuous control functions u : Θ → Rk with the uniform norm ∥ · ∥∞ ; ˜ denote by int (S1 ) the interior of S1 and by cl (S1 ) the closure of ▶ for any two subsets S1 , S2 of X˜ × U, S1 ; also, consider the set { } S1 \ S2 = (x, u) ∈ X˜ × U˜ : (x, u) ∈ S1 ∧ (x, u) ∈ / S2 ; (2.2)

▶ consider X × U as a nonempty open subset of X˜ × U˜ endowed with the inner product ∫ ⟨(x, u); (y, w)⟩ = [x(θ) · y(θ) + u(θ) · w(θ)]dθ1 · · · dθm , ∀(x, u), (y, w) ∈ X × U

(2.3)

Θ

and the induced norm; ▶ consider the real-valued continuously differentiable functions f : Θ × Rn × Rk → R, gβ : J 1 (Θ, Rn ) × k R → R, β = 1, q (where J 1 (Θ, Rn ) denotes the jet bundle of first-order associated to Θ and Rn ), and, for any (x, u) ∈ X × U, define the following continuous functionals: ∫ W (x, u) = f (θ, x(θ), u(θ)) dθ1 · · · dθm , (2.4) Θ

Gβ (x, u) = gβ (θ, x(θ), xν (θ), u(θ)) ,

θ ∈ Θ, β = 1, q.

(2.5)

S. Treanţă, M. Arana-Jiménez and T. Antczak / Nonlinear Analysis 191 (2020) 111640

3

Taking into account the previous mathematical tools, we formulate the PDE&PDI-constrained variational control problem considered in the paper as follows: (P )

min W (x, u)

(2.6)

(x,u)

subject to

Gβ (x, u) ≤ 0,

β ∈ Q := {1, . . . , q},

(2.7)

x|∂Θ = ϕ = given.

(2.8)

} { F = (x, u) ∈ X × U : Gβ (x, u) ≤ 0, β ∈ Q, x|∂Θ = ϕ = given

(2.9)

Further, denote by

the feasible set associated with the multidimensional variational control problem (P ) and, as well, define the index subset of active constraint functionals at (x, u) as follows Q(x,u) = {β ∈ Q | Gβ (x, u) = 0}.

(2.10)

Remark 2.1. For any β ∈ Q(x,u) , it results Gβ (x, u) = gβ (θ, x(θ), xν (θ), u(θ)) = 0,

θ ∈ Θ.

(2.11)

If, for any β ∈ Q(x,u) , the PDEs formulated in (2.11) can be rewritten in normal form (m-flow type PDEs) ∂xi (θ) = Yνi (θ, x(θ), u(θ)) , ∂θν

i = 1, n, ν = 1, m, θ ∈ Θ,

then we assume that the continuously differentiable functions ( ) Yν = Yνi : Θ × Rn × Rk → Rn , i = 1, n, ν = 1, m

(2.12)

(2.13)

fulfill the closeness conditions Dζ Yνi = Dν Yζi ,

ν, ζ = 1, m, ν ̸= ζ, i = 1, n,

(2.14)

where Dζ is the total derivative operator. Definition 2.1. A point (x0 , u0 ) ∈ F is said to be a global optimal solution of (P ) if the inequality W (x0 , u0 ) ≤ W (x, u) holds for all (x, u) ∈ F. Definition 2.2. A point (x0 , u0 ) ∈ F is said to be a local optimal solution of (P ) if there exists a neighborhood V(x0 ,u0 ) of the point (x0 , u0 ) such that the inequality W (x0 , u0 ) ≤ W (x, u) is fulfilled for all (x, u) ∈ F ∩ V(x0 ,u0 ) . Definition 2.3. For r ∈ R, r > 0, the set { } Br (x0 , u0 ) = (y, w) ∈ X × U : ∥ (y, w) − (x0 , u0 ) ∥< r

(2.15)

is an open ball with center (x0 , u0 ) ∈ X × U and radius r, where ∥ · ∥ is the induced norm by the inner product introduced in (2.3). Remark 2.2. (i) Obviously, any global optimal solution of (P ) is its local optimal solution. The reverse is false, in general. ) ( (ii) Since (x0 , u0 ) ∈ int V(x0 ,u0 ) , there exists r ∈ R, r > 0, such that Br (x0 , u0 ) ⊆ V(x0 ,u0 ) .

