A STUDY OF THE USE OF LAGRANGIAN MULTIPLIER’S AND KARUSH KUHN-TUCKER’S METHODS OF SOLVING NON LINEAR PROGRAMMING PROBLEMS (NLPP’S)

TABLE OF CONTENTS

Title page

Approval Page

Certification

Dedication

Acknowledgement

Table of Content

Abstract

CHAPTER ONE: INTRODUCTION

1.1       Aims and Objectives

1.2       Scope/Limitations

1.3       Significance of Study

CHAPTER TWO: LITERATURE REVIEW

CHAPTER THREE: NON LINEAR PROGRAMMING TECHNIQUES

3.1       Non Linear Programming Algorithms

3.2       Direct Search

3.3       Gradient Method

3.3.1 The Newton Raphson Method

3.4       Constrained

3.4.1 Direct Substitution

3.4.2  Langrangian Multiplier Method

3.4.3  Karush Kuhn Tucker

3.4.4 Graphical Method

3.4.5 The First Order Conditions of NLP

CHAPTER FOUR: PROBLEMS AND INTERPRETATION

4.1       Problem Solving with Lagrangian Multiplier Method

4.3       Interpretation of the Lagrange Multiplier

CHAPTER FIVE: CONCLUSION AND RECOMMENDATION

5.1       Conclusion

5.2       Recommendation

            Reference

ABSTRACT

The work attempts a fresh review into available techniques for solving non linear programming problems, NLPP’s with emphasis on the Lagrangian multipliers method and KKT method. A large focus is on the Lagrangain multiplier’s method and KKT, with the aim of reviewing their solvability ability as employed in solving general non linear programming problem. The KKT, admits all form of general non linear programming problem except the equality type, to give the optimal solutions.

In summary, we observe, that the KKT, remains the most viable method in solving general non linear programming problems.

CHAPTER ONE

INTRODUCTION

Non linear programming is similar to linear programming in the sense that it is made up of an objective function, general constraints, and variable bounds. The difference is that a non linear programming includes at least one non linear function, which could be the objective function, or some or all of the constraints.

However, non linear programming is the process of solving an optimization problem defined by a system of equalities and inequalities collectively termed constraints, over a set of unknown real variables, along with an objective function to be maximized or minimized, where some of the constraints or the objective function are non-linear.

Moreso, many real life problems are inherently non linear at the same time possess a great challenge on how to resolved them to determine the best ways resources are to be used efficiently. In answer to this, non-linear programming techniques become a valuable tool in handling any sort of non-linear problem, despite the fact that these methods are not absolute efficient method for the solution of general non linear problems, as it is still subject to research. That is to say that a specific techniques or method is employed for handling the non linearity programming.

A linear programming problem is expressed as Maximize/minimize Z = f(x1, x2, x3 …….. xn) Subject to the constraints: g¹ (x1, x2, x3 …….. xn) ≤, =, ≥ b1

g² (x1, x2, x3 …….. xn) ≤, =, ≥ b2

M

gⁿ (x1, x2, x3 …….. xn) ≤, =, ≥ bm xj ≥ 0, j = 1, 2, .………. n

Now if either the objective function or one or more of the constraints are non linear in X (x1, x2, x3 ….. xn) then the problem is a non linear programming problem. In general, non....

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