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Unformatted text preview: IE417: Nonlinear Programming: Lecture 12 Jeff Linderoth Department of Industrial and Systems Engineering Lehigh University 16th March 2006 Jeff Linderoth IE417:Lecture 12 Quiz Discussion Jeff Linderoth IE417:Lecture 12 Motivation We are interested in determining conditions under which we can verify that a solution is optimal. For constrained problems. For a very simple example, lets assume we are minimizing functions that are One-dimensional Continuous Differentiable Recall: a function f ( x ) is convex on a set S if for all a S and b S, f ( a + (1- ) b ) f ( a ) + (1- ) b . Jeff Linderoth IE417:Lecture 12 Why do we care? Algorithms for nonlinear programming work to find points that satisfy these conditions When faced with a problem that you dont know how to handle, write down the optimality conditions Often you can learn a lot about a problem, by examining the properties of its optimal solutions. Jeff Linderoth IE417:Lecture 12 (1-D) Constrained Optimization Now we consider the following problem for scalar variable x R 1 . z * = min x u f ( x ) There are three cases for where an optimal solution might be x = 0 < x < u x = u Jeff Linderoth IE417:Lecture 12 Breaking it down If < x < u , then the necessary and sufficient conditions for optimality are the same as the unconstrained case You should know these all too well! Namely, a necessary condition is that f ( x ) = 0 Jeff Linderoth IE417:Lecture 12 What if NOT < x < u If x = 0 , then we need f ( x ) (necessary), f > (sufficient) If x = u , then we need f ( x ) (necessary), f >...
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