lecture3 - Background Optimization Models Linear...

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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
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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 0 x 2 0 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
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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 0 You’ll (almost) never have to do this again!
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