Minimization of Vectors of Curvilinear Functionals on Second-Order Jet Bundle: Dual Program Theory

Abstract

A previous paper (2011), Pitea and Postolache, considered the problem of minimization of vectors of curvilinear functionals (well known as mechanical work), thought as multitime multiobjective variational problem, subject to PDE and/or PDI constraints. They have chosen the suitable framework offered by the second-order jet bundle, and initiated an optimization theory for this class of problems by introducing necessary conditions. As natural continuation of these results, the present work introduces a dual program theory, the general setting, and the theory which is new as a whole, containing our results.

1. Introduction

Fractional programming problems arise in wide areas of research in pure and applied sciences and in the new technology as well. We have in mind portfolio selection, stock cutting, game theory, and various decision problems in management science. The study of duality for this class of problems is an active area of research due to its wide scope of applications, see [1] by Chinchuluun and Pardalos, for an excellent paper in multiobjective optimization.

The duality theory for multiobjective optimization problems based on a vector-valued Lagrangian function were extensively investigated in [2] by Bitran, and [3] by Tanino and Sawaragi. Several authors were interested in this problem, see [4] by Egudo, [5] by Mond and Husain, [6] by Weir and Mond. Later, an interesting and rich duality theory for such problems was developed by Aghezzaf and Hachimi, [7], Preda [8, 9], and recently, duality problems were studied in [10] by Boţ and Grad, [1] by Chinchuluun and Pardalos, [11] by Pitea et al.

In the problems of our study, the objective function is of curvilinear integral type and could mould the mechanical work when we have in mind design problems in engineering. Also, such kind of problems proved to be useful tools for minimizing the cost or maximizing the profit when we discuss economical problems.

Motivated by the study reported above, in a very recent work [12], Pitea and Postolache considered the problem of minimization of vectors of curvilinear functionals (well known as mechanical work), thought as multitime multiobjective variational problem, subject to PDE and/or PDI constraints. They have chosen the suitable framework offered by the second-order jet bundle and initiated an optimization theory for this class of problems by introducing necessary conditions. As natural continuation of these results, the present work introduces a dual program theory. It is organized as follows. Next, in Section 2, we introduce the general setting, while in Section 3 we prove our results. Finally, we conclude the paper.

2. General Setting

Let (𝑇,ℎ) and (𝑀,𝑔) be Riemannian manifolds of dimensions 𝑝 and 𝑛, respectively. The local coordinates on 𝑇 and 𝑀 will be written 𝑡=(𝑡^𝛼) and 𝑥=(𝑥^𝑖), respectively. Let 𝐽²(𝑇,𝑀) be the second-order jet bundle associated to 𝑇 and 𝑀, see [13].

Throughout this work, we use the customary relations between two vectors of the same dimension, [11]. With the product-order relation on ℝ^𝑝, the hyperparallelepiped Ω_𝑡0,𝑡1, in ℝ^𝑝, with the diagonal opposite points 𝑡₀=(𝑡¹₀,…,𝑡^𝑝₀) and 𝑡₁=(𝑡¹₁,…,𝑡^𝑝₁), can be written as interval [𝑡₀,𝑡₁]. Suppose 𝛾_𝑡0,𝑡1 is a piecewise 𝐶²-class curve joining the points 𝑡₀ and 𝑡₁.

To simplify the notations, denote by𝑥_𝛾(𝑡)=𝜕𝑥𝜕𝑡^𝛾(𝑡),𝛾=1,𝑝,𝑥_𝜃𝜎(𝑡)=𝜕²𝑥𝜕𝑡^𝜃𝜕𝑡^𝜎(𝑡),𝜃,𝜎=1,𝑝, (2.1)the partial velocities and partial accelerations, respectively. Also, in our subsequent theory, we will set 𝜋_𝑥(𝑡)=(𝑡,𝑥(𝑡),𝑥_𝛾(𝑡),𝑥_𝜃𝜎(𝑡)).

On 𝐽²(𝑇,𝑀) with values on ℝ^𝑟, consider the closed Lagrange 1-forms densities of 𝐶^∞-class: 𝑓_𝛼=𝑓^ℓ_𝛼,𝑘_𝛼=𝑘^ℓ_𝛼,ℓ=1,𝑟,𝛼=1,𝑝, (2.2) which determine the following path-independent functionals: 𝐹^ℓ(𝑥(⋅))=_{𝛾𝑡01,𝑡}𝑓^ℓ_𝛼𝜋_𝑥(𝑡)𝑑𝑡^𝛼,𝐾^ℓ(𝑥(⋅))=_{𝛾𝑡01,𝑡}𝑘^ℓ_𝛼𝜋_𝑥(𝑡)𝑑𝑡^𝛼. (2.3) The closeness conditions (complete integrability conditions) are𝐷_𝛽𝑓^ℓ_𝛼=𝐷_𝛼𝑓^ℓ_𝛽,𝐷_𝛽𝑘^ℓ_𝛼=𝐷_𝛼𝑘^ℓ_𝛽,𝛼,𝛽=1,𝑝,𝛼≠𝛽,ℓ=1,𝑟, (2.4)

where 𝐷_𝛽 is the total derivative.

