Linear Programming Lecture Notes - Penn State Personal Web Server

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8.1 System 2 has a solution if (and only if) the vector c is contained inside the positive cone constructed from the rows of A. 124 8.2 System 1 has a solution if (and only if) the vector c is not contained inside the positive cone constructed from the rows of A. 124 8.3 An example of Farkas’ Lemma: The vector c is inside the positive cone formed by the rows of A, but c ′ is not. 125 8.4 The Gradient Cone: At optimality, the cost vector c is obtuse with respect to the directions formed by the binding constraints. It is also contained inside the cone of the gradients of the binding constraints, which we will discuss at length later. 130 8.5 This figure illustrates the optimal point of the problem given in Example 8.13. Note that at optimality, the objective function gradient is in the dual cone of the binding constraint. That is, it is a positive combination of the gradients of the left-hand-sides of the binding constraints at optimality. The gradient of the objective function is shown in green. 136 9.1 The dual feasible region in this problem is a mirror image (almost) of the primal feasible region. This occurs when the right-hand-side vector b is equal to the objective function coefficient column vector c T and the matrix A is symmetric. 146 9.2 The simplex algorithm begins at a feasible point in the feasible region of the primal problem. In this case, this is also the same starting point in the dual problem, which is infeasible. The simplex algorithm moves through the feasible region of the primal problem towards a point in the dual feasible region. At the conclusion of the algorithm, the algorithm reaches the unique point that is both primal and dual feasible. 148 9.3 Degeneracy in the primal problem causes alternative optimal solutions in the dual problem and destroys the direct relationship between the resource margin price that the dual variables represent in a non-degenerate problem. 153 x
Preface Stop! Stop right now! This is a set of lecture notes. It is not a book. Go away and come back when you have a real textbook on Linear Programming. Okay, do you have a book? Alright, let’s move on then. This is a set of lecture notes for Math 484–Penn State’s undergraduate Linear Programming course. Since I use these notes while I teach, there may be typographical errors that I noticed in class, but did not fix in the notes. If you see a typo, send me an e-mail and I’ll add an acknowledgement. There may be many typos, that’s why you should have a real textbook. The lecture notes are based on the first 6 chapters of Bazaraa et al.’s Linear Programming and Network Flows book. This is a reasonably good book, written primarily by and for Industrial Engineers. The only problem I have with the book is that it does not present major results in the standard theorem-proof style common to mathematical discourse. This set of notes corrects this problem by presenting the material in a format for presentation to a mathematics class. Many of the proofs in this set of notes are adapted from the textbook with some minor additions. (For example, each time Bazaraa et al. ask, “Why?” in a proof, I provide this information.) Additionally, I prefer to present maximization problems, while Linear Programming and Network Flows prefers the minimization format. I’ve modified all the proofs to operate on maximization problems. When used with the book, the student can obtain a complete set of proofs for elementary Linear Programming. In order to use these notes successfully, you should have taken courses in: (1) Vector calculus (Math 230/231 at Penn State) (2) Matrix algebra (Math 220 at Penn State) I review a substantial amount of the material you will need, but it’s always good to have covered prerequisites before you get to a class. That being said, I hope you enjoy using these notes! xi
Page 1: Linear Programming: Penn State Math
Page 4 and 5: 8. Caratheodory Characterization Th
Page 7 and 8: List of Figures 1.1 Goat pen with u
Page 9: 4.6 Convex Direction: Clearly every
Page 14 and 15: x Goat Pen Figure 1.1. Goat pen wit
Page 16 and 17: We have formulated the general maxi
Page 18 and 19: Figure 1.3. Contour Plot of z = x 2
Page 20 and 21: Figure 1.5. A Level Curve Plot with
Page 22 and 23: and thus our vector is v = (x1 −
Page 24 and 25: Figure 1.7. Gradients of the Bindin
Page 26 and 27: finishing toys and 120 hours per we
Page 28 and 29: 2. Graphically Solving Linear Progr
Page 30 and 31: S r x 0 Br( x0) Figure 2.2. A Bound
Page 32 and 33: S Every point on this line is an al
Page 34: 6. Problems with Unbounded Feasible
Page 37: to “prove” your example works.
Page 40 and 41: Definition 3.6 (Matrix Transpose).
Page 42 and 43: 3. Matrices and Linear Programming
Page 44 and 45: Remark 3.16. We can deal with const
Page 46 and 47: Theorem 3.20. Each elementary row o
Page 48 and 49: Gauss-Jordan Elimination Computing
Page 50 and 51: and right hand side vector b = [7,
Page 52 and 53: Gauss-Jordan elimination to solve t
Page 54 and 55: which clearly has a solution for al
Page 56 and 57: which is a contradiction. Case 2: S
Page 58 and 59: Definition 3.46 (Basic Variables).
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This leads to the following linear
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This is not in a form Matlab likes,
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Example 4.5. Figure 4.1 illustrates
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Figure 4.3. A hyperplane in 3 dimen
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4. Rays and Directions Recall the d
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for all λ > 0. This can only be tr
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Theorem 4.31. Let P ⊆ R n be a po
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Figure 4.9. A Polyhedral Set: This
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Figure 4.10. Visualization of the s
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It is clear that P is bounded. In f
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x1 x5 x2 x λx2 +(1− λ)x3 x4 x3
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If there is some i such that c T di
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Consider now the fact xj = 0 for al
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Example 5.9. Consider the Toy Maker
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using this information, we can comp
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Figure 5.1. The Simplex Algorithm:
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Using this information, we can see
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Based on this information, we can c
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Using the rule you developed in Exe
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Exercise 52. Consider the diet prob
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Suppose we choose s4 as the leaving
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CHAPTER 6 Simplex Initialization In
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A basic feasible solution for our a
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variables. This is because we start
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To remove the artificial variables
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Remark 6.6. In the case of a minimi
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Example 6.13. Suppose we solve the
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That is, s1 = −12 and s2 = −20
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The complete problem describing McL
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* data section */ data; set DAY :=
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This leads to a vector of reduced c
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2.1. Lexicographic Minimum Ratio Te
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Proof. The initial basis Im yieds a
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for i = 1, . . . , n. Then we have:
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Revised Simplex Algorithm (1) Ident
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simplex tableau). We obtain: z1 −
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Proof. We can prove Farkas’ Lemma
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A1· Half-space cy > 0 A2· c Am·
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3. The Karush-Kuhn-Tucker Condition
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Remark 8.9. The expressions: ∗ A
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(x ∗ 1,x ∗ 2) = (16, 72) 3x1 +
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Since w1, w2 ≥ 0, we know that w
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Example 8.13. Consider the followin
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Figure 8.5. This figure illustrates
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(9.4) Proof. Rewrite Problem D as:
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This vector will be related to c in
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3. Strong Duality Lemma 9.7. Proble
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Likewise beginning with the KKT con
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egions are polyhedral sets 1 . Cert
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Figure 9.2. The simplex algorithm b
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If a selfish profit maximizer wishe
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Recall the optimal full tableau for
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For the sake of space, we will prov
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We can choose either s1 or s2 as a
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