Optimization problems under generalized convexity

Abstract

Convex analysis has played an important role in mathematical programming problems since its development by Rockafellar, Fenchel and Moreau in the 1960s and 1970s. Linear programming problems come under convex programming due to their convex nature, which is the foundation of optimization problems with constraints. Whenever we have mathematical programming involving convex functions then every local solution is a global solution. Even though it has a wide range of applications in the fields of economics, business, operation research, optimization, etc. it lacks in solving most of the practical problems due to its non-convex nature. The main target of the optimization technique is double fold. On one hand, to generalize the optimization technique so that it can solve most of the practical problems which lead to the development of semi-infinite programming problems, mathematical programming problems with equilibrium constraints, interval programming problems, fractional programming problems, Bi-level Optimization, and stochastic optimization. On the other hand, generalization of convex function to generalized convex function viz. invex, (p,r)-invex , B(p,r)-invex, arcwise connected, B-arcwise connected functions, (h,and#981;)-(b,F,p)-convex, functions and generalized (h,and#981;)-(b,F,p)-convex, functions. newline newlineConsidering various types of convexity and differentiability with single and multi-objective functions has been studied for deriving optimum condition and duality results. The research provided in this thesis, in which I investigate the theoretical aspects of optimization problems, is arranged and divided into six chapters.