1 Introduction

Multiobjective optimization is a modern direction of study in science, from reasons related to their real world applications. In this regard, we mention the shortest path method, which involves the length of the paths and their costs. More than that, multiple criteria may refer to the length of a journey, its price, or the number of transfers. Also, the timetable information could be considered as a result of multiobjective optimization, if we have in view the unknown delays. Physics encounters many problems whose solutions can be found by using optimization approach, since a considerable number of them refer mainly to minimization principles. In this respect, there can be mentioned the study of interfaces and elastic manifolds, morphology evaluation of flow lines in high temperature superconductor or the analysis of X-ray data; for a detailed analysis, please see Hartman and Heiko [1], or Biswas et al. [2]. Another field which provides real world multiobjective optimization problems is material sciences, where an optimal estimation of the parameters of the materials is required. Further more such optimization problems can be found also in economics, or game theory, see Ehrgott et al. [3], Gal and Hanne [4] and the references therein.

One of the main directions of research in optimization refers to determining necessary or/and sufficient efficiency conditions for some vector optimization programs, and that of developing various duality results in connection to the primal multiobjective problem. These kinds of outcomes require the use of various types of generalized convexities, a direction of study started by Craven [5] and Hanson [6]. The pseudo-convexity and quasi-convexity provided to be appropriate tools for the development of duality results, please see Bector et al. [7]. Suneja and Srivastava [8] used generalized invexity in order to prove various duality results for multiobjective problems. Osuna-Gómez et al. [9] introduced optimality conditions and duality properties for a class of multiobjective programs under generalized convexity hypotheses. Antczak [10] used B-(p, r)-invexity functions to obtain sufficient optimality conditions for vector problems. Su and Hien [11] used Mordukhovich pseudoconvexity and quasiconvexity to prove strong Karush-Kuhn-Tucker optimality conditions for constrained multiobjective problems. The optimal power flow problem is solved by means of a characterization of the KT-invexity, by Bestuzheva and Hijazi [12]. Suzuki [13] joined quasiconvexity with necessary and sufficient optimality conditions in terms of Greenberg-Pierskalla subdifferential and Martínez-Legaz subdifferential. Jayswal et al. [14] developed duality results for semi-infinite problems in terms of (F, ρ)-V-invexity. The (F, ρ)-convexity introduced by Preda [15] allowed the study of efficiency of multiobjective programs. The same tool was used by Antczak and Pitea [16] to develop sufficient optimality conditions in a geometric setting, or by Antczak and Arana-Jiménez [17] who studied vector optimization problems by additional means of weighting.

The aim of this work is to develop sufficient optimality conditions and duality results, by the use of the generalized convexity introduced by Caristi et al. [18], and also one of the most effective tool in the study of multiobjective optimization, the parametric approach, whose basis were put by Saaty and Gass [19]. The class of problems which are to be proposed in the work refers to minimizing a vector of curvilinear integrals, where the integrand depends also on the velocities. This kind of problems are connected, for example, with Mechanical Engineering, considering that curvilinear integral objectives are frequently used because of their physical meaning as mechanical work, and there is a need to minimize simultaneously such kind of quantities, subject to some suitable constraints.

The paper is organized as follows. Section 2 presents preliminary issues on jet bundles, and the (Φ, ρ)-invexity, needed to develop our theory. Section 3 is dedicated to sufficient efficiency conditions for a multitime multiobjective minimization problem with constraints, by means of the generalized convexity. Section 4 consists of weak, strong, and converse duality results in the sense of Mond-Weir and Wolfe.

2 Preliminaries

2.1 On the First Order Jet Bundle

In order to make our work self contained, we recollect some basic facts on the first order jet bundle, J¹ (T, M), formed by the 1-jets

$j_t^1\varphi$

of the local sections ϕ ∈ Γ_t (ϖ). A 1-jet at the point t is an equivalence class of the sections which have the same value and the same first order partial derivatives at the point t.

