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Unformatted text preview: MIT OpenCourseWare http://ocw.mit.edu 16.323 Principles of Optimal Control Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms . 16.323 Lecture 1 Nonlinear Optimization Unconstrained nonlinear optimization Line search methods Figure by MIT OpenCourseWare. Spr 2008 16.323 11 Basics Unconstrained Typical objective is to minimize a nonlinear function F ( x ) of the parameters x . Assume that F ( x ) is scalar x = arg min x F ( x ) Define two types of minima: Strong : objective function increases locally in all directions A point x is a strong minimum of a function F ( x ) if a scalar > exists such that F ( x ) < F ( x + x ) for all x such that < x Weak : objective function remains same in some directions, and increases locally in other directions Point x is a weak minimum of a function F ( x ) if is not a strong minimum and a scalar > exists such that F ( x ) F ( x + x ) for all x such that < x Note that a minimum is a unique global minimum if the definitions hold for = . Otherwise these are local minima.21.510.5 0.5 1 1.5 2 1 2 3 4 5 6 x F(x) Figure 1.1: F ( x ) = x 4 2 x 2 + x + 3 with local and global minima June 18, 2008 Spr 2008 16.323 12 First Order Conditions If F ( x ) has continuous second derivatives, can approximate function in the neighborhood of an arbitrary point using Taylor series: F ( x + x ) F ( x ) + x T g ( x ) + 1 x T G ( x ) x + ... 2 where g gradient of F and G second derivative of F 2 F 2 F T F x 2 1 x 1 x n x 1 ,G = x 1 F . . . . . . . . . . . . . . x = , g = = . x 2 F 2 F F x n x n x 1 x 2 n x n Firstorder condition from first two terms (assume x 1 ) Given ambiguity of sign of the term x T g ( x ) , can only avoid cost decrease F ( x + x ) < F ( x ) if g ( x ) = 0 Obtain further information from higher derivatives g ( x ) = 0 is a necessary and sucient condition for a point to be a stationary point a necessary, but not sucient condition to be a minima. Stationary point could also be a maximum or a saddle point. June 18, 2008 Spr 2008 16.323 13 Additional conditions can be derived from the Taylor expansion if we set g ( x ) = 0 , in which case: 1 F ( x + x ) F ( x ) + x T G ( x ) x + ... 2 For a strong minimum, need x T G ( x ) x > for all x , which is sucient to ensure that F ( x + x ) > F ( x ) . To be true for arbitrary x = 0 , sucient condition is that G ( x ) > (PD)....
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