Optimization - AMSC/CMSC 660 Scientific Computing I Fall...

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AMSC/CMSC 660 Scientific Computing I Fall 2006 Unit 3: Optimization Dianne P. O’Leary c 2002,2004,2006 Optimization: Fundamentals Our goal is to develop algorithms to solve the problem Problem P: Given a function f : S → R , find min x S f ( x ) with solution x opt . The point x opt is called the minimizer , and the value f ( x opt ) is the minimum . For unconstrained optimization, the set S is usually taken to be R n , but sometimes we make use of upper or lower bounds on the variables, restricting our search to a box { x : x u } for some given vectors ‘, u ∈ R n . The plan 1. Basics of unconstrained optimization 2. Alternatives to Newton’s method 3. Fundamentals of constrained optimization 1
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Part 1: Basics of unconstrained optimization Plan for Part 1: The plan: How do we recognize a solution? Some geometry. Our basic algorithm for finding a solution. The model method: Newton. How close to Newton do we need to be? Making methods safe: Descent directions and line searches. Trust regions. How do we recognize a solution? What does it mean to be a solution? The point x opt is a local solution to Problem P if there is a δ > 0 so that if x S and k x - x opt k < δ , then f ( x opt ) f ( x ) . In other words, x opt is at least as good as any point in its neighborhood. The point x opt is a global solution to Problem P if for any x S , then f ( x opt ) f ( x ) . Note: It would be nice if every local solution was guaranteed to be global. This is true if f is convex . We’ll look at this case more carefully in the ”Geometry” section of these notes. Some notation 2
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We’ll assume throughout this unit that f is smooth enough that it has as many continuous derivatives as we need . For this section, that means 2 continuous derivatives plus one more, possibly discontinuous. The gradient of f at x is defined to be the vector g ( x ) = 5 f ( x ) = ∂f/∂x 1 . . . ∂f/∂x n . The Hessian of f at x is the derivative of the gradient: H ( x ) = 5 2 f ( x ) , with h ij = 2 f ∂x i ∂x j Note that the Hessian is symmetric, unless f fails to be smooth enough. How do we recognize a solution? Recall from calculus Taylor series : Suppose we have a vector p ∈ R n with k p k = 1 , and a small scalar h . Then f ( x opt + h p ) = f ( x opt ) + h p T g ( x opt ) + 1 2 h 2 p T H ( x opt ) p + O ( h 3 ) . First Order Necessary Condition for Optimality f ( x opt + h p ) = f ( x opt ) + h p T g ( x opt ) + 1 2 h 2 p T H ( x opt ) p + O ( h 3 ) . Now suppose that g ( x opt ) is nonzero. Then we can always find a descent or downhill direction p so that p T g ( x opt ) < 0 . (Take, for example, p = - g ( x opt ) / k g ( x opt ) k .) Therefore, for small enough h , we can make 1 2 h 2 p T H ( x opt ) p small enough that f ( x opt + h p ) < f ( x opt ) . Therefore, a necessary condition for x opt to be a minimizer is that g ( x opt ) = 0 . 3
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Second Order Necessary Condition for Optimality So we know that if x opt is a minimizer, then g ( x opt ) = 0 , so f ( x opt + h p ) = f ( x opt ) + 1 2 h 2 p T H ( x opt ) p + O ( h 3 ) .
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