If the local sections check the equality ϕ (t) = ψ (t), let (t^α, χⁱ) and (t^α′, χⁱ′) be two adapted coordinate systems around ϕ (t). Suppose the following equalities hold

$\frac{\partial {\varphi }^i}{\partial t^{\alpha }}\left(t\right)=\frac{\partial {\psi }^i}{\partial t^{\alpha }}\left(t\right).$

Then the next relations hold true

$\frac{\partial {\varphi }^{i^{\prime }}}{\partial t^{{\alpha }^{\prime }}}\left(t\right)=\frac{\partial {\psi }^{i^{\prime }}}{\partial t^{{\alpha }^{\prime }}}\left(t\right).$

Definition 1. Two local sections ϕ, ψ ∈ Γ_t (ϖ) are called 1-equivalent at the point t if

$\varphi \left(t\right)=\psi \left(t\right),\frac{\partial {\varphi }^i}{\partial t^{\alpha }}\left(t\right)=\frac{\partial {\psi }^i}{\partial t^{\alpha }}\left(t\right).$

The equivalence class containing the section ϕ is precisely the 1-jet associated with the local section ϕ, at the point t, denoted by

$j_t^1\varphi$

.

Definition 2. The set

$J^1(T,M)=\{j_t^1\varphi |t\in T,\varphi \in {\mathrm{\Gamma }}_t(\varpi )\}$

is called the first order jet bundle.If

$(\mathcal{U},u)$

, u = (t^α, χⁱ) is an adapted coordinate system on the product manifold T × M, the induced coordinate system,

$({\mathcal{U}}^1,u^1)$

, on J¹ (T, M), is defined as

${\mathcal{U}}^1=\left\{j_t^1\varphi |\varphi \left(t\right)\in \mathcal{U}\right\},u^1=\left(t^{\alpha },{\chi }^i,{\chi }_{\alpha }^i\right),$

where

$t^{\alpha }(j_t^1\varphi )=t^{\alpha }(t)$

, and

${\chi }^i(j_t^1\varphi )={\chi }^i(\varphi (t))$

.The pn functions

${\chi }_{\alpha }^i:{\mathcal{U}}^1\to \mathbb{R}$

form the coordinate derivatives.

Proposition 1. On the product manifold T × M, consider

$(\mathcal{U},u)$

the atlas of adapted charts. Then, the corresponding charts

$({\mathcal{U}}^1,u^1)$

form a finite dimensional atlas, of C^∞-class, on the first order jet bundle J¹(T, M).In order to make the presentation more readable, in the sequel we denote π_χ (t) = (t, χ (t), χ_γ (t)), where χ_γ is the derivative of χ with respect to t^γ.

2.2 Lagrange 1-Forms of the First Order

Any Lagrange 1-form of the first order, on the jet space J¹ (T, M), takes the form

$\omega =L_{\alpha }\left({\pi }_{\chi }\left(t\right)\right)d{t}^{\alpha }+M_i\left({\pi }_{\chi }\left(t\right)\right)d{\chi }^i+N_i^{\beta }\left({\pi }_{\chi }\left(t\right)\right)d{\chi }_{\beta }^i,$

where L_α, M_i, and

$N_i^{\beta }$

are Lagrangians of the first order, with the pullback

$\chi *\omega =\left(L_{\alpha }+M_i{\chi }_{\alpha }^i+N_i^{\beta }{\chi }_{\beta \alpha }^i\right)d{t}^{\alpha },$

a Lagrange 1-form of the second order on M. The coefficients

second order Lagrangians, are linear in the second order derivatives. The Pfaff equation ω = 0, and the partial differential equations

$L_{\alpha }+M_i{\chi }_{\alpha }^i+N_i^{\beta }{\chi }_{\beta \alpha }^i=0$

can be associated with the form ω.

Let L_β (π_χ(t)) dt^β be a closed Lagrange 1-form (completely integrable), that is D_βL_α = D_αL_β.

A closed 1-form in a simple-connected domain is an exact one. Its primitive can be expressed as a curvilinear integral,

$\varphi \left(t\right)={\int }_{{\mathrm{\Gamma }}_{t_0,t}}{L}_{\alpha }\left({\pi }_{\chi }\left(s\right)\right)d{s}^{\alpha },\varphi \left(t_0\right)=0,$

or as a system of partial derivative eqations,

$\frac{\partial \varphi }{\partial t^{\alpha }}\left(t\right)=L_{\alpha }\left({\pi }_{\chi }\left(t\right)\right),\varphi \left(t_0\right)=0.$

