Optimality and Duality Results for Some Bilevel Programming Problems

Abstract

The work exhibited in this thesis is an endeavor to achieve various optimality and duality results for bilevel programming problems. The proposed work encapsulates these results which are weaved into five chapters. The present thesis is assembled into chapters as described below: Chapter 1 is introductory and consists of definitions, notations and prerequisites of the present work. A brief account of the related work studied by various authors in the field and a summary of the thesis are also presented. Chapter 2 presents a Wolfe type dual corresponding to a multiobjective bilevel problem. Duality results are developed and with the help of a non-trivial example weak duality the- orem is demonstrated. Further we have studied a multi-objective bilevel problem where both the levels have multiple objectives. By using optimal value reformulation and a scalarization technique we reformulate the problem. We have developed sufficient opti- mality conditions for this model. We have proposed a Mond-Weir type dual corresponding to this model and developed the relevant duality theorems under ∂∗-pseudoconvex and ∂∗-quasiconvex assumptions. In Chapter 3, we examined a bilevel problem with multiple objectives at both lev- els. With the aid of kth-objective weighted constraint scalarization and objective value function reformulation, the problem is converted into a single-level mathematical pro- gramming problem. The necessary optimality conditions are obtained and an illustrative example is given to validate our result. In Chapter 4, we have considered a bilevel programming problem with uncertainty at the upper-level constraint. By using robust counterpart approach and optimal value reformulation we transform the robust counterpart bilevel problem into a single-level problem. We have developed the optimality conditions in terms of subdifferentials and convexifactors. Moreover we have considered a multi-objective robust bilevel problem and developed the necessary optimality conditions. Chapter 5 is devoted to the development of relationship between a fractional multi- objective bilevel programming problem and its Mond-Weir type dual under ∂∗-pseudoconvex and ∂∗-quasiconvex assumptions. An example is given to validate the weak duality the- orem.