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Unformatted text preview: Production Planning Separating Models and Data Yummy IE426: Optimization Models and Applications: Lecture 4 Jeff Linderoth Department of Industrial and Systems Engineering Lehigh University September 7, 2006 Jeff Linderoth IE426:Lecture 4 Production Planning Separating Models and Data Yummy HW #1 Student: I just think, like, Prof. Linderoth hates me. I really think he wants to kill me. Therapist: He doesn’t really want to kill you. Sometimes we just say that. Dr. Linderoth: No actually the boy is quite astute. I really am trying to kill him, but so far unsuccessfully. He’s quite wily like his ol’ Professor. Jeff Linderoth IE426:Lecture 4 Production Planning Separating Models and Data Yummy I Really AM Evil Actual Quotes From Actual Students “Prof. Linderoth’s homework assignments are !^&*(!^&!*(^! !^&*!^!&*^*&!^!*&^!*(! impossible.” “I started the homework at 8PM the night before it was due, and I still couldn’t finish it on time” “Once I figured out to start on the homework as soon as it was assigned, I was able to finish the homework on time.” “I loved the homework. It was much better than Cats. I want to do it again and again.” Jeff Linderoth IE426:Lecture 4 Production Planning Separating Models and Data Yummy Please Don’t Call On Me! Computer Geniuses!?!?!?!?! Was there any trouble getting XPRESSMP (student edition) to work? 1 What are the three components of an optimization model? 2 What is a linear program? 3 What is the best way to improve your modeling skills? Jeff Linderoth IE426:Lecture 4 Production Planning Separating Models and Data Yummy Background Model Steps Final Model Calcoollus Do you guys remember calculus? Calculus Recall : A function f : R ⊇ D → R is convex at x if and only if f ( x ) ≥ Thus, f is convex everywhere in its domain if and only if f ( x ) ≥ ∀ x ∈ D f : R ⊇ D → R is concave if and only if f is convex These definitions are great for one dimension. But I may want to solve problems with more than one variable. To generalize to higher dimensions, we need a few more definitions Jeff Linderoth IE426:Lecture 4 Production Planning Separating Models and Data Yummy Background Model Steps Final Model Multidimensional calculus The Hessian of a function f R n → R is an n × n matrix of its second partial derivatives. The Hessian of f is often denoted ∇ 2 ( f ) The ( i, j ) th entry of ∇ 2 ( f ) is [ ∇ 2 ( f )] ij = ∂ 2 ( f ) ∂x i ∂x j An i th principal minor of an n × n matrix is the determinant of any i × i matrix obtained by deleting n i rows and the corresponding n i columns of the matrix Key Convex Function Theorem f : R ⊇ D → R is convex if and only if all principal minors of ∇ 2 ( f ) are nonnegative....
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 Spring '08
 Linderoth
 Linear Programming, Optimization, Systems Engineering, Prof. Linderoth, Jeff Linderoth, Production Planning Separating, Planning Separating Models

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