Suppose there is a Lagrangian-like antiderivative

$L\left({\pi }_{\chi }\left(t\right)\right)={\int }_{{\mathrm{\Gamma }}_{t_0,t}}{L}_{\alpha }\left({\pi }_{\chi }\left(s\right)\right)d{s}^{\alpha },L\left.\left({\pi }_{\chi }\left(t_0\right)\right)\right)=0,$

or D_αL = L_α, where the foregoing pullback is the given closed 1-form,

$\frac{\partial L}{\partial t^{\beta }}+\frac{\partial L}{\partial {\chi }^i}\frac{\partial {\chi }^i}{\partial t^{\beta }}+\frac{\partial L}{\partial {\chi }_{\gamma }^i}\frac{\partial {\chi }_{\gamma }^i}{\partial t^{\beta }}+\frac{\partial L}{\partial {\chi }_{\mu \nu }^i}\frac{\partial {\chi }_{\mu \nu }^i}{\partial t^{\beta }}=L_{\beta },$

which is a completely integrable system of partial derivatives equations, with the unknown function χ(⋅).

Each smooth Lagrangian

$L\left({\pi }_{\chi }(t)\right)$

,

$t\in {\mathbb{R}}_{+}^m$

, leads to two smooth closed 1-forms:

– the differential

$dL=\frac{\partial L}{\partial t^{\gamma }}d{t}^{\gamma }+\frac{\partial L}{\partial {\chi }^i}d{x}^i+\frac{\partial L}{\partial {\chi }_{\gamma }^i}d{\chi }_{\gamma }^i,$

with the components

$\left(\frac{\partial L}{\partial t^{\gamma }},\frac{\partial L}{\partial {\chi }^i}\right)$

, with respect to the corresponding basis

$(d{t}^{\gamma },d{\chi }^i,d{\chi }_{\gamma }^i)$

;

– the restriction of dL to π_χ (t), namely the pullback

${\left.dL\right|}_{{\pi }_{\chi }\left(t\right)}=\left(\frac{\partial L}{\partial t^{\beta }}+\frac{\partial L}{\partial {\chi }^i}\frac{\partial {\chi }^i}{\partial t^{\beta }}+\frac{\partial L}{\partial {\chi }_{\gamma }^i}\frac{\partial {\chi }_{\gamma }^i}{\partial t^{\beta }}\right)d{t}^{\beta },$

of components

$D_{\beta }L=\frac{\partial L}{\partial t^{\beta }}\left({\pi }_{\chi }\left(t\right)\right)+\frac{\partial L}{\partial {\chi }^i}\left({\pi }_{\chi }\left(t\right)\right)\frac{\partial {\chi }^i}{\partial t^{\beta }}\left(t\right)+\frac{\partial L}{\partial {\chi }_{\gamma }^i}\left({\pi }_{\chi }\left(t\right)\right)\frac{\partial {\chi }_{\gamma }^i}{\partial t^{\beta }}\left(t\right),$

with respect to the basis dt^β.

For other important facts on jet bundles, we address the reader to the book of Saunders [20].

2.3 Generalized (Φ, ρ)-Invexity

Our results are developed by means of a suitable generalized convexity, introduced in the following.

Further, let Π = J¹ (T, M) be the first order jet bundle associated to T and M. By

$C^{\infty }\left({\mathrm{\Omega }}_{t_0,t_1},M\right)$

we denote the space of all functions

$\chi :{\mathrm{\Omega }}_{t_0,t_1}\to {\mathbb{R}}^n$

of C^∞-class.

Let

$A:C^{\infty }\left({\mathrm{\Omega }}_{t_0,t_1},M\right)\to {\mathbb{R}}^r$

be a path independent curvilinear vector functional

Now, we introduce the definition of the vectorial (Φ, ρ)-convexity for the vectorial functional A, which will be useful to state the results established in the paper. Before we do this, we give the definition of a convex functional.

Definition 3. The functional

$F:\mathrm{\Pi }\times \mathrm{\Pi }\times C^{\infty }{\mathrm{\Omega }}_{t_0,t_1},{\mathbb{R}}^n\times \mathbb{R}\to \mathbb{R}$

is convex with respect to the third component, if, for all χ (⋅),

$\bar{\chi }(\cdot )$

, η₁ (⋅), η₂ (⋅), the following inequality holds

for q, q₁,

$q_2\in {\mathbb{R}}^n$

, λ ∈ (0, 1).It can be easily proved that a similar property holds, if, instead of λ ∈ (0, 1), and 1−λ, we use λ₁, λ₂, … , λ_k ∈ (0, 1), with

${\sum }_{i=1}^k{\lambda }_i=1$

.Let S be a nonempty subset of

$C^{\infty }\left({\mathrm{\Omega }}_{t_0,t_1},M\right)$

, and

$\bar{\chi }(\cdot )\in S$

be given. Following the footsteps of [18], we have the following definition.

