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Unformatted text preview: Fall 2007 ARE211 Problem Set #05 Second Graphical Problem Set Due date: Oct 09 (1) Convexity and quasi-convexity: (a) Give a “diagrammatic proof” that if f is strictly convex then a local minimum of f is a strict global minimum. Make the argument that if f has a local minimum that isn’t a strict global minimum, then f isn’t strictly convex. Use the fact that f is strictly convex if the set above its graph is a “strictly convex” set. (b) Now establish the same result for the case in which f is strictly quasi-convex. Make the argument that if f has a local minimum that isn’t a strict global minimum, then f isn’t strictly quasi-convex. (2) Consider the constrained optimization problem: minimize a continuously differentiable func- tion f ( x ) s.t. x < 0. (a) In what way is this problem fundamentally different from the standard constrained optimization problem that we have been studying in lass? (b) Write down the necessary conditions for x * to be a solution to this problem. (c) Illustrate with a graphical example (i.e., construct an f ) with the property that at the unique solution to the problem, the constraint is satisfied with strict inequality and is linding but not binding (see lecture notes mathGraphical3 for a definition of linding). Explain carefully why the constraint is linding but not binding....
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- Fall '07
- Optimization, Constraint, 05 second, strict global minimum