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optimization - Chapter 6 Optimization Steepest descent...

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Chapter 6: Optimization Steepest descent method (gradient methods) Newton’s method Optimality conditions Convexity Example 1: strategic bidding in energy Example 2: classification, machine learning Example 3: The Netflix problem revisited
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Convexity
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Unconstrained Optimization min f ( x ) where x is a real valued function of n variables Constrained Optimization f ( x , y ) = (1 x ) 2 + 100( y x 2 ) 2 min f ( x ) s.t h ( x ) = 0 g ( x ) 0
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Chapter 2: Identifying a Local Solution Thm 1. First Order Necessary Conditions. If x * is a local minimizer of f and f is continuously differentiable, then f ( x *) = 0 Thm 2. Second Order Necessary Conditions. If x * is a local minimizer of f and 2 f is continuously differentiable, then f ( x *) = 0 and 2 f ( x *) 0 Necessary but not sufficient. Example: f ( x ) = x 3
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Gradient Vectors, Hessian Matrices f ( x , y ) = 2 x y = f / x f / y Example : f ( x , y ) = x 2 + 0.5 y 2 2 f ( x , y ) = 2 0 0 1 = 2 f / x 2 f / x y f / y x 2 f / y 2 If f is a continuously differentiable function, the Hessian matrix is symmetric
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Thm 4. Suppose that f is convex.
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