Definition 4. Let

$\rho =({\rho }_1,\dots ,{\rho }_r)\in {\mathbb{R}}^r$

,

$\mathrm{\Phi }:\mathrm{\Pi }\times \mathrm{\Pi }\times {\mathbb{R}}^r\to \mathbb{R}$

be convex with respect to the third component, and

$\mathrm{\Phi }({\pi }_{\chi }(t),{\pi }_{\bar{\chi }}(t);(0,{\rho }_i))\ge 0$

. The vectorial functional A is called (strictly) (Φ, ρ)-convex at the point

$\bar{\chi }(\cdot )$

on S if, for each i,

$i=\overline{1,r}$

, the following inequality

holds for all χ (⋅) ∈ S,

$(\chi (\cdot )\ne \bar{\chi }(\cdot ))$

). If these inequalities are satisfied at each

$\bar{\chi }(\cdot )\in S$

, then A is called (strictly) (Φ, ρ)-convex on S.This class of functionals entails that of (F, ρ)-convexity introduced in [15].

3 Sufficient Efficiency Conditions

The following well-known conventions for equalities and inequalities in case of vector optimization will be used in the sequel.

For any χ = (χ₁, χ₂, … , χ_p),

$\eta =\left({\eta }_1,{\eta }_2,\dots ,{\eta }_p\right)$

, consider.

1) χ = η if and only if χ_i = η_i, for all

$i=\overline{1,p}$

;

2) χ > η if and only if χ_i > η_i, for all

$\overline{1,p}$

;

3) χ ≧ η if and only if χ_i ≥ η_i, for all

$\overline{1,p}$

;

4) χ ≥ η if and only if χ ≧ η, and χ ≠ η.

This product order relation will be used on the hyperparallelepiped

${\mathrm{\Omega }}_{t_0,t_1}$

in

${\mathbb{R}}^p$

, with diagonal opposite points

$t_0=(t_0^1,\dots ,t_0^p)$

, and

$t_1=(t_1^1,\dots ,t_1^p)$

. Assume that

${\gamma }_{t_0,t_1}$

is a piecewise C¹-class curve joining the points t₀ and t₁, and that there exists an increasing piecewise smooth curve in

${\mathrm{\Omega }}_{t_0,t_1}$

which joins the points t₀ and t₁.

Let (T, h) and (M, g) be Riemannian manifolds of dimensions p and n, respectively, with the local coordinates t = (t^α),

$\alpha =\overline{1,p}$

, and χ = (χⁱ),

$i=\overline{1,n}$

, respectively, and Π = J¹ (T, M).

The closed Lagrange 1-forms densities of C^∞-class

$u_{\alpha }=\left(u_{\alpha }^i\right):\mathrm{\Pi }\to {\mathbb{R}}^r,i=\overline{1,r},\alpha =\overline{1,p},$

produce the following path independent curvilinear functionals

$\begin{array}{c}\hfill U^i\left(x\left(\cdot \right)\right)={\int }_{{\gamma }_{t_0,t_1}}{u}_{\alpha }^i\left({\pi }_{\chi }\left(t\right)\right)d{t}^{\alpha },i=\overline{1,r},\alpha =\overline{1,p},\hfill \end{array}$

where π_χ(t) = (t, χ(t), χ_γ(t)), and

${\chi }_{\gamma }(t)=\frac{\partial \chi }{\partial t^{\gamma }}(t)$

,

$\gamma =\overline{1,p}$

, are partial velocities.

Presume that the Lagrange densities matrix

$g=\left(g_a^j\right):\mathrm{\Pi }\to {\mathbb{R}}^{ms},a=\overline{1,s},j=\overline{1,m},m<n,$

of C^∞-class leads to the partial differential inequalities

and the Lagrange densities matrix

$h=\left(h_a^l\right):\mathrm{\Pi }\to {\mathbb{R}}^{ms},a=\overline{1,s},l=\overline{1,z},z<n,$

defines the partial differential equalities

In the paper, we consider the multitime multiobjective variational problem

$(\mathcal{CUP})$

of minimizing a vector of path independent curvilinear functionals defined by

$\begin{array}{c}\hfill \min U\left(\chi \left(\cdot \right)\right)=\left(U^1\left(\chi \left(\cdot \right)\right),\dots ,U^r\left(\chi \left(\cdot \right)\right)\right)\hfill \\ \hfill g\left({\pi }_{\chi }\left(\cdot \right)\right)\leqq 0,\hfill \\ \hfill h\left({\pi }_{\chi }\left(\cdot \right)\right)=0,\hfill \\ \hfill \chi \left(t_0\right)={\chi }_0,\chi \left(t_1\right)={\chi }_1.\hfill \end{array}\left(\mathcal{CUP}\right)$

