ON OPTIMALITY CONDITIONS AND DUALITY ASPECTS FOR DIFFERENT SEMI-INFINITE PROGRAMMING PROBLEMS

Abstract

The thesis devoted to the study of optimality and duality relations for di erent semiin nite programming problems having vanishing/equilibrium/cone constraints. It includes non-smooth, multiobjective, interval-valued and fractional semi-in nite optimization problems. The thesis is divided into seven chapters. Chapter 1 is introductory in nature. The chapter wise description of the thesis is as follows: Chapter 2 deals with a non-smooth semi-in nite optimization problem with cone constraints. An approximate necessary optimality condition for the said problem, with the help of the Abadie constraint quali cation is proved. A new class of functions namely Q-quasiconvex functions is introduced and further, an approximate su cient optimality condition is investigated. Additionally, we formulate the approximateWolfe's and approximate Mond-Weir's dual problems for the non-smooth semi-in nite optimization problem, and duality relations in terms of weak, strong and converse results between the conic non-smooth semi-in nite optimization model and the aforementioned dual problems are established under the approximate pseudoconvexity and Q-quasiconvexity assumptions. Moreover, to justify the main results of the chapter, numerical examples have been shown at the suitable places. Chapter 3 concentrates on investigating a non-smooth multiobjective semi-in nite programming problem having equilibrium constraints using the locally Lipschitzness property of functions. An approximate su cient optimality result for the semi-in nite optimization problem is developed using the approximate M-stationary point. Subsequently, a mixed type dual problem have been constructed for the semi-in nite optimization problem and approximate duality results have been proved using the convexity assumptions and M- stationary point conditions. The aim of Chapter 4 is to study a semi-in nite optimization model with vanishing constraints. Firstly, a necessary optimality condition using the limiting constraint quali- cation is established. Then, the su ciency results for the considered semi-in nite model are proved using the generalized Q-convexity/pseudoconvexity/Q-quasiconvexity assumptions. Additionally, Wolfe's and Mond-Weir types dual models for the problem have been formulated. Furthermore, weak, strong and converse duality results between the semiin nite model and its dual have been developed using convexity assumptions. Various numerical examples have been exempli ed at the appropriate places to support the results. Chapter 5 is devoted to study a multiobjective semi-in nite interval-valued optimization model with equilibrium constraints. A necessary optimality condition is proved by employing the Zangwill constraint quali cation for the problem. Then, a su cient optimality result under the convexity assumptions for the semi-in nite optimization model is developed. Further, Wolfe's and Mond{Weir type dual models are formulated for the considered semi-in nite optimization problem, and usual duality results are established under convexity assumptions. A real life application in the industrial sector for the waste management is given which demonstrates the use of the Wolfe type dual model and its duality results. The chapter incorporates numerical illustrations at suitable places to validate the results. In Chapter 6, the semi-in nite fractional optimization model over cones having multiple objectives is rst formulated. Due to presence of arbitrary cone, this work also generalize various well known results appeared in the literature. Three di erent types of dual models viz Mond-Weir, Wolfe and Schaible are presented and then, usual duality results are proved using higher-order (K Q)-(F; ; ; d)-type I convexity assumptions. To show the existence of such generalized convex functions, a non trivial example has also been exempli ed. Additionally, numerical examples illustrations are given to justify various results presented in the chapter. Chapter 7 focuses on a non-smooth fractional semi-in nite optimization model with equilibrium constraints. The chapter presents various ndings, starting with the development of E-necessary, E-su cient optimality conditions for the optimization model, considering E-convexity assumptions. Subsequently, E-Wolfe type dual model is constructed for the semi-in nite optimization problem and appropriate duality relations are proved under E- convexity assumptions. Additionally, numerical examples have been exempli ed at the appropriate place to support the results obtained in the chapter.