We accept that the Lagrange matrix densities:𝑔=𝑔^𝑏_𝑎∶𝐽²(𝑇,𝑀)⟶ℝ^𝑚𝑑,𝑎=1,𝑑,𝑏=1,𝑚,𝑚<𝑛, (2.5)

of 𝐶^∞-class define the partial differential inequations (PDI) of evolution 𝑔𝜋_𝑥(𝑡)≦0,𝑡∈Ω_𝑡0,𝑡1, (2.6) and the Lagrange matrix densities, ℎ=ℎ^𝑏_𝑎∶𝐽²(𝑇,𝑀)⟶ℝ^𝑞𝑑,𝑎=1,𝑠,𝑏=1,𝑞,𝑞<𝑛, (2.7) define the partial differential equations (PDE), of evolution ℎ𝜋_𝑥(𝑡)=0,𝑡∈Ω_𝑡0,𝑡1. (2.8)

For each ℓ=1,𝑟, suppose 𝐾^ℓ(𝑥(⋅))>0, and consider 𝐹(𝑥(⋅))𝐾(𝑥(⋅=𝐹))¹(𝑥(⋅))𝐾¹(𝑥(⋅𝐹)),…,^𝑟(𝑥(⋅))𝐾^𝑟(𝑥(⋅)). (2.9)

With conditions (2.6) and (2.8), we denote by ℱΩ_𝑡0,𝑡1=𝑥∈𝐶^∞Ω_𝑡0,𝑡1,𝑀∣𝑥𝑡₀=𝑥₀,𝑥𝑡₁=𝑥₁,𝑔𝜋_𝑥(𝑡)≦0,ℎ𝜋_𝑥(𝑡)=0,𝑡∈Ω_𝑡0,𝑡1 (2.10) the set of all feasible solutions of problem: ()𝐹(𝑥(⋅MFPmin))𝐾(𝑥(⋅,(⋅)Ω))subjectto𝑥∈ℱ_𝑡0,𝑡1, (2.11) a PDI- and/or PDE-constrained minimum problem.

In [12], we introduced necessary efficiency conditions for problem (MFP). The aim of this work is to introduce and study two dual programs:(1)the multitime multiobjective fractional variational problem (MFP) of minimizing a vector of quotients of path-independent curvilinear functionals;(2)the multiobjective variational dual problem: (MFD)max_𝑦(⋅)𝐹𝑦(⋅)𝐾𝑦(⋅Λ)subjectto_ℓ10𝜕𝑓^ℓ_𝛼𝜋𝜕𝑦^𝑦(𝑡)−Λ_ℓ²⁰𝜕𝑘^ℓ_𝛼𝜋𝜕𝑦^𝑦(𝑡)+𝜇_𝛼(𝑡),𝜕𝑔𝜋𝜕𝑦^𝑦(𝑡)+𝜈_𝛼(𝑡),𝜕ℎ𝜋𝜕𝑦^𝑦(𝑡)−𝐷_𝛾Λ_ℓ¹⁰𝜕𝑓^ℓ_𝛼𝜕𝑦_𝛾𝜋_𝑦(𝑡)−Λ_ℓ²⁰𝜕𝑘^ℓ_𝛼𝜕𝑦_𝛾𝜋_𝑦(𝑡)+𝜇_𝛼(𝑡),𝜕𝑔𝜕𝑦_𝛾𝜋_𝑦(𝑡)+𝜈_𝛼(𝑡),𝜕ℎ𝜕𝑦_𝛾𝜋_𝑦(𝑡)+𝐷²_𝜃𝜎𝜆,𝜕𝑓_𝛼𝜕𝑥_𝜃𝜎𝜋_𝑦(𝑡)+𝜇_𝛼(𝑡),𝜕𝑔𝜕𝑥_𝜃𝜎𝜋_𝑦(𝑡)+𝜈_𝛼(𝑡),𝜕ℎ𝜕𝑥_𝜃𝜎𝜋_𝑦(𝑡)=0,𝑡∈Ω_𝑡0,𝑡1,𝛼=𝜇1,𝑝_𝛼(𝑡),𝑔𝜋_𝑦(𝑡)+𝜈_𝛼(𝑡),ℎ𝜋_𝑦(𝑡)≧0,𝛼=1,𝑝,𝑡∈Ω_𝑡0,𝑡1Λ¹⁰≥0, (2.12) taking into account that the functions 𝑥(𝑡) and 𝑦(𝑡) have to satisfy the boundary conditions 𝑥(𝑡₀)=𝑥₀, 𝑥(𝑡₁)=𝑥₁, or 𝑥(𝑡)|_{𝜕Ω𝑡01,𝑡}=𝜒= given, respectively, 𝑦(𝑡₀)=𝑥₀, 𝑦(𝑡₁)=𝑥₁, or 𝑦(𝑡)|_{𝜕Ω𝑡01,𝑡}=𝜒= given, the partial differential inequations of evolution (2.6), and the partial differential equations of evolution (2.8).