S. Treanţă, M. Arana-Jiménez and T. Antczak / Nonlinear Analysis 191 (2020) 111640

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3. Main results In this section, an efficient condition is formulated and proved such that any local optimal solution associated with the considered PDE&PDI-constrained variational control problem (P ) is also a global optimal solution of (P ). The next result provides a sufficient condition for a local optimal solution of the variational problem (P ) to be its global minimizer. ( )2 Theorem 3.1. For any (x, u), (y, w) ∈ F, with W (x, u) − W (y, w) < 0, if there exists γ : X × U → S, where S = {s : [0, 1] → X × U | (∃) limτ →0+ ∥ s(τ ) ∥=∥ s(0) ∥= 0}, such that (y, w) + γ ((x, u), (y, w)) (τ ) ∈ X × U, and, for all τ ∈ (0, 1), it follows γ ((x, u), (y, w)) (τ )|∂Θ = (0, u), u ∈ U, and: W ((y, w) + γ ((x, u), (y, w)) (τ )) < W (y, w), Gβ ((y, w) + γ ((x, u), (y, w)) (τ )) ≤ 0,

(3.1)

β ∈ Q,

(3.2)

then any local optimal solution associated with the considered PDE&PDI-constrained variational control problem (P ) is also its global optimal solution. Proof . Let (x0 , u0 ) ∈ F be a local optimal solution of (P ). We proceed by contradiction. Suppose that there ( )2 exists (x, u) ∈ F such that W (x, u) − W (x0 , u0 ) < 0. By hypothesis, there exists γ : X × U → S, where S = {s : [0, 1] → X × U | (∃) limτ →0+ ∥ s(τ ) ∥=∥ s(0) ∥= 0}, such that ( ) (x0 , u0 ) + γ (x, u), (x0 , u0 ) (τ ) ∈ X × U, (3.3) ( ) ( ( ) ) γ (x, u), (x0 , u0 ) (τ )|∂Θ = (0, u), W (x0 , u0 ) + γ (x, u), (x0 , u0 ) (τ ) < W (x0 , u0 ),

(3.4)

for u ∈ U and τ ∈ (0, 1). Further, since X × U is an open set and (x0 , u0 ) ∈ X × U, there exists r0 ∈ R, r0 > 0, such that Br0 (x0 , u0 ) ⊂ X × U. Let β ∈ Q \ Q(x0 ,u0 ) . Consequently, it follows that Gβ (x0 , u0 ) < 0 and, by using the continuity property of the functionals Gβ , there exist rβ ∈ R, rβ > 0, such that Brβ (x0 , u0 ) ⊂ X × U and Gβ (y, w) < 0,

∀(y, w) ∈ Brβ (x0 , u0 ).

(3.5)

Since ( ) ( ) lim ∥ γ (x, u), (x0 , u0 ) (τ ) ∥=∥ γ (x, u), (x0 , u0 ) (0) ∥= 0,

τ →0+

(3.6)

therefore, for any r ∈ R, r > 0, there exists τ0 ∈ (0, 1) such that ( ) ∥ γ (x, u), (x0 , u0 ) (τ ) ∥< min{r, r0 , rβ }, β ∈ Q \ Q(x0 ,u0 ) , τ ∈ [0, τ0 ]. (3.7) ( ) For (z, µ) = (x0 , u0 ) + γ (x, u), (x0 , u0 ) ( τ20 ), we get (z, µ) ∈ Brβ (x0 , u0 ) ⊂ X × U, β ∈ Q \ Q(x0 ,u0 ) , and, by using (3.5), it follows Gβ (z, µ) < 0, β ∈ Q \ Q(x0 ,u0 ) . (3.8) By hypothesis (3.2), we obtain ( ( ) τ0 ) Gβ (z, µ) = Gβ (x0 , u0 ) + γ (x, u), (x0 , u0 ) ( ) = 0, 2

β ∈ Q(x0 ,u0 ) .

(3.9)

Taking into account (3.4), (3.8) and (3.9), it follows that (z, µ) ∈ F and W (z, µ) < W (x0 , u0 ), that is, (x , u0 ) is not a local optimal solution of (P ). Consequently, we obtain a contradiction and the proof is complete. □ 0

S. Treanţă, M. Arana-Jiménez and T. Antczak / Nonlinear Analysis 191 (2020) 111640

5

Now, we prove that the previous condition in not only sufficient but also necessary such that any local optimal solution associated with the variational control problem (P ) is its global optimal solution. Theorem 3.2. If any local optimal solution associated with the considered constrained variational control problem (P ) is also its global optimal solution, then, for any (x, u), (y, w) ∈ F, with W (x, u) − W (y, w) < 0, ( )2 there exists γ : X × U → S, with S = {s : [0, 1] → X × U | (∃) limτ →0+ ∥ s(τ ) ∥=∥ s(0) ∥= 0}, such that (y, w) + γ ((x, u), (y, w)) (τ ) ∈ X × U, and, for all τ ∈ (0, 1), it is verified γ ((x, u), (y, w)) (τ )|∂Θ = (0, u), u ∈ U, and: W ((y, w) + γ ((x, u), (y, w)) (τ )) < W (y, w), (3.10) Gβ ((y, w) + γ ((x, u), (y, w)) (τ )) ≤ 0,

β ∈ Q.