Let

$\begin{array}{c}\hfill D=\left\{\chi \in C^{\infty }\left({\mathrm{\Omega }}_{t_0,t_1},M\right):t\in {\mathrm{\Omega }}_{t_0,t_1},\chi \left(t_0\right)={\chi }_0,\chi \left(t_1\right)={\chi }_1,g\left({\pi }_{\chi }\left(t\right)\right)\leqq 0,h\left({\pi }_{\chi }\left(t\right)\right)=0\right\}\hfill \end{array}$

denote the set all feasible solutions of problem

$(\mathcal{CUP})$

.

Definition 5. A feasible solution

$\bar{\chi }(\cdot )\in D$

is called an efficient solution to the problem

$(\mathcal{CUP})$

if there is no other feasible solution χ (⋅) ∈ D such that

If, in this relation, we use the strict inequality, then

$\bar{\chi }(\cdot )$

is called a weakly efficient solution to the problem

$(\mathcal{CUP})$

.In [21] were proved necessary optimality conditions for a problem similar to

$(\mathcal{CUP})$

; for our case we obtain the next theorem.

Theorem 1. Let

$\bar{\chi }(\cdot )\in D$

be a normal efficient solution in multitime multiobjective problem

$(\mathcal{CUP})$

. Then there exist the vector

$\mathrm{\Lambda }\in {\mathbb{R}}^r$

and the smooth functions

$M:{\mathrm{\Omega }}_{t_0,t_1}\to {\mathbb{R}}^{msp}$

,

$N:{\mathrm{\Omega }}_{t_0,t_1}\to {\mathbb{R}}^{rsp}$

such that

The following theorem establishes sufficient conditions of efficiency for the problem

$(\mathcal{CUP})$

.

Theorem 2. Presume that the following conditions are fulfilled:

1)

$\bar{\chi }(\cdot )\in D$

, Λ, M (⋅) and N (⋅) satisfy the necessary conditions of efficiency (Eqs 1–3).

2) The objective functional U is (Φ, ρ_U)-convex with regard to its third argument at

$\bar{\chi }(\cdot )$

on D.

3)

${\int }_{{\gamma }_{t_0,t_1}}⟨M_{\alpha j}(\cdot ),g^j({\pi }_{\chi }(\cdot ))⟩d{t}^{\alpha }$

,

$j=\overline{1,m}$

, are

$(\mathrm{\Phi },{\rho }_{g_j})$

-convex with regard to its third argument at

$\bar{\chi }(\cdot )$

on D;

4)

${\int }_{{\gamma }_{t_0,t_1}}⟨N\alpha l(\cdot ),h^l({\pi }_{\chi }(\cdot ))⟩d{t}^{\alpha }$

,

$l=\overline{1,z}$

, are

$(\mathrm{\Phi },{\rho }_{h_l})$

-convex with regard to its third argument at

$\bar{\chi }(\cdot )$

on D;

5)

$⟨\mathrm{\Lambda },{\rho }_U⟩+{\sum }_{j=1}^m{\rho }_{g_j}+{\sum }_{l=1}^z{\rho }_{h_l}\geqq 0$

.

Then

$\bar{\chi }(\cdot )$

is an efficient solution to the problem

$(\mathcal{CUP})$

.