To develop the theory in our main section, we need the notion of efficient solution.

Definition 2.1. A feasible solution 𝑥^∘(⋅)∈ℱ(Ω_𝑡0,𝑡1) is called efficient solution for the program (MFP) if and only if for any feasible solution 𝑥(⋅)∈ℱ(Ω_𝑡0,𝑡1), one has the implication:

𝐹(𝑥(⋅))𝐾(𝑥(⋅≧𝐹𝑥))^∘(⋅)𝐾(𝑥^∘(⋅⟹𝐹(𝑥(⋅))))𝐾(𝑥(⋅=𝐹𝑥))^∘(⋅)𝐾(𝑥^∘(⋅)). (2.13)

3. Main Results

To state our results, we have to introduce an appropriate generalized convexity. For this purpose, consider 𝜌 be a real number, 𝑏∶𝐶^∞(Ω_𝑡0,𝑡1,𝑀)×𝐶^∞(Ω_𝑡0,𝑡1,𝑀)→[0,∞) a functional, and 𝑎=(𝑎_𝛼), 𝛼=1,𝑝, a closed 1-form. To 𝑎 we associate the curvilinear integral 𝐴(𝑥(⋅))=_{𝛾𝑡01,𝑡}𝑎_𝛼𝜋_𝑥(𝑡)𝑑𝑡^𝛼. (3.1)

Definition 3.1. The functional 𝐴 is called [strictly] (𝜌,𝑏)-quasiinvex at the point 𝑥^∘(⋅) if there exists a vector function 𝜂∶𝐽²(Ω_𝑡0,𝑡1,𝑀)×𝐽²(Ω_𝑡0,𝑡1,𝑀)→ℝ^𝑛, vanishing at the point (𝜋_𝑥∘(𝑡),𝜋_𝑥∘(𝑡)), and the function 𝜃 defined on 𝐶^∞(Ω_𝑡0,𝑡1,𝑀)×𝐶^∞(Ω_𝑡0,𝑡1,𝑀) to ℝ^𝑛, such that for any 𝑥(⋅)[𝑥(⋅)≠𝑥^∘(⋅)], the following implication holds: 𝐴(𝑥(⋅))≤𝐴𝑥^∘(⋅)⟹𝑏𝑥(⋅),𝑥^∘(⋅)_{𝛾𝑡01,𝑡}𝜂𝜋_𝑥(𝑡),𝜋_𝑥∘(𝑡),𝜕𝑎_𝛼𝜋𝜕𝑥^𝑥∘(𝑡)+𝐷_𝛾𝜂𝜋_𝑥(𝑡),𝜋_𝑥∘(𝑡),𝜕𝑎_𝛼𝜕𝑥_𝛾𝜋_𝑥∘(𝑡)+𝐷²_𝜃𝜎𝜂𝜋_𝑥(𝑡),𝜋_𝑥∘(𝑡),𝜕𝑎_𝛼𝜕𝑥_𝜃𝜎𝜋_𝑥∘(𝑡)𝑑𝑡^𝛼[<]≤−𝜌𝑏𝑥(⋅),𝑥^∘(⋅)𝜃𝑥(⋅),𝑥^∘(⋅)². (3.2)
The notion of quasiinvexity is used, in appropriate forms, in recent works for studies of some multiobjective programming problems, for example, see [14] by Nahak and Mohapatra.
Let us denote by 𝜛(𝑥^∘(⋅)) the minimizing functional vector of problem (MFP) at the point 𝑥^∘(⋅)∈ℱ(Ω_𝑡0,𝑡1) and by (𝛿(𝑦(⋅),𝑦_𝛾(⋅),Λ¹⁰,Λ²⁰,𝜇(⋅),𝜈(⋅)) the maximizing functional vector of dual (MFD) at the point (𝑦(⋅),𝑦_𝛾(⋅),Λ¹⁰,Λ²⁰,𝜇(⋅),𝜈(⋅))∈Δ, where Δ is the domain of problem (MFD).

