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.
TABLE OF CONTENTS Title Page i Approval page ii Dedication iii Acknowledgement iv Abstract v Table of Contents iv 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 4.2 General Quadratic Problems 70 4.3 Methods 75 4.4 Ways/Procedures of Obtaining the optimal Solution from the Kuhn-Tucker Conditions method 76 The Two-Phase Method 76 The Elimination Method 77 4.5 Summary 117 CHAPTER FIVE Conclusion 118 References 120
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