(3.11)

Proof . Assume that any local optimal solution of (P ) is also global. In accordance with (3.10), consider (x, u), (y, w) ∈ F such that W (x, u) − W (y, w) < 0. (3.12) Since X × U is an open set, there exists r1 ∈ R, r1 > 0, such that Br1 (y, w) ⊂ X × U. By using the (x,u) continuity property of F on X × U, for ϵ := W (y,w)−W > 0, there exists r2 ∈ (0, r1 ) such that 2 ∥ W (z, µ) − W (y, w) ∥< ϵ,

∀(z, µ) ∈ Br2 (y, w).

(3.13)

Further, for any r ∈ (0, r2 ), define the following variational control problem associated to (P ) (Pr′ ) subject to Denote by

Fr′

min W (z, µ)

(3.14)

(z,µ)

(z, µ) ∈ cl (Br (y, w)) ∩ F.

the set of optimal solutions associated to

(Pr′ ).

(3.15)

Since

(cl (Br (y, w)) ∩ F) ⊂ X × U

(3.16)

˜ it follows that F ′ ̸= ∅. Now, let us consider the following two cases. is a compact subset of X˜ × U, r ∗ (i) There exists r ∈ (0, r2 ) such that (z ∗ , µ∗ ) ∈ int (cl (Br∗ (y, w)) ∩ F) and (z ∗ , µ∗ ) ∈ Fr′ ∗ . This case involves (z ∗ , µ∗ ) is a local optimal solution of (P ) and, by hypothesis, it is a global optimal solution for (P ). But, in accordance with (3.13) and using the inequality (3.12), we get W (x, u) < W (y, w) − ϵ = W (y, w) − W (z ∗ , µ∗ ) + W (z ∗ , µ∗ ) − ϵ ∗

∗

∗

(3.17)

∗

< ϵ + W (z , µ ) − ϵ = W (z , µ ), which means that (z ∗ , µ∗ ) is not a global optimal solution of (P ) and, therefore, we obtain a contradiction. In consequence, we shall consider the next case. (ii) For any r∗ ∈ (0, r2 ), if (z ∗ , µ∗ ) ∈ Fr′ ∗ , then (z ∗ , µ∗ ) ∈ (cl (Br∗ (y, w)) ∩ F) \ int (cl (Br∗ (y, w)) ∩ F) ,

(3.18)

0 <∥ (y, w) − (z ∗ , µ∗ ) ∥≤ r∗ .

(3.19)

involving Further, we shall construct the function γ. Consider the function a : [0, 1] → R, a(τ ) = τ · r∗ . Also, for each τ ∈ (0, 1), introduce the following optimization problem ′ (Pa(τ ))

min W (z, µ) (z,µ)

(3.20)

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S. Treanţă, M. Arana-Jiménez and T. Antczak / Nonlinear Analysis 191 (2020) 111640

subject to

( ) (z, µ) ∈ cl Ba(τ ) (y, w) ∩ F.

(3.21)

As in the previous case, we get ( ( ) ) ( ( ) ) ′ ∅= ̸ Fa(τ ) ⊂ cl Ba(τ ) (y, w) ∩ F \ int cl Ba(τ ) (y, w) ∩ F ,

(3.22)

′ ′ where Fa(τ ) denotes the set of optimal solutions for (Pa(τ ) ). Define

γ ((x, u), (y, w)) : [0, 1] → X × U,

γ ((x, u), (y, w)) (τ ) = (z, µ)τ − (y, w),

(3.23)

′ where (z, µ)τ is an optimal solution in (Pa(τ ) ). We get

∥ γ ((x, u), (y, w)) (τ ) ∥=∥ (z, µ)τ − (y, w) ∥≤ a(τ ) = τ · r∗ ,

(3.24)

lim ∥ γ ((x, u), (y, w)) (τ ) ∥=∥ γ ((x, u), (y, w)) (0) ∥= 0.