Theorem 3.2 (Weak Duality). Let 𝑥^∘(⋅) be a feasible solution of the problem (𝑀𝐹𝑃) and let 𝑦(⋅) be an efficient solution of problem (𝑀𝐹𝐷). Assume that the following conditions are fulfilled: (a)Λ_ℓ¹⁰>0, Λ_ℓ²⁰>0, Λ_ℓ¹⁰𝐹^ℓ(𝑦(⋅))−Λ_ℓ²⁰𝐾^ℓ(𝑦(⋅))=0, ℓ=1,𝑟; (b)for any ℓ=1,𝑟, the functional 𝐹^ℓ(𝑥(⋅)) is (𝜌^′ℓ,𝑏)-quasiinvex at the point 𝑦(⋅) and the functional −𝐾^ℓ(𝑥(⋅)) is (𝜌^′′ℓ,𝑏)-quasiinvex at the point 𝑦(⋅) with respect to 𝜂 and 𝜃;(c)the functional,_{𝛾𝑡01,𝑡}𝜇_𝛼(𝑡),𝑔𝜋_𝑥(𝑡)+𝜈_𝛼(𝑡),ℎ𝜋_𝑥(𝑡)𝑑𝑡^𝛼, (3.3) is (𝜌^′′′,𝑏)-quasiinvex at the point 𝑦(⋅) with respect to 𝜂 and 𝜃;(d)either the functional of (b) or the functional of (c) is strictly quasiinvex;(e)𝜌^′ℓΛ_ℓ¹⁰+𝜌^′′ℓΛ_ℓ²⁰+𝜌^′′′≥0.Then, the inequality 𝜛(𝑥^∘(⋅))≤𝛿(𝑦(⋅),𝑦_𝛾(⋅),Λ¹⁰,Λ²⁰,𝜇(⋅),𝜈(⋅)) is false.

