Multi Objective Fractional Programming Problem Involving Differentiable Vector Convex Functions Including Certain Classes of Nonlinear Composite Problems

Multi Objective Fractional Programming Problem Involving Differentiable Vector Convex Functions Including Certain Classes of Nonlinear Composite Problems

Dr. G. Varalakshmi

 Lecturer in statistics, PRR & VS Government College, Vidavalur

DOI: https://doi.org/10.51584/IJRIAS.2025.100700031

Received: 07 July 2025; Accepted: 15 July 2025; Published: 02 August 2025

ABSTRACT

Differentiable multiobjective fractional programming problems under V-convexity with special emphasis on specific classes of nonlinear composite problems

INTRODUCTION

In the differentiable case, Jeyakumar and Mond [10] introduced the concept of vector invexity, which effectively addresses a key challenge in invex analysis—namely, the difficulty of verifying inequality conditions for the same function. They formulated sufficient optimality conditions under V-pseudo invexity and established corresponding duality results. Building upon this foundation, Egudo and Hanson [6], inspired by Zhao’s work [15], extended the notion of V-invexity to the nonsmooth setting by replacing classical gradients with Clarke’s generalized gradients [2]. While Jeyakumar and Mond’s framework was developed within the context of generalized convex mathematical programming, it was not applied to multiobjective fractional programming. This paper seeks to fill that gap by extending the theory to differentiable multiobjective fractional programming problems under V-convexity, with particular attention to specific classes of nonlinear composite problems that often arise in practical applications.

Notations & Preliminaries

Consider constrained multiobjective fractional optimization problem.

the symbol V- minimize stands for vector minimization. This is the problem of finding the set of weak minimum for points (VFP). When P=1, the problem (VPF) reduces to a Scalar optimization problem and it is denoted by (FP). Convexity of the Scalar problem (FP) is characterized by the inequalities.

The functional form (x-a)  here plays no role in establishing the following two well-known properties in scalar convex programming:

(s). Every feasible Kuhn – Tucker point is a global minimum (w) weak duality holds between (FP) and its associated dual problem. Having this in mind, considered problem (FP) for which there exists a function η:X_∘*X_∘→R^n such that

and showed that such problems (known now as invex problem) also. Possess properties (s) and (w). Since then, various generalization of conditions ( I ) to multiobjective problems and many properties of functions that satisfy  ( I ) have been established in the literature. However, the major difficulty is that the invex problems require the same function η(“x,a” ) for the objective function and the constraints. This requirement turns out to be a severe restriction in applications. Because of this restriction, pseudo liner multiobjective, problems and certain non-linear multiobjective fractional programming problems require separate treatment as far as optimality and duality properties are concerned. In this chapter we show how this situation can be improved and how the properties (s) and (w) can be extended to hold for generalized convex multiobjective problems and certain multi- objective fraction problems. To this, we modify the condition (I ) in the next section as follows:

New Classes & generalized convex vector functions:

We now show that the V-Convexity is preserved under smooth convex transformation.

Necessary Theorem:-

This is a contradiction.

Numerical Example

Let:

Decision variable: x∈R

Objective functions:

f₁(x) = x²+1/ x+2,

f₂(x) = x²+2/x+3

These are both fractional functions and convex under appropriate domains.

Let    f_i(x ) = [f₁(x)/ f₂(x)]

Let ϕ(t) = e^t which is convex and differentiable with positive derivative ϕ^t(t)=e^t>0

Now consider the composition:

h_j(x) = [ϕ(f1(x)) / ϕ(f2(x))] = [e^f₁^(x)  / e^f₂^(x)]

Let’s pick x = 1  :

f₁(1) =1²+1/ 1+2 = 2/3

So, ϕ ( f₁(1)) = e^2/3      ≈1.948

f₂ (1) = 1²+2 / 1+3= ¾,

So ϕ (  f₂(1))=e3/4 ≈ 2.117

So, h (1) ≈ [1.948 / 2.117]

This example demonstrates that composing a differentiable convex function ϕ\phi with positive derivative with a vector of fractional V-convex functions f(x), yields another V-convex vector-valued function ϕ (f(x)).

Sufficiency theorem:- 

From the V-quasi-convexity condition, we get

Duality:-

Dual Formulation:-

V-maximize ,

(VFD) Subject to

Contradicting weak duality.

Practical Applications

Multi objective fractional programming with differentiable vector convexity appears in several real-world contexts:

Energy and Power Systems

Objectives like:

Fuel cost per MW → Cost/ Power output

Emission per MW → CO2/ Power output

Subject to grid balance, generation capacity constraints.

Telecommunications / Network Optimization

Objective ratios:

Delay per bandwidth →Latency/Bandwidth​

Cost per coverage → Cost/Coverage ​

Nonlinear constraints on routing flow conservation.

Transportation and Logistics

Objectives like:

Fuel per kilometer → Fuel usage/Distance

Delivery cost per product → Total cost/ Items delivered

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