(3.25)

which implies τ →0+

′ As well, for any τ ∈ (0, 1), the relation (3.22) is fulfilled and, moreover, (y, w) ∈ / Fa(τ ) (see (3.19)). Thus,

W ((y, w) + γ ((x, u), (y, w)) (τ )) = W ((y, w) + (z, µ)τ − (y, w))

(3.26)

= W ((z, µ)τ ) < W (y, w). By (3.23), it follows (y, w) + γ ((x, u), (y, w)) (τ ) ∈ F and, in consequence, the condition (3.11) is satisfied for all τ ∈ (0, 1). Also, (3.23) involves γ ((x, u), (y, w)) (τ )|∂Θ = (0, u),

u ∈ U,

(3.27)

for all τ ∈ (0, 1). The proof is complete. □ Corollary 3.1. Any local optimal solution associated with the considered constrained variational control problem (P ) is also its global optimal solution if and only if, for any (x, u), (y, w) ∈ F, with W (x, u) − ( )2 W (y, w) < 0, there exists γ : X × U → S, with S = {s : [0, 1] → X × U | (∃) limτ →0+ ∥ s(τ ) ∥=∥ s(0) ∥= 0}, such that (y, w) + γ ((x, u), (y, w)) (τ ) ∈ X × U, and, for all τ ∈ (0, 1), it is verified γ ((x, u), (y, w)) (τ )|∂Θ = (0, u), u ∈ U, and: W ((y, w) + γ ((x, u), (y, w)) (τ )) < W (y, w), Gβ ((y, w) + γ ((x, u), (y, w)) (τ )) ≤ 0,

β ∈ Q.

(3.28) (3.29)

Proof . Applying Theorems 3.1 and 3.2, the result is proved. 4. Illustrative application In order to illustrate the effectiveness of the aforementioned result, for m = 2, n = k = 1, q = 4 (see Section 2) and Θ a square fixed by the diagonally opposite points θ0 = (0, 0) and θ1 = (1, 1) in R2 , consider the non-convex variational control problem { ] } ∫ [ 2 u (θ) 1 2 (BP ) min W (x, u) = − u(θ) + 2 dθ dθ (4.1) 4 (x,u) Θ subject to

S. Treanţă, M. Arana-Jiménez and T. Antczak / Nonlinear Analysis 191 (2020) 111640

∂x (θ) = u2 (θ) + u(θ) − 1, ∂θ1 ∂x (θ) = u2 (θ) + u(θ) − 1, ∂θ2

(4.2) (4.3)

x2 (θ) − 2x(θ) ≤ 0,

(4.4)

x2 (θ) + 2x(θ) − 3 ≤ 0,

(4.5)

x(0, 0) = 0, (

7

x(1, 1) = 2,

(4.6)

) 2

where θ = θ1 , θ ∈ Θ. Further, we assume in the considered variational control problem that we have interest only for affine state functions. In the previous application, we have f : Θ × R × R → R,

gβ : J 1 (Θ, R) × R → R,

with f (θ, x(θ), u(θ)) =

β = 1, 4,

u2 (θ) − u(θ) + 2, 4

∂x (θ), ∂θ1 ∂x g2 (θ, x(θ), xν (θ), u(θ)) = u2 (θ) + u(θ) − 1 − 2 (θ), ∂θ g1 (θ, x(θ), xν (θ), u(θ)) = u2 (θ) + u(θ) − 1 −

(4.7)

(4.8) (4.9) (4.10)

g3 (θ, x(θ), xν (θ), u(θ)) = x2 (θ) − 2x(θ),

(4.11)

g4 (θ, x(θ), xν (θ), u(θ)) = x2 (θ) + 2x(θ) − 3,

(4.12)

accompanied by the boundary conditions x(0, 0) = 0, x(1, 1) = 2. With the above mathematical objects and X × U, F having the same meaning as in Section 2, for (x, u), (y, w) ∈ F with W (x, u) − W (y, w) < 0, define γ ((x, u), (y, w)) : [0, 1] → X × U, γ ((x, u), (y, w)) (τ ) = (z, µ)τ − (y, w), ( 1 ) where (z, µ)τ = θ + θ2 , 1 is an optimal solution in (BPτ ) subject to

min W (z, µ)

(4.13)

(4.14)

(z,µ)

(z, µ) ∈ cl (Bτ (y, w)) ∩ F,

(4.15)

with τ ∈ (0, 1). We get 0 <∥ γ ((x, u), (y, w)) (τ ) ∥=∥ (z, µ)τ − (y, w) ∥≤ τ,

(4.16)

lim ∥ γ ((x, u), (y, w)) (τ ) ∥=∥ γ ((x, u), (y, w)) (0) ∥= 0.