Proof. From condition (b), it follows that 𝐹^ℓ𝑥^∘(⋅)𝐹^ℓ𝑦(⋅⟹𝑏𝑥_∘)(⋅),𝑦(⋅)_{𝛾𝑡01,𝑡}𝜂𝜋_𝑥∘(𝑡),𝜋_𝑦(𝑡),𝜕𝑓^ℓ_𝛼𝜋𝜕𝑦^𝑦(𝑡+𝐷_𝛾)𝜂𝜋_𝑥∘(𝑡),𝜋_𝑦(𝑡),𝜕𝑓^ℓ_𝛼𝜕𝑦_𝛾𝜋_𝑦(𝑡)+𝐷²_𝜃𝜎𝜂𝜋_𝑥∘(𝑡),𝜋_𝑦(𝑡),𝜕𝑓^ℓ_𝛼𝜕𝑦_𝜃𝜎𝜋_𝑦(𝑡)𝑑𝑡^𝛼≦−𝜌^′ℓ𝑏𝑥^∘(⋅),𝑦(⋅)𝜃𝑥^∘(⋅),𝑦(⋅)²,ℓ=1,𝑟,−𝐾^ℓ𝑥^∘(⋅)−𝐾^ℓ𝑦(⋅)⟹𝑏𝑥^∘(⋅),𝑦(⋅)_{𝛾𝑡01,𝑡}−𝜂𝜋_𝑥∘(𝑡),𝜋_𝑦(𝑡),𝜕𝑘^ℓ_𝛼𝜋𝜕𝑦^𝑦(𝑡)−𝐷_𝛾𝜂𝜋_𝑥∘(𝑡),𝜋_𝑦(𝑡),𝜕𝑘^ℓ_𝛼𝜕𝑦_𝛾𝜋_𝑦(𝑡)−𝐷²_𝜃𝜎𝜂𝜋_𝑥∘(𝑡),𝜋_𝑦(𝑡),𝜕𝑘^ℓ_𝛼𝜕𝑦_𝜃𝜎𝜋_𝑦(𝑡)𝑑𝑡^𝛼≦𝜌^′′ℓ𝑏𝑥^∘(⋅),𝑦(⋅)𝜃𝑥^∘(⋅),𝑦(⋅)²,ℓ=1,𝑟. (3.4)(3.5) We multiply (3.4) by Λ_ℓ¹⁰>0 and (3.5) by Λ_ℓ²⁰>0. We make the sum and we obtain the following implications: Λ_ℓ¹⁰𝐹^ℓ𝑥^∘(⋅)−Λ_ℓ²⁰𝐾^ℓ𝑥^∘(⋅⟹𝑏𝑥)≦0_∘(⋅),𝑦(⋅)_{𝛾𝑡01,𝑡}𝜂𝜋_𝑥∘(𝑡),𝜋_𝑦(𝑡),Λ_ℓ¹⁰𝜕𝑓^ℓ_𝛼𝜋𝜕𝑦^𝑦(𝑡)−Λ_ℓ²⁰𝜕𝑘^ℓ_𝛼𝜋𝜕𝑦^𝑦(𝑡+𝐷_𝛾)𝜂𝜋_𝑥∘(𝑡),𝜋_𝑦(𝑡),Λ_ℓ¹⁰𝜕𝑓^ℓ_𝛼𝜕𝑦_𝛾𝜋_𝑦(𝑡)−Λ_ℓ²⁰𝜕𝑘^ℓ_𝛼𝜕𝑦_𝛾𝜋_𝑦(𝑡)+𝐷²_𝜃𝜎𝜂𝜋_𝑥∘(𝑡),𝜋_𝑦(𝑡),Λ_ℓ¹⁰𝜕𝑓^ℓ_𝛼𝜕𝑦_𝜃𝜎𝜋_𝑦(𝑡)−Λ_ℓ²⁰𝜕𝑘^ℓ_𝛼𝜕𝑦_𝜃𝜎𝜋_𝑦(𝑡)𝑑𝑡^𝛼≦−𝑏𝑥^∘(⋅),𝑦(⋅)𝜃𝑥^∘(⋅),𝑦(⋅)²𝜌^′ℓΛ_ℓ¹⁰+𝜌^′′ℓΛ_ℓ²⁰. (3.6) According to hypothesis (c), we have _{𝛾𝑡01,𝑡}𝜇_𝛼(𝑡),𝑔𝜋_𝑥∘(𝑡)+𝜈_𝛼(𝑡),ℎ𝜋_𝑥∘(𝑡)𝑑𝑡^𝛼≦_{𝛾𝑡01,𝑡}𝜇_𝛼(𝑡),𝑔𝜋_𝑦(𝑡)+𝜈_𝛼(𝑡),ℎ𝜋_𝑦(𝑡)𝑑𝑡^𝛼⟹𝑏𝑥^∘(⋅),𝑦(⋅)_{𝛾𝑡01,𝑡}𝜂𝜋_𝑥∘(𝑡),𝜋_𝑦(𝑡),𝜇_𝛼(𝑡),𝜕𝑔𝜋𝜕𝑦^𝑦(𝑡)+𝜈_𝛼(𝑡),𝜕ℎ𝜋𝜕𝑦^𝑦(𝑡)+𝐷_𝛾𝜂𝜋_𝑥∘(𝑡),𝜋_𝑦(𝑡),𝜇_𝛼(𝑡),𝜕𝑔𝜕𝑦_𝛾𝜋_𝑦(𝑡)+𝜈_𝛼(𝑡),𝜕ℎ𝜕𝑦_𝛾𝜋_𝑦(𝑡)+𝐷²_𝜃𝜎𝜂𝜋_𝑥∘(𝑡),𝜋_𝑦(𝑡),𝜇_𝛼(𝑡),𝜕𝑔𝜕𝑦_𝜃𝜎𝜋_𝑦(𝑡)+𝜈_𝛼(𝑡),𝜕ℎ𝜕𝑦_𝜃𝜎𝜋_𝑦(𝑡)𝑑𝑡^𝛼≦−𝜌^′′′𝑏𝑥^∘(⋅),𝑦(⋅)𝜃𝑥^∘(⋅),𝑦(⋅)². (3.7) Making the sum of the implications (3.6) and (3.7), it follows that