Authors: Peter Kwasi Sarpong1, Andrew Owusu-Hemeng2, , Joseph Ackora-Prah3
1,2,3Department of Mathematics, Kwame Nkrumah University of Science and Technology,Kumasi,Ghana
Email: kp.sarp@yahoo.co.uk, owusuhemengandrew@gmail.com, ackph@yahoo.co.uk
Abstract
In recent years, convex optimization has become a computational tool of central importance in mathematics and economics, thanks to its ability to solve very large, practical mathematical problems reliably and efficiently. The goal of this project is to give an overview of the basic concepts of convex sets, functions and convex optimization problems, so that the reader can more readily recognize and formulate basic problems using modern convex optimization. This helps in solving real world problems.
Convex functions appear in many problems in pure and applied mathematics. They play an extremely important role in the study of both linear and non linear programming problems. It is very important in the study of optimization. The solutions to these problems lie on their vertices.
The theory of convex functions is part of the general subject of convexity, since a convex function is one whose epigraph is a convex set. Nonetheless it is an important theory which touches almost all branches of mathematics. Graphical analysis is one of the first topics in mathematics which requires the concept of convexity. Calculus gives us a powerful tool in recognizing convexity, the second-derivative test. Miraculously, this has a natural generalization for the several variables case, the Hessian test.
This project is intended to study the basic properties, some definitions, proofs of theorems and some examples of convex functions. Some definitions like convex and concave sets, affine sets, conical sets, concave functions shall be known. It will also prove that the negation of a convex function will generate a concave function and a concave set is a convex set. There is also the preposition that the intersection of convex sets is a convex set but the union of convex sets is not necessarily a convex set.
This work is intended to help students acquire more knowledge on convex and concave functions of single variables. This will be done by differentiating the given function twice. If the second differential of the function is positive then we have a convex function. On the other hand if the second differential is negative then that function will be considered as concave. Examples will be solved to elaborate more on this. The convexity of functions of several variables will also be determined. This will be done by the use of the Hessian matrix. This will generate the idea of principal minors and leading principal minors. Firms can also use the idea of convex functions to know how they are doing in the market. Equations can be generated and with the help of curve sketching they will know if they are maximizing profits or making losses.
Keywords: Concave Function, Convex Functions, Linear Programming
