anay_cen_web_art

anay_cen_web_art - Notes for Class: Analytic Center,...

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Notes for Class: Analytic Center, Newton’s Method, and Web-Based ACA Robert M. Freund April 8, 2004 c ± 2004 Massachusetts Institute of Technology. 1
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± 1 The Analytic Center of a Polyhedral System Given a polyhedral system of the form: Ax b, Mx = g, the analytic center is the solution of the following optimization problem: m (ACP:) maximize x,s s i i =1 s.t. Ax + s = b s 0 = g. This is easily seen to be the same as: m (ACP:) minimize x,s ln( s i ) i =1 s.t. Ax + s = b s 0 = The analytic center possesses a very nice “centrality” property. Suppose that s ) is the analytic center. Define the following matrix: x, ˆ s 1 ) 2 0 ... 0 0 ( ˆ s 2 ) 2 0 ˆ S 2 := . . . . . . . . . . . . . 0 0 s m ) 2 2
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± ² ³ ± ² ³ x ^ P E out E in Figure 1: Illustration of the Ellipsoid construction at the analytic center. Next define the following sets: P := { x | Mx = g, Ax b } x ) T A T S ˆ 2 A ( x ˆ E IN := x | = ( x ˆ x ) 1 x ) T A T S ˆ 2 A ( x ˆ E OUT := x | = ( x ˆ x ) m x, ˆ Theorem 1.1 If s ) is the analytic center, then: E IN P E OUT . This theorem is illustrated in Figure 1. The theorem is actually pretty easy to prove. 3
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± ² Proof: Suppose that x E IN ,and let s = b Ax .S in c e Mx = g ,weneed only prove that s 0 to show that x P . By construction of E IN , s satisfies ( s s ˆ) T S ˆ 2 ( s s 1, where ˆ = b A ˆ s x . This in turn can be written as: m ³ ( s i s ˆ i ) 2 1 , 2 i =1 s ˆ i whereby we see that each s i must satisfy s i 0. Therefore Ax b and so x P . We can write the optimality conditions (KKT conditions) for problem ACP as: S ˆ 1 e + λ = 0 0+ A T λ + M T u =0 A ˆ s x = b M ˆ x = g, where e =(1 ,..., 1) T , i.e., the e is the vector of ones. From this we can derive the following fact: if ( x, s ) is feasible for problem ACP, then T S ˆ 1 T ˆ T e s = e S 1 ( b Ax )= λ T b + u T = λ T b + u g. Since this is true for any ( x, s ) feasible for it will also be true for s ) x, ˆ (where ˆ = b A ˆ s x ), and so T S ˆ 1 e s = λ T b + u T g = e T S ˆ 1 s ˆ = m.
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This note was uploaded on 12/04/2011 for the course ESD 15.094 taught by Professor Jiesun during the Spring '04 term at MIT.

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anay_cen_web_art - Notes for Class: Analytic Center,...

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