Λ_ℓ¹⁰𝐹^ℓ𝑥^∘(⋅)−Λ_ℓ²⁰𝐾^ℓ𝑥^∘(⋅+_{𝛾𝑡01,𝑡})𝜇_𝛼(𝑡),𝑔𝜋_𝑥∘(𝑡)+𝜈_𝛼(𝑡),ℎ𝜋_𝑥∘(𝑡)𝑑𝑡^𝛼−_{𝛾𝑡01,𝑡}𝜇_𝛼(𝑡),𝑔𝜋_𝑦(𝑡)+𝜈_𝛼(𝑡),ℎ𝜋_𝑦(𝑡)𝑑𝑡^𝛼⟹𝑏𝑥≦0_∘(⋅),𝑦(⋅)_{𝛾𝑡01,𝑡}𝜂𝜋_𝑥∘(𝑡),𝜋_𝑦(𝑡),Λ_ℓ¹⁰𝜕𝑓^ℓ_𝛼𝜋𝜕𝑦^𝑦(𝑡)−Λ_ℓ²⁰𝜕𝑘^ℓ_𝛼𝜋𝜕𝑦^𝑦(𝑡)+𝜇_𝛼(𝑡),𝜕𝑔𝜋𝜕𝑦^𝑦(𝑡)+𝜈_𝛼(𝑡),𝜕ℎ𝜋𝜕𝑦^𝑦(𝑡)+𝐷_𝛾𝜂𝜋_𝑥∘(𝑡),𝜋_𝑦(𝑡),Λ_ℓ¹⁰𝜕𝑓^ℓ_𝛼𝜕𝑦_𝛾𝜋_𝑦(𝑡)−Λ_ℓ²⁰𝜕𝑘^ℓ_𝛼𝜕𝑦_𝛾𝜋_𝑦(𝑡)+𝜇_𝛼(𝑡),𝜕𝑔𝜕𝑦_𝛾𝜋_𝑦(𝑡)+𝜈_𝛼(𝑡),𝜕ℎ𝜕𝑦_𝛾𝜋_𝑦(𝑡)+𝐷²_𝜃𝜎𝜂𝜋_𝑥∘(𝑡),𝜋_𝑦(𝑡),Λ_ℓ¹⁰𝜕𝑓^ℓ_𝛼𝜕𝑦_𝜃𝜎𝜋_𝑦(𝑡)−Λ_ℓ²⁰𝜕𝑘^ℓ_𝛼𝜕𝑦_𝜃𝜎𝜋_𝑦(𝑡)+𝜇_𝛼(𝑡),𝜕𝑔𝜕𝑦_𝜃𝜎𝜋_𝑦(𝑡)+𝜈_𝛼(𝑡),𝜕ℎ𝜕𝑦_𝜃𝜎𝜋_𝑦(𝑡)𝑑𝑡^𝛼<−𝑏𝑥^∘(⋅),𝑦(⋅)𝜃𝑥^∘(⋅),𝑦(⋅)²𝜌^′ℓΛ_ℓ¹⁰+𝜌^′′ℓΛ_ℓ²⁰+𝜌^′′′. (3.8) Since 𝑏(𝑥^∘(⋅),𝑦(⋅))>0, we obtain _{𝛾𝑡01,𝑡}𝜂𝜋_𝑥∘(𝑡),𝜋_𝑦(𝑡),𝜕𝑉_𝛼𝜕𝑦𝜋_𝑦(𝑡)(𝑡)(𝑡+𝐷,𝜆,𝜇,𝜈)_𝛾𝜂𝜋_𝑥∘(𝑡),𝜋_𝑦(𝑡),𝜕𝑉_𝛼𝜕𝑦_𝛾𝜋_𝑦(𝑡)(𝑡)(𝑡+𝐷,𝜆,𝜇,𝜈)_2𝜃𝜎𝜂𝜋_𝑥∘(𝑡),𝜋_𝑦(𝑡),𝜕𝑉_𝛼𝜕𝑦_𝜃𝜎𝜋_𝑦(𝑡),𝜆,𝜇(𝑡),𝜈(𝑡)𝑑𝑡^𝛼<−𝜃𝑥^∘(⋅),𝑦(⋅)²𝜌^′ℓΛ_ℓ¹⁰+𝜌^′′ℓΛ_ℓ²⁰+𝜌^′′′, (3.9) where 𝑉_𝛼𝜋_𝑦(⋅),𝜆,𝜇(⋅),𝜈(⋅)=Λ_ℓ¹⁰𝑓^ℓ_𝛼𝜋_𝑦(𝑡)−Λ_ℓ²⁰𝑘^ℓ_𝛼𝜋_𝑦(𝑡+𝜇_𝛼)𝑦(𝑡),𝑔𝜋_𝑦(𝑡)+𝜈_𝛼(𝑡),ℎ𝜋_𝑦(𝑡),𝑡∈Ω_𝑡0,𝑡1,𝛼=1,𝑝. (3.10) The following relations hold: 𝐷_𝛾𝜂𝜋_𝑥∘(𝑡),𝜋_𝑦(𝑡),𝜕𝑉_𝛼𝜕𝑦_𝛾𝜋_𝑦(𝑡),𝜆,𝜇(𝑡),𝜈(𝑡)=𝐷_𝛾𝜂𝜋_𝑥∘(𝑡),𝜋_𝑦(𝑡),𝜕𝑉_𝛼𝜕𝑦_𝛾𝜋_𝑦(𝑡)(𝑡)(𝑡)−𝜂𝜋,𝜆,𝜇,𝜈_𝑥∘(𝑡),𝜋_𝑦(𝑡),𝐷_𝛾𝜕𝑉_𝛼𝜕𝑦_𝛾𝜋_𝑦(𝑡)(𝑡)(𝑡),𝐷,𝜆,𝜇,𝜈_2𝜃𝜎𝜂𝜋_𝑥∘(𝑡),𝜋_𝑦(𝑡),𝜕𝑉_𝛼𝜕𝑦_𝜃𝜎𝜋_𝑦(𝑡),𝜆,𝜇(𝑡),𝜈(𝑡)=𝐷_𝜃𝐷_𝜎𝜂𝜋_𝑥∘(𝑡),𝜋_𝑦(𝑡),𝜕𝑉_𝛼𝜕𝑦_𝜃𝜎𝜋_𝑦(𝑡),𝜆,𝜇(𝑡),𝜈(𝑡)−𝐷_𝜃𝜂𝜋_𝑥∘(𝑡),𝜋_𝑦(𝑡),𝐷_𝜎𝜕𝑉_𝛼𝜕𝑦_𝜃𝜎𝜋_𝑦(𝑡),𝜆,𝜇(𝑡),𝜈(𝑡)−𝐷_𝜎𝜂𝜋_𝑥∘(𝑡),𝜋_𝑦(𝑡),𝐷_𝜃𝜕𝑉_𝛼𝜕𝑦_𝜃𝜎𝜋_𝑦(𝑡)(𝑡)(𝑡+𝜂𝜋,𝜆,𝜇,𝜈)_𝑥∘(𝑡),𝜋_𝑦(𝑡),𝐷²_𝜎𝜃𝜕𝑉_𝛼𝜕𝑦_𝜃𝜎𝜋_𝑦(𝑡),𝜆,𝜇(𝑡),𝜈(𝑡). (3.11)(3.12) By replacing the relations (3.11) and (3.12), and using Euler-Lagrange PDE, the relation (3.9) becomes _{𝛾𝑡01,𝑡}𝐷_𝛾𝜂𝜋_𝑥∘(𝑡),𝜋_𝑦(𝑡),𝜕𝑉_𝛼𝜕𝑦_𝛾𝜋_𝑦(𝑡),𝜆,𝜇(𝑡),𝜈(𝑡)+𝐷_𝜃𝐷_𝜎𝜂𝜋_𝑥∘(𝑡),𝜋_𝑦(𝑡),𝜕𝑉_𝛼𝜕𝑦_𝜃𝜎𝜋_𝑦(𝑡),𝜆,𝜇(𝑡),𝜈(𝑡)−𝐷_𝜃𝜂𝜋_𝑥∘(𝑡),𝜋_𝑦(𝑡),𝜕𝑉_𝛼𝜕𝑦_𝜃𝜎𝜋_𝑦(𝑡),𝜆,𝜇(𝑡),𝜈(𝑡)−𝐷_𝜎𝜂𝜋_𝑥∘(𝑡),𝜋_𝑦(𝑡),𝐷_𝜃𝜕𝑉_𝛼𝜕𝑦_𝜃𝜎𝜋_𝑦(𝑡),𝜆,𝜇(𝑡),𝜈(𝑡)𝑑𝑡^𝛼<−𝜃𝑥^∘(⋅),𝑦(⋅)²𝜌^′ℓΛ_ℓ¹⁰+𝜌^′′ℓΛ_ℓ²⁰+𝜌^′′′. (3.13) According to [15], §9, a total divergence is equal to a total derivative, therefore, the left-hand side of (3.13) becomes null, and replacing into the inequality (3.13), it follows that 0<−𝜃𝑥^∘(⋅),𝑦(⋅)²𝜌^′ℓΛ_ℓ¹⁰+𝜌^′′ℓΛ_ℓ²⁰+𝜌^′′′. (3.14) From hypothesis (e), the previous relation becomes 0<0, which is false. Next, from relation (3.8), it follows that 0≦Λ_ℓ¹⁰𝐹^ℓ𝑥^∘(⋅)−Λ_ℓ²⁰𝐾^ℓ𝑥^∘(⋅)+_{𝛾𝑡01,𝑡}𝜇_𝛼(𝑡),𝑔𝜋_𝑥∘(𝑡)+𝜈_𝛼(𝑡),ℎ𝜋_𝑥∘(𝑡)𝑑𝑡^𝛼−_{𝛾𝑡01,𝑡}𝜇_𝛼(𝑡),𝑔𝜋_𝑦(𝑡)+𝜈_𝛼(𝑡),ℎ𝜋_𝑦(𝑡)𝑑𝑡^𝛼. (3.15) Taking into account the inequality: _{𝛾𝑡01,𝑡}𝜇_𝛼(𝑡),𝑔𝜋_𝑦(𝑡)+𝜈_𝛼(𝑡),ℎ𝜋_𝑦(𝑡)𝑑𝑡^𝛼≧0, (3.16) the above-mentioned relation becomes Λ_ℓ¹⁰𝐹^ℓ𝑥^∘(⋅)−Λ_ℓ²⁰𝐾^ℓ𝑥^∘(⋅)>0, (3.17) that is, 𝐾^ℓ𝑥^∘(⋅)𝐾^ℓ𝑦(⋅)𝐹^ℓ𝑥^∘(⋅)𝐾^ℓ(𝑥^∘(⋅−𝐹))^ℓ𝑦(⋅)𝐾^ℓ𝑦(⋅)>0. (3.18) Because 𝐾^ℓ(𝑥^∘(⋅))𝐾^ℓ(𝑦(⋅))>0, ℓ=1,𝑟, we conclude that 𝐹𝑥^∘(⋅)𝐾(𝑥^∘(⋅−𝐹𝑦(⋅)))𝐾𝑦(⋅)≰(0,…,0), (3.19) or 𝐹𝑥^∘(⋅)𝐾(𝑥^∘(⋅≰𝐹𝑦(⋅)))𝐾𝑦(⋅). (3.20) Therefore, the relation 𝜛(𝑥^∘(⋅))≤𝛿(𝑦(⋅),𝑦_𝛾(⋅),Λ¹⁰,Λ²⁰,𝜇(⋅),𝜈(⋅)) is not satisfied, and this completes the proof.