(4.17)

which implies τ →0+

Also, for any τ ∈ (0, 1), it results (see Theorem 3.2) that the following relation ∅= ̸ Fτ′ ⊂ (cl (Bτ (y, w)) ∩ F) \ int (cl (Bτ (y, w)) ∩ F)

(4.18)

is fulfilled, where Fτ′ denotes the set of optimal solutions for (BPτ ). Moreover, by using (4.16), we have (y, w) ∈ / Fτ′ . In consequence, W ((y, w) + γ ((x, u), (y, w)) (τ )) = W ((y, w) + (z, µ)τ − (y, w)) = W ((z, µ)τ ) < W (y, w).

(4.19)

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S. Treanţă, M. Arana-Jiménez and T. Antczak / Nonlinear Analysis 191 (2020) 111640

By direct computation, it follows (y, w) + γ ((x, u), (y, w)) (τ ) ∈ F and, in consequence, the condition Gβ ((y, w) + γ ((x, u), (y, w)) (τ )) ≤ 0,

β ∈ Q = {1, 2, 3, 4} ,

(4.20)

is satisfied for all τ ∈ (0, 1). Also, (4.13) involves γ ((x, u), (y, w)) (τ )|∂Θ = (0, u),

τ ∈ (0, 1),

(4.21)

where u = 1 − w|∂Θ . In the case when (x, u), (y, w) ∈ F and W (x, u)−W (y, w) < 0 is not verified, then we set γ ((x, u), (y, w)) : [0, 1] → X × U, γ ((x, u), (y, w)) (τ ) = 0. 5. Conclusion In this paper, in accordance with the connections between mathematical programming and classical calculus of variations, we have investigated optimality of PDE&PDI-constrained variational control problems. Namely, we have established an efficient condition such that any local optimal solution of the considered PDE&PDI-constrained variational control problem is also its global optimal solution. The effectiveness of the established condition is illustrated by a suitable example of nonconvex variational control problem. References [1] T. Antczak, M. Arana-Jim´ enez, KT-G-invexity in multiobjective programming, Int. J. Math. Comput. 27 (2016) 23–39. [2] M. Arana-Jim´ enez, T. Antczak, The minimal criterion for the equivalence between local and global optimal solutions in nondifferentiable optimization problem, Math. Methods Appl. Sci. 40 (2017) 6556–6564. [3] F.H. Clarke, Functional. Analysis, Functional Analysis Calculus of Variations and Optimal Control, in: Graduate Texts in Mathematics, vol. 264, Springer, London, 2013. [4] F. Giannessi, Constrained Optimization and Image Space Analysis. I: Separation of Sets and Optimality Conditions, Springer, New York, 2005, pp. 1–395. [5] J.-B. Hiriart-Urruty, C. Lemar´ echal, Fundamentals of Convex Analysis, Springer, Berlin, 2001. [6] R. Horst, A note on functions whose local minima are global, J. Optim. Theory Appl. 36 (1982) 457–463. [7] V.I. Ivanov, Second-order Kuhn–Tucker invex constrained problems, J. Global Optim. 50 (2011) 519–529. [8] D.H. Martin, The essence of invexity, J. Optim. Theory Appl. 47 (1985) 65–76. [9] S ¸ t. Mititelu, S. Treant¸a ˘, Efficiency conditions in vector control problems governed by multiple integrals, J. Appl. Math. Comput. 57 (2018) 647–665. [10] B.T. Polyak, Introduction to Optimization, Optimization Software, Publications Division, New York, 1987. [11] S. Treant¸a ˘, On a modified optimal control problem with first-order PDE constraints and the associated saddle-point optimality criterion, Eur. J. Control, http://dx.doi.org/10.1016/j.ejcon.2019.07.003. [12] S. Treant¸˘ a, On a new class of vector variational control problems, Numer. Funct. Anal. Optim. 39 (2018) 1594–1603. [13] S. Treant¸˘ a, Variational Analysis with Applications in Optimisation and Control, Cambridge Scholars Publishing, ISBN: 978-1-5275-3728-6, 2019. [14] S. Treant¸˘ a, M. Arana-Jim´ enez, KT-pseudoinvex multidimensional control problem, Optim. Control Appl. Meth. 39 (2018) 1291–1300. [15] S. Treant¸a ˘, M. Arana-Jim´ enez, On generalized KT-pseudoinvex control problems involving multiple integral functionals, Eur. J. Control 43 (2018) 39–45. [16] I. Zang, M. Avriel, On functions whose local minima are global, J. Optim. Theory Appl. 16 (1975) 183–190. [17] I. Zang, E.U. Choo, M. Avriel, A note on functions whose local minima are global, J. Optim. Theory Appl. 18 (1976) 555–559. [18] I. Zang, E.U. Choo, M. Avriel, On functions whose stationary points are global minima, J. Optim. Theory Appl. 22 (1977) 195–208.