MULTIVARIABLE OPTIMIZATION WITH CONSTRAINTS
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
CHAPTER ONE
1.0    Introduction                        
1.1    Basic definitions                        
1.2    Layout of work                            
CHAPTER TWO    
2.0    Introduction                            
2.1    Lagrange Multiplier Method                    
2.2    Kuhn Tucker Conditions                    
2.3    Sufficiency of the Kuhn-Tucker Conditions        
2.4    Kuhn Tucker Theorems                    
2.5    Definitions – Maximum and minimum of a function        
2.6    Summary                                
CHAPTER THREE
3.0    Introduction                        
3.1    Newton Raphson Method                
3.2    Penalty Function                        
3.3    Method of Feasible Directions            
3.4    Summary                            
CHAPTER FOUR
4.0    Introduction                            
4.1    Definition – Quadratic Programming        
4.2    General Quadratic Problems                
4.3    Methods                            
4.4    Ways/Procedures of Obtaining the optimal
Solution from the Kuhn-Tucker Conditions
method                        
4.4.1    The Two-Phase Method                
4.4.2    The Elimination Method                
4.5    Summary                                
CHAPTER FIVE
Conclusion                                
References                            
CHAPTER ONE
1.0    INTRODUCTION
There are two types of optimization problems:
    Type 1
    Minimize or maximize          Z = f(x)        
                XE Rn
    Type 2
    Minimize or maximize     Z  =  f(x)        
            Subject to      g(x)   ~ bi,  i, = 1, 2, -----, m   
        where x E Rn
    and for each i, ~ can be either <, > or =.
    Type 1 is called unconstrained optimization problem.  It has an objective function without constraints. The methods used in solving such problem are the direct search methods and the gradient method (steepest ascent method).
    In this project, we shall be concerned with optimization problems with constraints.
    The type 2 is called the constrained optimization problem.  It has an objective function and constraints.  The constraints can either be equality (=) or inequality constraints (<, >).
    Moreover, in optimization problems with inequality constraints, the non-negativity conditions, X >0 are part of the constraints.
    Also, at least one of the functions f(x) and g(x) is non linear and all the functions are continuously differentiable.
    There are five methods of solving the constrained multivariable optimization.  These are:
1.    The Lagrange multiplier method.
2.    The Kuhn-Tucker conditions
3.    Gradient methods
a.    Newton-Raphson method
b.    Penalty function
4.    Method of feasible directions.
The Lagrange multiplier method is used in solving optimization problems with equality constraints, while the Kuhn-Tucker conditions are used in solving optimization problems with inequality constraints, though they play a major role in a type of constrained multivariable optimization called “Quadratic programming”.
The gradient methods include:
The Newton-Raphson method and the penalty function.  They are used in solving optimization problems with equality constraints while the method of feasible directions are used in solving problems with inequality constraints.
BASIC DEFINITIONS
1.    NEGATIVE DEFINITE: