ON CONVERGENCE OF EXPONENTIAL PENALTY FOR THE MULTI-DIMENSIONAL VARIATIONAL PROBLEMS

. In this article, we describe a method to deal with a multi-dimensional variational problem with inequality constraints using an exponential penalty function. We formulate an unconstrained multi-dimensional variational problem and examine the relationships between the optimal solution to the considered multi-dimensional variational problem and the sequence of minimizers of the unconstrained multi-dimensional variational problem. The convergence of the proposed exponential penalty approach is also investigated, which shows that a convergent subsequence of the sequence of minimizers of the unconstrained multi-dimensional variational problem approaches an optimal solution to the multi-dimensional variational problem. Further, an illustrative application (to minimize a manufacturing cost functional of a production firm) is also presented to confirm the effectiveness of the proposed outcomes


Introduction
Variational problems arising from the calculus of variations have received much attention in various fields, including science, engineering, pure and applied mathematics, decision-making, economics, etc. and are defined as the task of determining a function that minimizes a given functional.The mathematical association between the optimization problems and the calculus of variation was first investigated and developed by Hanson [10].Afterward, several researchers have made their contributions to this field of study.Mond and Husain [19] have given sufficient optimality and duality results for the variational problem under generalized invexity.Jiang [17] proved the existence and uniqueness of the solution to a variational problem with inequality constraints by applying the saddle-point theory.Jiménez et al. [4] studied optimality conditions and duality results for a class of variational problems under generalized invexity.Recently, Jayswal et al. [15] have investigated an equivalence relation between the solution of the variational problem and its associated modified variational problem using the notion of convexity.For more insights readers are referred to see [3,12,22,23] and references therein.
The theoretical and practical views of variational problems encouraged authors to investigate this field intensively.In 2012, studied the multi-dimensional variational problem with curvilinear integral functional and presented the Mond-Weir type duality results under some generalized convexity.Treant ¸ȃ [25] investigated the well-posedness of variational problems with inequality constraints by defining the set of approximating solutions.Later on, Jayswal et al. [11] established the equivalence relatiuon between an optimal solution to a class of multi-dimensional variational problem and its associated variational inequality under the hypotheses of convexity.For more insights in this field, readers are referred to see [6,7,21] and references therein.
On the other hand, the penalty function method is a systematic technique for solving constrained optimization problems.In this method, the constrained problem is transformed into a sequence of unconstrained problems using a penalty function.If the sequence of solutions to unconstrained problems converges towards the solution of the constrained optimization problem, then the technique is said to be effective.Many authors have explored this method to solve different types of constrained optimization problems; see, for example, [1,14,16,24] and references therein.
The exponential penalty function method is one of an efficient approach to solve the constrained optimization problem and in this method no need to restrict the involved functions to have convexity or some generalized convexity properties.Murphy [20] first introduced to a class of exponential penalty functions to solve constrained optimization problems.In 2011, Antczak [2] solved nonconvex mathematical programming problems using the  1 exact exponential penalty function method.Later on, Mandal et al. [18] transferred a constrained variational problem into an unconstrained one by using the exponential penalty function and presented optimality conditions and duality results.Recently, Jayswal and Choudhury [13] utilize the exponential penalty function method to address a constrained fractional problem and investigate the convergence of the utilized method.
Motivated by the works mentioned earlier, in this paper we consider a multi-dimensional variational problem with inequality constraints and obtain the optimal solution of the problem using the concept of exponential penalty function method.The method presented in this paper does not require the convexity or differentiability of the involved functionals to establish an association between minimizers and optimal solution of the unconstrained and constrained problem, respectively.The management of this paper is as follows: Section 2 consists of some notations and preliminaries which are helpful in the presentation of the main results.In Section 3, we formulate an unconstrained multi-dimensional variational problem (MVP)  with the help of exponential penalty function associated with the variational problem with inequality constraints (MVP).In Section 4, we show that the sequence of minimizer of problem (MVP)  converges towards an optimal solution to the problem (MVP).Furthermore, a suitable example has been fabricated to validate the main results.Finally, Section 5 concludes the paper.