Proof 1. Assume that

$\bar{\chi }(\cdot )$

, Λ, M, and N fulfill the conditions from relations (Eqs 1–3), and that

$\bar{\chi }(\cdot )$

is not an efficient solution to problem

$(\mathcal{CUP})$

. In this case, there can be found

$\tilde{\chi }(\cdot )\in \mathrm{\Gamma }({\mathrm{\Omega }}_{t_0,t_1})$

such that

more precisely

with at least one index for which the inequality is a strict one.Taking advantage of the hypothesis 2), and the (Φ, ρ)-invexity, the previous relations compel

which, by inequalities (Eq. 4), imply that

where at least one inequality is a strict one. Multiplying the previous inequality by Λ_i accordingly,

$i=\overline{1,r}$

, and dividing by

$L={\sum }_{i=1}^r{\mathrm{\Lambda }}_i+m+z$

, we get

On the other hand,

which leads, by the (Φ, ρ)-invexity, to

Now, by the properties of h,

$\bar{\chi }(\cdot )$

, and

$\tilde{\chi }(\cdot )$

, we get

which leads to

Using the convexity of the functional F in the third component, and adding inequalities (Eqs 5, 6), it follows that

By the equality from (Eq. 1), this inequality implies

which is a contradiction with the properties of the function Φ.Therefore, our assumption was false, and

$\bar{\chi }(\cdot )$

is an efficient solution to the problem

$(\mathcal{CUP})$

.

4 Dual Programming Theory

Consider the dual problem to

$(\mathcal{CUP})$

in the sense of Mond-Weir

$\begin{array}{ccc}\hfill & \hfill & \max U\left(\chi \left(\cdot \right)\right)\hfill \\ \hfill & \hfill & ⟨\mathrm{\Lambda },\frac{\partial u_{\alpha }}{\partial \chi }\left({\pi }_{\bar{\chi }}\left(t\right)\right)⟩+⟨M_{\alpha }\left(t\right),\frac{\partial g}{\partial \chi }\left({\pi }_{\bar{\chi }}\left(t\right)\right)⟩+⟨N_{\alpha }\left(t\right),\frac{\partial h}{\partial \chi }\left({\pi }_{\bar{\chi }}\left(t\right)\right)⟩\left(\mathcal{DCUP}\right)\hfill \\ \hfill & \hfill & -D_{\gamma }\left(⟨\mathrm{\Lambda },\frac{\partial u_{\alpha }}{\partial {\chi }_{\gamma }}\left({\pi }_{\bar{\chi }}\left(t\right)\right)⟩+⟨M_{\alpha }\left(t\right),\frac{\partial g}{\partial {\chi }_{\gamma }}\left({\pi }_{\bar{\chi }}\left(t\right)\right)⟩+⟨N_{\alpha }\left(t\right),\frac{\partial h}{\partial {\chi }_{\gamma }}\left({\pi }_{\bar{\chi }}\left(t\right)\right)⟩\right)=0,\hfill \\ \hfill & \hfill & ⟨M_{\alpha j}\left(t\right),g^j\left({\pi }_{\bar{\chi }}\left(t\right)\right)⟩+⟨N_{\alpha j}\left(t\right),h^j\left({\pi }_{\chi }\left(t\right)\right)⟩\ge 0,\hfill \\ \hfill & \hfill & \mathrm{\Lambda }\geqq 0,t\in {\mathrm{\Omega }}_{t_0,t_1},\alpha =\overline{1,p},j=\overline{1,m}\left(m=z\right).\hfill \end{array}$

Let ΔD be the set of the feasible solutions to the dual problem

$(\mathcal{DCUP})$

, and Δ = {η (⋅):[η (⋅), λ, M(⋅), ν(⋅)] ∈ ΔD}. By using (Φ, ρ)-convexity hypothesis, weak, strong, and converse duality results may be stated and proved, as in the sequel.

We start with a weak duality result, as follows.

Theorem 3. Suppose that

$\bar{\chi }(\cdot )$

and [η (⋅), λ, M (⋅), N (⋅)] are feasible solutions to the problems

$(\mathcal{CUP})$

, and

$(\mathcal{DCUP})$

, respectively. Additionally, presume that the next hypotheses are satisfied:

1) The objective functional U is (Φ, ρ_U)-convex with regard to its third argument at η(⋅).

2)

${\int }_{{\gamma }_{t_0,t_1}}\left(⟨M_{\alpha j}(t),g^j({\pi }_{\bar{\chi }}(t))⟩+⟨N_{\alpha j}(t),h^j({\pi }_{\chi }(t))⟩\right)d{t}^{\alpha }$

,

$j=\overline{1,m}$

, are

$(\mathrm{\Phi },{\rho }_{g{h}_j})$

-convex with regard to its third argument at

$\bar{\chi }(\cdot )$

;

3)

$⟨\mathrm{\Lambda },{\rho }_U⟩+{\sum }_{j=1}^m{\rho }_{g{h}_j}\geqq 0$

.

