# Multivariable optimization with constraints

Complete Chapter One

MULTIVARIABLE OPTIMIZATION WITH CONSTRAINTS

BY
DEPARTMENT OF MATHEMATICS

ABSTRACT
It has been proved that in non linear programming, there are five methods of solving multivariable optimization with constraints.
In this project, the usefulness of some of these methods (Kuhn – Tucker conditions and the Lagrange multipliers) as regards quadratic programming is unveiled.
Also, we found out how the other methods are used in solving constrained optimizations and all these are supported with examples to aid better understanding.

Title Page i
Approval page ii
Dedication iii
Acknowledgement iv
Abstract v
CHAPTER ONE
1.0 Introduction 1
1.1 Basic definitions 3
1.2 Layout of work 6
CHAPTER TWO
Introduction 9
2.1 Lagrange Multiplier Method 9
2.2 Kuhn Tucker Conditions 19
2.3 Sufficiency of the Kuhn-Tucker Conditions 24
2.4 Kuhn Tucker Theorems 30
2.5 Definitions – Maximum and minimum of a function 34
2.6 Summary 38
CHAPTER THREE
Introduction 39
3.1 Newton Raphson Method 39
3.2 Penalty Function 53
3.3 Method of Feasible Directions 57
3.4 Summary 67
CHAPTER FOUR
4.0 Introduction 68
4.1 Definition – Quadratic Programming 69