On equivalence between a variational problem and its modified variational problem with the η‐objective function under invexity

• DOI: http://doi.org/10.1111/itor.12377
• Date: 01 September 2019
• Author: Tadeusz Antczak,Shalini Jha,Anurag Jayswal
• Published date: 01 September 2019

Intl. Trans. in Op. Res. 26 (2019) 2053–2070

DOI: 10.1111/itor.12377

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On equivalence between a variational problem and its modified

variational problem with the η-objective function under invexity

Anurag Jayswala, Tadeusz Antczakband Shalini Jhaa

aDepartment of Applied Mathematics, Indian Institute of Technology (Indian School of Mines), Dhanbad-826 004,

Jharkhand, India

bFaculty of Mathematics and Computer Science, University of Lodz, Banacha 22, 90-238 Lodz, Poland

E-mail: anurag_jais123@yahoo.com [Jayswal]; antczak@math.uni.lodz.pl [Antczak];

jhashalini.rash89@gmail.com [Jha]

Received 2 October 2015; receivedin revised form 2 October 2016; accepted 10 October 2016

Abstract

In this paper, a new approach to analyze optimality and saddle-point criteria for a new class of nonconvex

variational problems involving invex functions is studied. Namely, the modified objective function method is

used for the considered variational problem in order to characterize its optimal solution. In this method, for

the considered variationalproblem, corresponding modified variational problem with the η-objectivefunction

is constructed. The equivalence in the original variational problem and its associated modified variational

problemwith the η-objective function is proved under invexityhypotheses. Furthermore,by using the notion of

a Lagrangian function, the connection between a saddle-point in the modified objective function variational

problem and an optimal solution in the considered variational problem is presented. Some examples of

nonconvex variational problems are also given to verify the results established in the paper.

Keywords:optimal solution; variational problem; saddle-point criteria; optimality condition

1. Introduction

In the paper, the following variational problem is considered:

(P)minimize F(x)=b

a

φ(t,x(t), ˙

x(t))dt

subject to g(t,x(t), ˙

x(t)) ≤0,t∈I,

x(a)=α, x(b)=β,

where I=[a,b] is a real interval, φ:I×Rn×Rn→ R,and g:I×Rn×Rn→ Rmare continu-

ously differentiable functions with respect to their arguments. Let Xbe the space of continuously

C

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2054 A. Jayswal et al. / Intl. Trans. in Op. Res.26 (2019) 2053–2070

differentiable functions x:I→ Rnwith the norm x=x∞+Dx∞, where the differential

operator Dis defined as

u=Dx ⇔x(t)=x(a)+t

a

u(s)ds,

where x(a)is a given boundary value. Therefore, D=d

dt except at points of discontinuities.

Let denote the feasible set of the variational problem (P), that is,

={x∈X:x(a)=α, x(b)=βand gj(t,x(t), ˙

x(t)) ≤0,j∈J={1,...,m},t∈I}.

The concept of invexity was first given by Hanson (1981) based on the generalization of the

definition of a differentiable convex function. Further, many authors used the concept of invexity

and various generalized invexity in proving results for optimization problems (see, e.g., Ahmad and

Gulati, 2005; Ahmad et al., 2014; Ahmad and Sharma, 2010; Antczak, 2009, 2014; Arana-Jim´

enez

et al., 2008, 2009; Chandra et al., 1985; Hachimi and Aghezzaf, 2006; Husain and Ahmad, 2006;

Mond et al., 1988; Mond and Smart, 1988; Nahak and Nanda, 1996; Sharma, 2015; Zhian and

Qingkai, 2001; among others). Mond and Hanson (1967) considered control variational problems

as continuous analogues of the usual primal and dual problems in mathematical programming

problems and they first obtained the duality results for control problems under convexity. Chandra

et al. (1985) established optimality conditions and duality results for nondifferentiable continuous

programming problems using Fritz John (FJ) conditions. Mond et al. (1988) extended the work of

Mond and Hanson (1967) to the class of variational control problems involving invex functions

whose definition was originally introduced by Hanson (1981) for scalar differentiable optimization

problems. Thereafter, Mond and Husain (1989) established Kuhn–Tucker (KT) type sufficient

optimality criteria and duality results forvariational problems under a variety of generalized invexity

assumptions. Husain and Ahmed (2006) formulated a mixed-type dual for a variational control

problem and establishedvarious duality results under strong pseudo-invexity of constraints.Arana-

Jim´

enez et al. (2008) introduced the concept of KT-invexity for a control problem and showed

that a KT-invex control problem is characterized in order that a Kuhn–Tucker point is an optimal

solution. Further, Arana-Jim´

enez et al. (2009) proved that the introduced concept of FJ-invexity

is both necessary and sufficient in order to characterize the optimal solution set using Fritz John

points for a control problem.

Lagrange multipliers and saddle-points, which play a vital role in solving optimization problems,

have been analyzed by various authors (Ghosh and Shaiju, 2004; Antczak, 2007; Li et al., 2007).

Antczak (2003) introduced a new approach for solving the considered multiobjective programming

probleminvolving invex functions byconstructing an equivalent vector programming problemwith a

modification of the objective function. Further, Antczak (2007) established the saddle-pointcriteria

for a nonlinear mathematical programming problem by defining an η-approximated optimization

problem.

Motivated by the above research works, we study the optimality conditions and saddle-point

criteria for variational problems involving invex and generalized invex functions. In our considera-

tions, we use the modified objective function method introduced by Antczak (2003). Therefore, for

the considered variational control problem, we construct in this approach the modified objective

function control problem. Using invexity and generalized invexity hypotheses, we connect optimal

C

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