Then

$U(\bar{\chi }(\cdot ))\nleqq U(\eta (\cdot ))$

.

Proof 2. Presume that

$U(\bar{\chi }(\cdot ))\le U(\eta (\cdot ))$

, that is

$U_i\left(\bar{\chi }\left(\cdot \right)\right)\le U_i\left(\eta \left(\cdot \right)\right),i=\overline{1,r},$

where the inequality is strict for at least one of the indices.By the use of the (Φ, ρ)-invexity related to U, the previous relations imply

$\begin{array}{ccc}\hfill & \hfill & {\int }_{{\gamma }_{t_0,t_1}}\mathrm{\Phi }\left({\pi }_{\bar{\chi }}\left(t\right),{\pi }_{\eta }\left(t\right);\frac{\partial u_{\alpha }^i}{\partial \eta }\left({\pi }_{\eta }\left(t\right)\right)-D_{\gamma }\left(\frac{\partial u_{\alpha }^i}{\partial {\eta }_{\gamma }}\left({\pi }_{\eta }\left(t\right)\right)\right),{\rho }_{U_i}\right)d{t}^{\alpha }\hfill \\ \hfill & \leqq \hfill & U^i\left(\bar{\chi }\left(\cdot \right)\right)-U^i\left(\eta \left(\cdot \right)\right)\hfill \\ \hfill & \leqq \hfill & 0,i=\overline{1,r},\hfill \end{array}$

We multiply each relation by Λ_i,

$i=\overline{1,r}$

, and then dividing by

$L={\sum }_{i=1}^r{\mathrm{\Lambda }}_i+m$

, it follows that

$\sum _{i=1}^r\frac{{\mathrm{\Lambda }}_i}{L}{\int }_{{\gamma }_{t_0,t_1}}\mathrm{\Phi }\left({\pi }_{\bar{\chi }}\left(t\right),{\pi }_{\eta }\left(t\right);\frac{\partial u_{\alpha }^i}{\partial \eta }\left({\pi }_{\eta }\left(t\right)\right)-D_{\gamma }\left(\frac{\partial u_{\alpha }^i}{\partial {\eta }_{\gamma }}\left({\pi }_{\eta }\left(t\right)\right)\right),{\rho }_{U_i}\right)d{t}^{\alpha }<0.(8)$

Having in mind assumption (Eq. 2) from the theorem, we get, by the (Φ, ρ)-invexity, that

The properties of F, jointly with inequalities (Eq. 8), and (Eq. 10), imply

By the constraints of the dual problem

$(\mathcal{DCUP})$

, this inequality leads to

which is a contradiction with the properties of the function Φ.Therefore, our assumption was false, and U (χ(⋅))≰U (η(⋅)).In the following, we provide a strong duality result and also a converse duality one.

Theorem 4. Consider that χ (⋅) is an efficient solution to the primal problem

$(\mathcal{CUP})$

. Then there exists λ, M, N so that [χ (⋅), λ, M (⋅), N (⋅)] ∈ ΔD. More than that, if assumptions (Eqs 2–5) from Theorem 2 are fulfilled. then [χ (⋅), λ, M (⋅), N (⋅)] is an efficient solution to the dual problem

$(\mathcal{DCUP})$

.

Theorem 5. Let

$(\eta (\cdot ),\lambda ,M(\cdot )),\left.N(\cdot )\right)$

be an efficient solution to the dual problem

$(\mathcal{DCUP})$

. Assume that conditions 2)-5) from Theorem 2 are satisfied. Then η (⋅) is an efficient solution to the primal problem

$(\mathcal{CUP})$

.In a similar manner, a dual problem in the sense of Wolfe can be associated to our vector problem

$(\mathcal{CUP})$

. First, we introduce the objective of this problem.

where

$e={(1,\dots ,1)}^T\in {\mathbb{R}}^r$

.The associated multitime multiobjective problem dual to

$(\mathcal{CUP})$

in the sense of Wolfe is

$(\mathcal{WDCUP})$

, as in the following.

Again, by the use of the notion of (Φ, ρ)-convexity, some weak, strong and converse duality results can be stated and proved, in a similar manner.

Data Availability Statement

The original contributions presented in the study are included in the article/Supplementary Material, further inquiries can be directed to the corresponding author.

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Reference