PPubMed7 Jul 2014Cited 3×

Second Order Duality in Multiobjective Fractional Programming with Square Root Term under Generalized Univex Function

Arun Kumar Tripathy

Abstract

Three approaches of second order mixed type duality are introduced for a nondifferentiable multiobjective fractional programming problem in which the numerator and denominator of objective function contain square root of positive semidefinite quadratic form. Also, the necessary and sufficient conditions of efficient solution for fractional programming are established and a parameterization technique is used to establish duality results under generalized second order ρ-univexity assumption.

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Introduces three second‑order duality approaches for nondifferentiable multi‑objective fractional programs with square‑root terms, proving optimality and duality under generalized ρ‑univexity.

What’s new

i.Three second‑order mixed type duality approaches are introduced for nondifferentiable multiobjective fractional programming with square‑root terms
ii.Necessary and sufficient optimality conditions for the fractional programming problem are established
iii.A parameterization technique is used to derive duality results under generalized ρ‑univexity
iv.The work extends earlier approaches of Dinklebaeh and Jagannathan to both primal and dual formulations
v.The paper suggests extensions to higher‑order mixed type fractional problems and to nondifferentiable minimax mixed fractional programming

How they did it

The author adapts three approaches originally given by Dinklebaeh and Jagannathan for both primal and second‑order mixed type dual formulations. Necessary and sufficient optimality conditions are derived, and a parameterization technique is applied to obtain duality results under a generalized second‑order ρ‑univexity assumption.

What they found

Theorem 11 provides a sufficient optimality condition for MFP1 under second‑order ρ‑pseudounivexity
Theorem 13 proves weak duality: Inf(λF(x)) ≥ Sup(λG(u)) under ρ‑pseudounivex and σ‑quasiunivex assumptions
Theorem 14 establishes strong duality with equal primal and dual objective values (zero) when the dual feasible set is non‑empty

Caveats

✕The study is limited to second‑order mixed type duality; higher‑order extensions are not addressed
✕Applications to nondifferentiable minimax mixed fractional programming are suggested but not developed

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