Notations and preliminaries
This section contains some notations that will assist in framing the problem and presenting the significant results.
-Consider the finite dimensional Euclidean spaces   and   .Let  = (  ),  = 1,  and  = (  ),  = 1,  be the local coordinates of   and   respectively.-Let Ω = Ω 0,1 ⊂   be a hyperparallelepiped, fixed by the diagonally opposite points  0 = (  0 ) and  1 = (  1 ),  = 1,  in   , as well as d = d   9]).A family of functions is said to be uniformly equicontinuous if all the functions are continuous and they have equal variation over a given interval.Mathematically, the family of piecewise smooth (that implies continuity) functions () : Ω ↦ →  is said to be uniformly equicontinuous, if for each  > 0,  = ( 1 ,  2 , . . .,   ) there exists a  > 0,  = ( 1 ,  2 , . . .,   ), such that ||( -For any two points,  = ( ℓ ) and  = ( ℓ ), ℓ = 1,  in   , the following convention will be used through paper -Let  and  are two subsets in Considering the above mathematical tools, we formulate the following multi-dimensional variational problem with inequality constraints (MVP) as: where Remark 2.1.(i) If we take the variable  of single dimension in the problem (MVP), then it reduces to the problems (VP) and (CVP) given Jimenez et al. [4,5], respectively.(ii) If we take the variable  of single dimension and assume the interval-valued objective functional in the problem (MVP), then it becomes the problem (IVP) given by Debnath and Pokharna [8].
Definition 3.1.A point â() ∈  is said to be a minimizer of (MVP)  , if Also, consider all across the article, D denote the the set of all minimizer to (MVP)  .From now onwards, we utilize the following notions for the convenience of presentation:  = (),   =   (),  = (, (),   ()), ρ = (, â(), â ()),  0 = (,  0 (),  0  ()),   = (,   (),   ()), etc.The following lemma proves that a feasible solution to the constrained multi-dimensional variational problem predicts the limit point of the exponential penalty function concerning the penalty parameter   .The lemma is essential as it is used to prove some of the significant results of this paper.Lemma 3.1.Let  be the set of feasible solutions to (MVP) Proof.(i) Let  ∈  be a feasible solution to (MVP), and so (by L-Hospital rule) = 0.
(ii) Let  is not a feasible solution to (MVP), and so Hence the proof is complete.Or On contrary we assume that â / ∈ , so from Lemma 3.1, we get lim Again, from Lemma 3.1, for any  ∈ , we obtain From the above two relations, for sufficiently large  (say  >  0 ), we get which contradicts the inequality (3).This completes the proof.

Convergence of the exponential penalty function method
In this section, we show the convergence of the exponential penalty function method for the multi-dimensional variational problem with inequality constraints (MVP).
Also,  ∈ D , yields On taking limit  → ∞ and using Lemma 3.1, the above inequality becomes which contradicts the inequality (4).Case 2. Let  / ∈  and thus from Lemma 3.1, we get Again, from Lemma 3.1, we have lim Hence we have come to conclusion that for sufficiently large which contradicts the inequality (6).This completes proof.
Now, since , â ∈ , then using Lemma 3.1, the above inequality becomes Again, since  ∈ D , it yields which contradicts the inequality (8).Case 2. Let  / ∈  and thus from Lemma 3.1, we get lim Again, from Lemma 3.1, we have lim Hence we have come to conclusion that for sufficiently large which contradicts the inequality ( 9).This completes the proof.Proof.This proof follows from Theorems 4.1 and 4.2.
The following theorem shows that the limit point of the penalty function at each point of subsequence will be zero if the subsequence of the minimizers of the unconstrained multi-dimensional variational problem (MVP)  converges towards a feasible solution to the constrained multi-dimensional variational problem (MVP).Theorem 4.4.Let â ∈  be a minimizer to the probem (MVP)  and {â  } be the convergent subsequence of {â  } with lim →∞ â ∈ .Further, if  is measurable and   =  (  ) be a sequence of non-negative measurable functions, then Proof.We begin with the contradiction and assume that lim Then the sequence {â  } has a subsequence {â   },  = {1, 2, . ..} such that lim Since â  is a minimizer to the (MVP)   , we get From Arzelà-Ascoli theorem [9], there exist a convergent subsequence {â  } of {â  } with lim →∞ â = â, then the above inequality becomes )︂]︂ .
On taking limit inferior with  → ∞ and using inequality (11), we get Since   is a sequence of non-negative measurable functions and lim →∞ â = â, It follow from Fatou's lemma Again, since lim →∞ â = â, from the continuity of  , we obtain On the other hand, since â ∈ , from Lemma 3.1, we have Therefore from the inequality (12) and equation ( 13), we conclude that  ≤ 0, Hence the theorem.Proof.This proof follows from Lemma 3.2.
In next theorem, we shall establish that a convergent subsequence of the sequence of minimizers to the unconstrained multi-dimensional variational problem (MVP)  can approach an optimal solution to the constrained multi-dimensional variational problem (MVP).Theorem 4.6.Let â ∈  be a minimizer to the probem (MVP)  and {â  } is a convergent subsequence of {â  } with lim →∞ â = â ∈ .Further, if  is measurable and   =  (  ) be a sequence of non-negative measurable functions, then â is an optimal solution to (MVP).
Proof.Since, â ∈ D , so for all  0 ∈ , we get From the feasibility of  0 and Lemma 3.1, we get lim Also, from Theorem 4.4, we have lim )︂]︂ = 0.
On taking limit inferior as  → ∞ and in agreement of the above two equations, inequality ( 14 From Arzelà-Ascoli theorem [9], there exist a convergent subsequence{â  } of {â  } with lim →∞ â = â, then the above inequality becomes Therefore â is an optimal solution to (MVP).This completes the proof.Now we'll give an example to demonstrate the effectiveness of the exponential penalty function method in solving the multi-dimensional constrained variational problem.

Conclusion
In this article, the exponential penalty function method has been used to solve a multi-dimensional variational problem and then demonstrated the convergence of the method.An example has also been included to understand the applicability of the findings.Consequently, we suggest that the exponential penalty function method effectively solves constrained multi-dimensional variational problems.So, the results presented in this work give a powerful technique for solving the problems mentioned above that do not rely on the characterization of convex functions or generalized convex.Therefore, the multi-dimensional variational problems with inequality constraints that naturally emerge in various mathematical models may be handled more quickly and efficiently utilizing this method.