We will finish our considerations by giving the statements of two results on direct duality and converse duality, respectively. Their proofs follow directly from the weak duality.

Theorem 3.3 (Direct Duality). Let 𝑥^∘(⋅) be an efficient solution of (𝑀𝐹𝑃). Suppose that the hypotheses of Theorem are satisfied. Then, there are Λ¹⁰ and Λ²⁰ in ℝ^𝑟, and the smooth functions 𝜇^∘∶Ω_𝑡0,𝑡1→ℝ^𝑚𝑠𝑝 and 𝜈^∘∶Ω_𝑡0,𝑡1→ℝ^𝑞𝑠𝑝, such that (𝑥^∘(⋅),𝑥^∘_𝛾(⋅),Λ¹⁰,Λ²⁰,𝜇^∘(⋅),𝜈^∘(⋅)) is an efficient solution of the dual (𝑀𝐹𝐷), and 𝜛(𝑥^∘(⋅))=𝛿(𝑥^∘(⋅),𝑥^∘_𝛾(⋅),Λ¹⁰,Λ²⁰,𝜇^∘(⋅),𝜈^∘(⋅)).

We will present now a theorem concerning the converse duality, by changing some of the hypotheses.

Theorem 3.4 (Converse Duality). Let (𝑥^∘(⋅),𝑥^∘_𝛾(⋅),Λ¹⁰,Λ²⁰,𝜇^∘(⋅),𝜈^∘(⋅)) be an efficient solution of the dual problem (𝑀𝐹𝐷). Suppose the following conditions hold: (a)𝑥(⋅) is an efficient solution of the primal problem (𝑀𝐹𝑃);(b)for any ℓ=1,𝑟, one has𝐹^ℓ𝑥^∘(⋅)>0,𝐾^ℓ𝑥^∘(⋅)>0,Λ_ℓ¹⁰𝐹^ℓ𝑥^∘(⋅)−Λ_ℓ²⁰𝐾^ℓ𝑥^∘(⋅)=0; (3.21)(c)for any ℓ=1,𝑟, the functional 𝐹^ℓ(𝑥(⋅)) is (𝜌^′ℓ,𝑏)-quasiinvex at the point 𝑥^∘(⋅) and the functional −𝐾^ℓ(𝑥(⋅)) is (𝜌^′′ℓ,𝑏)-quasiinvex at the point 𝑥^∘(⋅), with respect to 𝜂 and 𝜃;(d)the functional,_{𝛾𝑡01,𝑡}𝜇_𝛼(𝑡),𝑔𝜋_𝑥(𝑡)+𝜈_𝛼(𝑡),ℎ𝜋_𝑥(𝑡)𝑑𝑡^𝛼, (3.22) is (𝜌^′′′,𝑏)-quasiinvex at the point 𝑥^∘(⋅) with respect to 𝜂 and 𝜃;(e)one of the functionals of (c), (d) is strictly (𝜌^′ℓ,𝑏)-, (𝜌^′′ℓ,𝑏)-, or (𝜌^′′′,𝑏)-quasiinvex with respect to 𝜂 and 𝜃, respectively;(f)𝜌^′ℓΛ_ℓ¹⁰+𝜌^′′ℓΛ_ℓ²⁰+𝜌^′′′≥0. Then, 𝑥(⋅)=𝑥^∘(⋅), and 𝜛(𝑥^∘(⋅))=𝛿(𝑥^∘(⋅),𝑥^∘_𝛾(⋅),Λ¹⁰,Λ²⁰,𝜇^∘(⋅),𝜈^∘(⋅)).

4. Conclusions

In our previous work [12], we initiated an optimization theory for the second-order jet bundle. We considered the problem of minimization of vectors of curvilinear functionals (well known as mechanical work), thought as multitime multiobjective variational problem, subject to PDE and/or PDI constraints (limited resources). Within this framework, we introduced necessary conditions. As natural continuation of our results in [12], and strongly motivated by its possible applications in mechanics, the present work introduced a dual program theory for this class of problems (for related but complementary research, see [16, 17]).