Abstract
In this paper, we consider a class of interval-valued variational optimization problem. We extend the definition ofB-(p,r)-invexity which was originally defined for scalar optimization problem to the interval-valued variational problem. The necessary and sufficient optimality conditions for the problem have been established underB-(p,r)-invexity assumptions. An application, showing utility of the sufficiency theorem in real-world problem, has also been provided. In addition to this, for the interval-optimization problem Mond–Weir and Wolfe type duals are presented and related duality theorems have been proved. Non-trivial examples verifying the results have also been presented throughout the paper.
Highlights
Interval-valued optimization has a huge number of real-world applications such as in fuzzy logic, artificial intelligence, robotics, genetic algorithm, optimal control, neural computing, image restoration problem, optimal shape design problems, robust optimization, power unit problems, molecular distance geometry problems, engineering designs etc
Till the times when Hanson [16] introduces the concept of invexity, optimality conditions and duality results were proved for the optimization problems where the functions involved were convex
If the hypotheses of weak duality theorem are satisfied for all feasible points (v, σ, ξ), (u, σ, ξ) is an LU optimal point for (IMD)
Summary
Interval-valued optimization has a huge number of real-world applications such as in fuzzy logic, artificial intelligence, robotics, genetic algorithm, optimal control, neural computing, image restoration problem, optimal shape design problems, robust optimization, power unit problems, molecular distance geometry problems, engineering designs etc. Many researchers have driven attention towards the study of optimality conditions and duality results for interval-valued optimization problems under various generalized convexity assumptions. Wu [23] studied Wolfe-type dual and derived duality theorems for an interval valued optimization problem using nondominated solution concept [24,25,26]. For an interval-valued optimization problem, Ahmad et al [2] obtained optimality conditions and duality results by using generalized (p, r)-ρ-(η, θ) invexity. Till the times when Hanson [16] introduces the concept of invexity, optimality conditions and duality results were proved for the optimization problems where the functions involved were convex.
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