Optimizationcrashcourse

# Optimizationcrashcourse - A Crash Course in Optimization...

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1 Examples Why optimize? 1. Microeconomics: The consumer wants to choose the best bundle that she can afford. 2. Finance: Portfolio management (minimize risk subject to return) 3. Diet Problem: Find the cheapest combination of food that satisﬁes all the daily nutri- tional requirements. 4. Engineering: Design of aircrafts and aerospace structures of minimum weight given constraints, or ﬁnd the optimal trajectories of space vehicles (dynamic optimization). 5. Physics: Thermodynamics (heat bath), quantum mechanics, etc. 2 Optimization Optimize = Maximize or Minimize 2.1 Without Constraint 2.1.1 One Variable Maximize f where f : R R . Definition 1 A stationary point x is a point such that f 0 ( x ) = 0 . If x * is an optimizer, then it is a stationary point . To ﬁnd an optimizer, we take the ﬁrst-order condition and solve f 0 ( x ) = 0 for x . To determine whether an optimizer is a maximizer or a minimizer, we take the second-order condition; i.e. we look at the sign of

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Optimizationcrashcourse - A Crash Course in Optimization...

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