Abstract
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 of the local sections ϕ ∈ Γt (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 (tα, χi) and (tα′, χi′) be two adapted coordinate systems around ϕ (t). Suppose the following equalities holdThen the next relations hold true
Definition 1Two local sections ϕ, ψ ∈ Γt (ϖ) are called 1-equivalent at the point t ifThe equivalence class containing the section ϕ is precisely the 1-jet associated with the local section ϕ, at the point t, denoted by .
Definition 2The set is called the first order jet bundle.If , u = (tα, χi) is an adapted coordinate system on the product manifold T × M, the induced coordinate system, , on J1 (T, M), is defined aswhere , and .The pn functions form the coordinate derivatives.
Proposition 1On the product manifold T × M, consider the atlas of adapted charts. Then, the corresponding charts 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 πχ (t) = (t, χ (t), χγ (t)), where χγ is the derivative of χ with respect to tγ.
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 formwhere Lα, Mi, and are Lagrangians of the first order, with the pullbacka Lagrange 1-form of the second order on M. The coefficientssecond order Lagrangians, are linear in the second order derivatives. The Pfaff equation ω = 0, and the partial differential equationscan 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 antiderivativeor 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 , , leads to two smooth closed 1-forms:
- the differentialwith the components , with respect to the corresponding basis ;
- the restriction of dL to πχ (t), namely the pullbackof componentswith 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 we denote the space of all functions of C∞-class.
Let 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 3The functional is convex with respect to the third component, if, for all χ (⋅), , η1 (⋅), η2 (⋅), the following inequality holdsfor q, q1, , 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, … , λk ∈ (0, 1), with .Let S be a nonempty subset of , and be given. Following the footsteps of [18], we have the following definition.
Definition 4Let , be convex with respect to the third component, and . The vectorial functional A is called (strictly) (Φ, ρ)-convex at the point on S if, for each i, , the following inequalityholds for all χ (⋅) ∈ S, ). If these inequalities are satisfied at each , 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, … ,
χp),
, consider.
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 in , with diagonal opposite points , and . Assume that is a piecewise C1-class curve joining the points t0 and t1, and that there exists an increasing piecewise smooth curve in 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 = (tα), , and χ = (χi), , respectively, and Π = J1 (T, M).
The closed Lagrange 1-forms densities of C∞-classproduce the following path independent curvilinear functionalswhere πχ(t) = (t, χ(t), χγ(t)), and , , are partial velocities.
Presume that the Lagrange densities matrixof C∞-class leads to the partial differential inequalitiesand the Lagrange densities matrixdefines the partial differential equalities
In the paper, we consider the multitime multiobjective variational problem of minimizing a vector of path independent curvilinear functionals defined by
Letdenote the set all feasible solutions of problem .
Definition 5A feasible solution is called an efficient solution to the problem if there is no other feasible solution χ (⋅) ∈ D such thatIf, in this relation, we use the strict inequality, then is called a weakly efficient solution to the problem .In [21] were proved necessary optimality conditions for a problem similar to ; for our case we obtain the next theorem.
Let be a normal efficient solution in multitime multiobjective problem . Then there exist the vector and the smooth functions , such that
The following theorem establishes sufficient conditions of efficiency for the problem .
Presume that the following conditions are fulfilled:
Then is an efficient solution to the problem .
Proof 1Assume that , Λ, M, and N fulfill the conditions from relations (Eqs 1–3), and that is not an efficient solution to problem . In this case, there can be found such thatmore preciselywith 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 compelwhich, by inequalities (Eq. 4), imply thatwhere at least one inequality is a strict one. Multiplying the previous inequality by Λi accordingly, , and dividing by , we getOn the other hand,which leads, by the (Φ, ρ)-invexity, toNow, by the properties of h, , and , we getwhich leads toUsing the convexity of the functional F in the third component, and adding inequalities (Eqs 5, 6), it follows thatBy the equality from (Eq. 1), this inequality implieswhich is a contradiction with the properties of the function Φ.Therefore, our assumption was false, and is an efficient solution to the problem .
4 Dual Programming Theory
Consider the dual problem to in the sense of Mond-Weir
Let ΔD be the set of the feasible solutions to the dual problem , 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.
Suppose that
and [η (⋅), λ, M (⋅), N (⋅)] are feasible solutions to the problems
, and
, respectively. Additionally, presume that the next hypotheses are satisfied:
1) The objective functional U is (Φ, ρU)-convex with regard to its third argument at η(⋅).
2) ,, are -convex with regard to its third argument at ;
3) .
Then .
Proof 2Presume that , that iswhere 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 implyWe multiply each relation by Λi, , and then dividing by , it follows thatHaving in mind assumption (Eq. 2) from the theorem, we get, by the (Φ, ρ)-invexity, thatThe properties of F, jointly with inequalities (Eq. 8), and (Eq. 10), implyBy the constraints of the dual problem , this inequality leads towhich 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.
Consider that χ (⋅) is an efficient solution to the primal problem . 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 .
Let be an efficient solution to the dual problem . Assume that conditions 2)-5) from Theorem 2 are satisfied. Then η (⋅) is an efficient solution to the primal problem .
In a similar manner, a dual problem in the sense of Wolfe can be associated to our vector problem . First, we introduce the objective of this problem.where .
The associated multitime multiobjective problem dual to in the sense of Wolfe is , 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.
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Summary
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
Volume
10 - 2022
Edited by
Josef Mikes, Palacký University, Olomouc, Czechia
Reviewed by
Dana Smetanová, Institute of Technology and Business, Czechia
Sayantan Choudhury, National Institute of Science Education and Research (NISER), India
Updates
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© 2022 Pitea.
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*Correspondence: Ariana Pitea, arianapitea@yahoo.com
This article was submitted to Statistical and Computational Physics, a section of the journal Frontiers in Physics
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All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.