MULTIVARIABLE OPTIMIZATION WITH CONSTRAINTS


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

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   

2.0    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

3.0    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

The solution from the Kuhn-Tucker Conditions

method                                76

4.4.1    The Two-Phase Method                    76   

4.4.2    The Elimination Method                    77   

4.5    Summary                                117   

CHAPTER FIVE

Conclusion                                    118

References                                    120                           CHAPTER ONE

1.0    INTRODUCTION

There are two types of optimization problems:

    Type 1

    Minimize or maximize          Z = f(x)            (1)

                XE Rn

    Type 2

    Minimize or maximize     Z  =  f(x)            (2)

            Subject to      g(x)   ~ bi,  i, = 1, 2, -----, m   (3)

        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:

The quadratic form XT Rx is negative definite if (-1)i+1 Ri<0, i = 1(1)m.

Using (-1)i+1 Ri<0.

When i = 1  à  (-12 R1     i = 2 à (-1)3 R2 < 0  à  R2 < 0: R2 > 0

    i = 3 à (-1)4 R3 < 0   à  R3 < 0

  R1 < 0, R2 > 0, R3 < 0, R4 > 0, -------

2.    NEGATIVE SEMI-DEFINITE

The quadratic form XT Rx is negative semi-definite if (-1)i+1 Ri < 0 and at least one (-1)i+1 Ri ¹ 0

3.    POSITIVE DEFINITE

The quadratic form XT Rx is positive definite if Ri > 0, i = 1 (1)m.

        Example:

    R  =          r11    r12    r13  - - - - - - -    r1m

            r21    r22    r23  - - - - - - -    r2m

            r31    r32    r33  - - - - - - -    r3m

            rm1    rm2    rm3  - - - - - - -    rmm

    where

        R1  =    r11     > 0

        R2  =

            r11    r12    > 0

            r21    r22

4.    POSITIVE SEMI DEFINITE

The quadratic form XT Rx is positive semi definite if Ri > 0, i = 1 (1)m and at least one Ri ¹ 0

    5.    CONVEX

The function f is convex if the matrix R positive definite.  Example is f(x).

    6.    CONCAVE

A function f is said to be concave if its negative is convex.  Example is   -f (x).

    NOTE:

Whether the objective function is convex or concave, it means the matrix is positive definite or negative definite.  When the matrix is positive definite or positive semi-definite, it should be minimized, but when it is negative definite or negative semi-definite, then it should be maximized.

LAYOUT OF WORK

    There are five chapters in this project.

Chapter two is dedicated to two methods of solving constrained optimization.  These methods are the Lagrange multiplier method and the Kuhn-Tucker conditions.  This section clearly shows how the Kuhn-Tucker conditions are derived from the Lagrange multiplier method, in an optimization problem with inequality constraints.  As part of this chapter, the global maximum, local maximum and the global minimum of an optimization problem was also derived.

Chapter three presents the gradient methods and the method of feasible directions.  The gradient methods are the Newton Raphson method and the penalty function.

The gradient methods are used in solving optimization problems with equality constraints while the method of feasible directions is used in solving optimization problems with inequality constraints.

Chapter four is specifically on a type of multivariable optimization with constraints.  This is called “Quadratic programming”.  This chapter comprises of quadratic forms, general quadratic problems and it shows the importance of two methods called the Lagrange multiplier method and the Kuhn-Tucker conditions.  This section explains how we can arrive at an optimal solution through two different methods after the Kuhn-Tucker conditions have been formed.  These are the two-phase method and the elimination method.

Chapter 5 is the concluding part of this project.

Each chapter starts with an introduction that facilitates the understanding of the section and also contains useful examples.

In conclusion, this research will make us understand the different methods of solving constrained optimization and how some of these methods are applied in quadratic programming.

.

MULTIVARIABLE OPTIMIZATION WITH CONSTRAINTS



TYPE IN YOUR TOPIC AND CLICK SEARCH.






