# lect22 - ISE 536Fall03 Linear Programming and Extensions...

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ISE 536–Fall03: Linear Programming and Extensions November 24, 2003 Lecture 22: IPM, Path Following Methods Lecturer: Fernando Ord´o˜nez 1 A few ideas from convex optimization For a convex function f : < n 7→ < , the point that minimizes f ( x ), satisﬁes f ( x ) = 0, i.e.: Assume now that there also is a function g : < n 7→ < m , and that you are interested in the minimizer of f ( x ) constrained to g ( x ) = 0. How do you ﬁnd the point that solves: min f ( x ) s . t . g ( x ) = 0 1.1 Newton’s method To obtain the minimizer to optimization problems we need to ﬁnd x such that h ( x ) = 0 for some system of equations. Newton’s method does just this! If x ∈ < , to ﬁnd h ( x ) = 0 Newton’s method constructs the following iteration x k +1 = x k - h ( x k ) h 0 ( x k ) . 1

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For n equations and n unknowns, that is if h : < n 7→ < n and x ∈ < n , then Newton’s method is: x k +1 = x k - J ( x k ) - 1 h ( x k ) , where J ( x k ) is the n × n matrix of partial derivatives:
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lect22 - ISE 536Fall03 Linear Programming and Extensions...

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