Linear Programming Lecture Notes - Penn State Personal Web Server

More documents

Recommendations

Info

Contents List of Figures vii Preface xi Chapter 1. Introduction to Optimization 1 1. A General Maximization Formulation 2 2. Some Geometry for Optimization 4 3. Gradients, Constraints and Optimization 10 Chapter 2. Simple Linear Programming Problems 13 1. Modeling Assumptions in Linear Programming 14 2. Graphically Solving Linear Programs Problems with Two Variables (Bounded Case) 16 3. Formalizing The Graphical Method 17 4. Problems with Alternative Optimal Solutions 18 5. Problems with No Solution 20 6. Problems with Unbounded Feasible Regions 22 Chapter 3. Matrices, Linear Algebra and Linear Programming 27 1. Matrices 27 2. Special Matrices and Vectors 29 3. Matrices and Linear Programming Expression 30 4. Gauss-Jordan Elimination and Solution to Linear Equations 33 5. Matrix Inverse 35 6. Solution of Linear Equations 37 7. Linear Combinations, Span, Linear Independence 39 8. Basis 41 9. Rank 43 10. Solving Systems with More Variables than Equations 45 11. Solving Linear Programs with Matlab 47 Chapter 4. Convex Sets, Functions and Cones and Polyhedral Theory 51 1. Convex Sets 51 2. Convex and Concave Functions 52 3. Polyhedral Sets 53 4. Rays and Directions 56 5. Directions of Polyhedral Sets 57 6. Extreme Points 59 7. Extreme Directions 62 iii
Page 1: Linear Programming: Penn State Math
Page 5: ‘ v
Page 8 and 9: combination of (x1, x2) on the line
Page 10 and 11: 8.1 System 2 has a solution if (and
Page 13 and 14: CHAPTER 1 Introduction to Optimizat
Page 15 and 16: connected set (as it is in a neighb
Page 17 and 18: Definition 1.7 (Graph). Let z : D
Page 19 and 20: Theorem 1.15. If z : R n → R is d
Page 21 and 22: Then the slope of the tangent line
Page 23 and 24: Example 1.22 (Continuation of Examp
Page 25 and 26: CHAPTER 2 Simple Linear Programming
Page 27 and 28: Proportionality Assumption: A probl
Page 29 and 30: draw parallel lines in the directio
Page 31 and 32: Example 2.7. Suppose the toy maker
Page 33 and 34: The level sets of the objective and
Page 36 and 37: Based on these two examples, we can
Page 39 and 40: CHAPTER 3 Matrices, Linear Algebra
Page 41 and 42: Exercise 21. By analogy define the
Page 43 and 44: Theorem 3.14. Every linear programm
Page 45 and 46: (3) If xi is unrestricted in any wa
Page 47 and 48: Definition 3.21 (Row Equivalence).
Page 49 and 50: 6. Solution of Linear Equations Let
Page 51 and 52: 7. Linear Combinations, Span, Linea
Page 53 and 54:
The last equation is tautological (
Page 55 and 56:
We can rearrange the terms in this
Page 57 and 58:
Example 3.45. Again consider the ma
Page 59 and 60:
11. Solving Linear Programs with Ma
Page 61 and 62:
Matlab Solution Figure 3.1. The fea
Page 63 and 64:
CHAPTER 4 Convex Sets, Functions an
Page 65 and 66:
f(x1)+(1− λ)f(x2) f(λx1 +(1−
Page 67 and 68:
a T x2 = b Then we have: (4.10) a T
Page 69 and 70:
Example 4.23. Consider the unbounde
Page 71 and 72:
Clearly d ≥ 0 and d = 0. 6. Extre
Page 73 and 74:
These two facts hold since Gd = 0 a
Page 75 and 76:
here we are interested only in dire
Page 77 and 78:
generality, suppose that γ is dete
Page 79 and 80:
If δi = 0 if i > k, then we have s
Page 81 and 82:
CHAPTER 5 The Simplex Method 1. Lin
Page 83 and 84:
was called a basic solution (See De
Page 85 and 86:
Expression 5.20 is called the minim
Page 87 and 88:
Using this information, we can comp
Page 89 and 90:
In this case, we see that index 2 (
Page 91 and 92:
Here 0 is the vector of zeros of ap
Page 93 and 94:
5. Identifying Unboundedness We hav
Page 95 and 96:
2x1 + x2 =6 Extreme direction x1
Page 97 and 98:
The cost information becomes: c T B
Page 99 and 100:
Figure 5.4. An optimization problem
Page 101:
Simplex Algorithm in Algebraic Form
Page 104 and 105:
Since bi ≥ 0, we will require xai
Page 106 and 107:
At this point, we have eliminated b
Page 108 and 109:
Two-Phase Simplex Algorithm (1) Giv
Page 110 and 111:
6.3 then we obtain the sequence of
Page 112 and 113:
point in Y we know that at most m e
Page 114 and 115:
4. The Single Artificial Variable T
Page 116 and 117:
he wants to know how many shamrocks
Page 118 and 119:
# # This finds the optimal solution
Page 121 and 122:
CHAPTER 7 Degeneracy and Convergenc
Page 123 and 124:
Tableau V: ⎡ z x6 x7 x3 ⎢ ⎣ T
Page 125 and 126:
Again, we chose to enter variable x
Page 127 and 128:
if i ∈ I1. This confirms (since b
Page 129 and 130:
CHAPTER 8 The Revised Simplex Metho
Page 131 and 132:
Let: (8.6) y3 = 1 8 (80 − 3x1 −
Page 133 and 134:
This is the ā1 column that is appe
Page 135 and 136:
Thus the fact that zj − cj ≤ 0
Page 137 and 138:
[0 1] [1 2] System 2 has a solution
Page 139 and 140:
The (negative) identity rows in E e
Page 141 and 142:
⎧ ⎪⎨ Dual Feasibility ⎪⎩
Page 143 and 144:
with A ∈ Rm×n , b ∈ Rm and (ro
Page 145 and 146:
This simplifies to: wB − vB = cB
Page 147 and 148:
the portion of the matrix where the
Page 149 and 150:
CHAPTER 9 Duality In the last chapt
Page 151 and 152:
CONSTRAINTS VARIABLES MINIMIZATION
Page 153 and 154:
2. Weak Duality There is a deep rel
Page 155 and 156:
Thus, we have shown that the KKT co
Page 157 and 158:
Thus it follows from Lemma 9.7 that
Page 159 and 160:
We can also illustrate the process
Page 161 and 162:
That is, the i th element of w repr
Page 163 and 164:
In this case, it is more difficult
Page 165 and 166:
(a) Original Problem (b) RHS Decrea
Page 167 and 168:
Then the standard form problem is g
Page 169:
Bibliography [BJS04] Mokhtar S. Baz
show all

Linear Programming Lecture Notes - Penn State Personal Web Server

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?