Optimality and Duality for Variational and Bilevel Optimization Problems

Abstract

Optimization is a fundamental aspect of mathematical modeling and decision-making, playing a crucial role in several areas such as economics, engineering, and operations research. In many practical situations, decisions must be made to optimize multiple conflicting objectives simultaneously, leading to the field of multiobjective optimization. Multiobjective optimization seeks to find solutions that balance these competing objectives, providing a set of optimal trade-offs rather than a single optimal solution. The study of various models in optimization and multiobjective optimization is essential because it allows researchers and practitioners to explore different mathematical formulations and algorithms tailored to specific problem characteristics. By studying these models, we gain insights into how different optimization approaches perform under varying conditions, enabling us to choose the most suitable methods for addressing complex decision-making challenges in diverse application domains. The main objective of this thesis is to address three significant problems in the field of optimization. One major focus is achieving higher-order duality results for fractional variational problems under generalized convexity conditions. This involves developing and proving advanced duality theorems that can handle complex fractional and variational formulations. Secondly, the thesis aims to derive optimality conditions and duality results for interval-valued variational programming problems, where uncertainty in parameters is represented as intervals. Lastly, using a reformulation approach, the thesis concentrates on obtaining optimality conditions and duality results for bilevel optimization problems. Keeping these objectives in mind, we will address the optimality and duality of developed mathematical models. In what follows, the whole work has been examined by dividing it into six chapters. Chapter 1 provides a brief overview of the theory of variational problems, interval-valued optimization problems, and bilevel