RESEARCHWAP.ORG

Researchwap.org is an online repository for free project topics and research materials, articles and custom writing of research works. We’re an online resource centre that provides a vast database for students to access numerous research project topics and materials. Researchwap.org guides and assist Postgraduate, Undergraduate and Final Year Students with well researched and quality project topics, topic ideas, research guides and project materials. We’re reliable and trustworthy, and we really understand what is called “time factor”, that is why we’ve simplified the process so that students can get their research projects ready on time. Our platform provides more educational services, such as hiring a writer, research analysis, and software for computer science research and we also seriously adhere to a timely delivery.

TESTIMONIES FROM OUR CLIENTS


Please feel free to carefully review some written and captured responses from our satisfied clients.

  • "Exceptionally outstanding. Highly recommend for all who wish to have effective and excellent project defence. Easily Accessable, Affordable, Effective and effective."

    Debby Henry George, Massachusetts Institute of Technology (MIT), Cambridge, USA.
  • "I saw this website on facebook page and I did not even bother since I was in a hurry to complete my project. But I am totally amazed that when I visited the website and saw the topic I was looking for and I decided to give a try and now I have received it within an hour after ordering the material. Am grateful guys!"

    Hilary Yusuf, United States International University Africa, Nairobi, Kenya.
  • "Researchwap.org is a website I recommend to all student and researchers within and outside the country. The web owners are doing great job and I appreciate them for that. Once again, thank you very much "researchwap.org" and God bless you and your business! ."

    Debby Henry George, Massachusetts Institute of Technology (MIT), Cambridge, USA.
  • "Great User Experience, Nice flows and Superb functionalities.The app is indeed a great tech innovation for greasing the wheels of final year, research and other pedagogical related project works. A trial would definitely convince you."

    Lamilare Valentine, Kwame Nkrumah University, Kumasi, Ghana.
  • "I love what you guys are doing, your material guided me well through my research. Thank you for helping me achieve academic success."

    Sampson, University of Nigeria, Nsukka.
  • "researchwap.com is God-sent! I got good grades in my seminar and project with the help of your service, thank you soooooo much."

    Cynthia, Akwa Ibom State University .
  • "Sorry, it was in my spam folder all along, I should have looked it up properly first. Please keep up the good work, your team is quite commited. Am grateful...I will certainly refer my friends too."

    Elizabeth, Obafemi Awolowo University
  • "Am happy the defense went well, thanks to your articles. I may not be able to express how grateful I am for all your assistance, but on my honour, I owe you guys a good number of referrals. Thank you once again."

    Ali Olanrewaju, Lagos State University.
  • "My Dear Researchwap, initially I never believed one can actually do honest business transactions with Nigerians online until i stumbled into your website. You have broken a new legacy of record as far as am concerned. Keep up the good work!"

    Willie Ekereobong, University of Port Harcourt.
  • "WOW, SO IT'S TRUE??!! I can't believe I got this quality work for just 3k...I thought it was scam ooo. I wouldn't mind if it goes for over 5k, its worth it. Thank you!"

    Theressa, Igbinedion University.
  • "I did not see my project topic on your website so I decided to call your customer care number, the attention I got was epic! I got help from the beginning to the end of my project in just 3 days, they even taught me how to defend my project and I got a 'B' at the end. Thank you so much researchwap.com, infact, I owe my graduating well today to you guys...."

    Joseph, Abia state Polytechnic.
  • "My friend told me about ResearchWap website, I doubted her until I saw her receive her full project in less than 15 miniutes, I tried mine too and got it same, right now, am telling everyone in my school about researchwap.com, no one has to suffer any more writing their project. Thank you for making life easy for me and my fellow students... Keep up the good work"

    Christiana, Landmark University .
  • "I wish I knew you guys when I wrote my first degree project, it took so much time and effort then. Now, with just a click of a button, I got my complete project in less than 15 minutes. You guys are too amazing!."

    Musa, Federal University of Technology Minna
  • "I was scared at first when I saw your website but I decided to risk my last 3k and surprisingly I got my complete project in my email box instantly. This is so nice!!!."

    Ali Obafemi, Ibrahim Badamasi Babangida University, Niger State.
  • To contribute to our success story, send us a feedback or please kindly call 2348037664978.
    Then your comment and contact will be published here also with your consent.

    Thank you for choosing researchwap.com.