- Department of Mathematics and Informatics, University Politehnica of Bucharest, Bucharest, Romania
The aim of this work is to study constrained optimization problems by means of (Φ, ρ)-convexity. We provide some sufficient conditions of optimality for a class of vectors of cuvilinear integrals by means of an adequate generalized convexity. Dual problems associated with this one are stated and developed, in terms of weak, strong, and converse duality results. The framework chosen here is one specific to the Riemannian geometry, namely that of first order jet bundles.
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
If the local sections check the equality ϕ (t) = ψ (t), let (tα, χi) and (tα′, χi′) be two adapted coordinate systems around ϕ (t). Suppose the following equalities hold
Then the next relations hold true
Definition 1. Two local sections ϕ, ψ ∈ Γt (ϖ) are called 1-equivalent at the point t if
The equivalence class containing the section ϕ is precisely the 1-jet associated with the local section ϕ, at the point t, denoted by
Definition 2. The set
where
Proposition 1. On the product manifold T × M, consider
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
where Lα, Mi, and
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
can be associated with the form ω.
Let Lβ (πχ(t)) dtβ be a closed Lagrange 1-form (completely integrable), that is DβLα = DαLβ.
A closed 1-form in a simple-connected domain is an exact one. Its primitive can be expressed as a curvilinear integral,
or as a system of partial derivative eqations,
Suppose there is a Lagrangian-like antiderivative
or DαL = Lα, where the foregoing pullback is the given closed 1-form,
which is a completely integrable system of partial derivatives equations, with the unknown function χ(⋅).
Each smooth Lagrangian
- the differential
with the components
- the restriction of dL to πχ (t), namely the pullback
of components
with respect to the basis dtβ.
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
Let
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
for q, q1,
Definition 4. Let
holds for all χ (⋅) ∈ S,
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, … , χp),
1) χ = η if and only if χi = ηi, for all
2) χ > η if and only if χi > ηi, for all
3) χ ≧ η if and only if χi ≥ ηi, for all
4) χ ≥ η if and only if χ ≧ η, and χ ≠ η.
This product order relation will be used on the hyperparallelepiped
Let (T, h) and (M, g) be Riemannian manifolds of dimensions p and n, respectively, with the local coordinates t = (tα),
The closed Lagrange 1-forms densities of C∞-class
produce the following path independent curvilinear functionals
where πχ(t) = (t, χ(t), χγ(t)), and
Presume that the Lagrange densities matrix
of C∞-class leads to the partial differential inequalities
and the Lagrange densities matrix
defines the partial differential equalities
In the paper, we consider the multitime multiobjective variational problem
Let
denote the set all feasible solutions of problem
Definition 5. A feasible solution
If, in this relation, we use the strict inequality, then
Theorem 1. Let
The following theorem establishes sufficient conditions of efficiency for the problem
Theorem 2. Presume that the following conditions are fulfilled:
1)
2) The objective functional U is (Φ, ρU)-convex with regard to its third argument at
3)
4)
5)
Then
Proof 1. Assume 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 Λi accordingly,
On the other hand,
which leads, by the (Φ, ρ)-invexity, to
Now, by the properties of h,
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
4 Dual Programming Theory
Consider the dual problem to
Let ΔD be the set of the feasible solutions to the dual problem
We start with a weak duality result, as follows.
Theorem 3. Suppose that
1) The objective functional U is (Φ, ρU)-convex with regard to its third argument at η(⋅).
2)
3)
Then
Proof 2. Presume that
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
We multiply each relation by Λi,
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
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
Theorem 5. Let
where
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.
Author Contributions
The author confirms being the sole contributor of this work and has approved it for publication.
Conflict of Interest
The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
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Keywords: Riemannian mainfold, jet bundle, multiobjective optimization problem, efficiency, duality, generalized convexity
Citation: Pitea A (2022) Multiobjective Optimization Problems on Jet Bundles. Front. Phys. 10:875847. doi: 10.3389/fphy.2022.875847
Received: 14 February 2022; Accepted: 28 March 2022;
Published: 04 May 2022.
Edited by:
Josef Mikes, Palacký University, Olomouc, CzechiaReviewed by:
Dana Smetanová, Institute of Technology and Business, CzechiaSayantan Choudhury, National Institute of Science Education and Research (NISER), India
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*Correspondence: Ariana Pitea, arianapitea@yahoo.com