Conjugacy in convex analysis

The concept of duality appears in several areas of mathematics.When forced with a problem, a mathematician wants to solve it byconverting that into another problem. This new problem appears tobe quite different, yet it mirrors all aspects of the originalproblem and is easier to solve. The duality involves with twomathematical theories, each of which includes different theorems.In this work, which is consisted of five chapters, an introductionto duality is made; the relationship between convex functions andconvex conjugate is studied by corresponding each convex functionto another convex function which is called the conjugate of thegiven function.In the first section of this work, some basic definitions andtheorems, necessary for this work, are given. In the secondchapter, sublinearity is defined and some properties of sublinearfunctions are investigated.In the third chapter, by defining the subdifferentials of finiteconvex functions in three different ways, equivalence of thesethree definitions is given. After that, local properties ofsubdifferential are investigated and subdifferentials of somespecial functions are evaluated.In the fourth chapter, conjugate functions of the functions whichare defined from IR, IR^n and any topological vector space toIR are given, respectively. The relationship betweensubdifferential and conjugate function is given by investigatingthe fundamental properties of conjugate functions.In the last chapter, the dual problem of convex optimizationproblem is constructed by using conjugate functions. In this waythe relationship between primal problems and dual problems aregiven.