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Unformatted text preview: Background Optimization Models Linear Programming IE426: Optimization Models and Applications: Lecture 3 Jeff Linderoth Department of Industrial and Systems Engineering Lehigh University September 5, 2006 Jeff Linderoth IE426:Lecture 3 Background Optimization Models Linear Programming Today’s Outline Review Linear Programming Definition Canonical Models Geometry Our First Model XPRESS and Mosel Jeff Linderoth IE426:Lecture 3 Background Optimization Models Linear Programming Please don’t call on me!!! Which of the following are “easy” and which are “hard”? 1 Minimize a convex function? 2 Minimize a concave function? 3 Minimize a nonconvex function? 4 Maximize a convex function? 5 Maximize a concave function? 6 Maximize a nonconvex function? Jeff Linderoth IE426:Lecture 3 Background Optimization Models Linear Programming Please don’t call on me!!! True or False Discrete Sets are Convex? Polyhedral Sets are Convex? ∀ means “in”? ∀ means “for all”? My son’s name is Matilda? Jeff Linderoth IE426:Lecture 3 Background Optimization Models Linear Programming Review Terminology Linear Programming (LP) Here we wish to optimize a linear function over a polyhedral constraint set. max x ∈ R n f ( x ) def = c T x subject to Ax ≤ b l ≤ x ≤ u Vector inequalities are always meant componentwise : Ax ≤ b ⇔ j ∈ N a ij x j ≤ b i ∀ i ∈ M l ≤ x ≤ u ⇔ l i ≤ x i ≤ u i ∀ j ∈ N Jeff Linderoth IE426:Lecture 3 Background Optimization Models Linear Programming Review Terminology An LP Instance maximize 2 x 1 + 4 x 2 subject to 3 x 1 2 x 2 ≤ 6 x 1 + x 2 ≤ 8 x 1 ≥ x 2 ≥ x1 x2 Jeff Linderoth IE426:Lecture 3 Background Optimization Models Linear Programming Review Terminology Facts About Linear Programs 1 Their objective functions are convex and concave 2 Their feasible region is a convex (polyhedral) set 3 If there is an optimal solution, then there is an optimal solution that occurs at an extreme point of the feasible region Jeff Linderoth IE426:Lecture 3 Background Optimization Models Linear Programming Review Terminology Massaging Problems to Standard Form Lots of problems can be put into this “standard” form minimize? Negate the objective function: max f ( x ) ⇔ min f ( x ) ≥ constraint? Multiply constraint by 1: j ∈ N a j x j ≥ b ⇔ j ∈ N a j x j ≤  b Jeff Linderoth IE426:Lecture 3 Background Optimization Models Linear Programming Review Terminology Massaging Problems to Standard Form = constraint? Write as two constraints: ( ≤ and ≥ ) : j ∈ N a j x j = b ⇔ j ∈ N a j x j ≤ b, j ∈ N a j x j ≤  b Bounds l and u We allow l j ∈ {∞∪ R } , u j ∈ { R ∪∞} Almost always, we will have x j ≥ You’ll (almost) never have to do this again!...
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This note was uploaded on 02/29/2008 for the course IE 426 taught by Professor Linderoth during the Spring '08 term at Lehigh University .
 Spring '08
 Linderoth
 Optimization, Systems Engineering

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