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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 doesnt 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. Hes 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. Linderoths homework assignments are !^&*(!^&!*(^! !^&*!^!&*^*&!^!*&^!*(! impossible. I started the homework at 8PM the night before it was due, and I still couldnt 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 Dont Call On Me! Computer Geniuses!?!?!?!?! Was there any trouble getting XPRESS-MP (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 Cal-cool-lus 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 non-negative....
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