Optimality and Duality for Variational & 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 optimization problems, tracing their evolution and significance. It presents essential definitions, notations and a comprehensive review of the foundational concepts necessary to understand these areas. This chapter also includes a concise survey of relevant work conducted by other researchers, highlighting key findings and advancements that have contributed to the current body of knowledge. Thus, it provides readers with the necessary background to comprehend the contributions of the subsequent chapters to these fields. Chapter 2 explores the multiobjective fractional variational problems, emphasizing support functions. It introduces the definitions of higher-order pseudoinvex, higher-order (F,α,ρ,d)-convex, and higher-order (F,α,ρ,d)-pseudoconvex functions, enriching the theoretical framework for optimization problems. Further, Mond-Weir-type primal-dual pairs are formulated under both inequality and cone constraints. A rigorous theoretical analysis is then conducted to establish the relationship between the values of the dual pairs, providing insights into the underlying structure and connections between these models. Finally, the chapter concludes by validating the weak duality theorems, reinforcing the theoretical findings and demonstrating their practical implications for optimization problems. Chapter 3 investigates multiobjective fractional variational problems involving support functions over cones, introducing the concept of higher-order K-η convexity. Within this context, the study explored duality results that relate to the values of primal and dual problems. To enhance understanding, a numerical example of a functional that is higher-order K-η convex but not first-order K-η convex is included. Various models are further shown to be special cases of our proposed framework under certain parametric values. Additionally, a real-world example is presented to validate the findings of the weak duality theorem, demonstrating the practical relevance of the theoretical framework developed in this work. Chapter 4 deals with the interval-valued variational problems for both second-order and higher-order objective functions. The chapter introduces definitions of η -bonvexity and higher-order invexity, providing illustrative examples that satisfy these definitions while not conforming to existing ones. Thus, this necessitates a theoretical analysis of interval-valued problems. Furthermore, primal-dual pairs for both second-order and higher-order interval-valued variational objective functions are formulated, along with the governing optimality conditions for the models. The exploration of duality theorems elucidates the relationship between primal and dual problem values, offering insights into the underlying structure and connections between these models. We further discuss under what environment and how the existing model can be derived from the presented model. Numerical examples are presented to demonstrate the effectiveness and applicability of the proposed model. Moreover, an example is employed to validate the weak duality theorem, showcasing the practical relevance of the theoretical framework developed in this research. Chapter 5 aims to study the robust bilevel programming problems with interval-valued objective functions and constraint-wise uncertainty at the upper-level. By utilizing an optimal value reformulation, the bilevel problem is remodified into a single-level problem, which allows for the application of robust counterpart optimization techniques to handle uncertainty effectively. The study establishes necessary optimality conditions for robust LU-optimal solutions, employing an extended robust nonsmooth Abadie constraint qualification (EACQ) based on convexificators. Furthermore, an example is provided with a detailed explanation to enhance understanding of the theoretical optimality conditions. Additionally, the chapter discusses duality results for the original problem and its Mond-Weir dual. Chapter 6 focuses on the class of multiobjective interval-valued bilevel optimization problem. Initially, we state a nonsmooth constraint qualification for this class. Following this, necessary and sufficient optimality conditions for the optimization model are developed. Further, the Mond–Weir-type dual model is formulated for considered bilevel interval-valued multiobjective optimization problems, and weak, strong, and converse duality results are established under generalized ∂^*-convexity assumptions. To understand the established necessary and sufficient conditions proposed in the theorem, a detailed discussion is provided through numerical examples.