This preview shows pages 1–4. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Lecture 12 Optimality Conditions UCSD Math 171A: Numerical Optimization Philip E. Gill http://ccom.ucsd.edu/~peg/math171 Friday, February 6th, 2009 Convex set Definition (Convex set) A set S is convex if, for every x , y S , it holds that z = (1 ) x + y S for all 0 1 UCSD Center for Computational Mathematics Slide 2/40, Friday, February 6th, 2009 Convex set x y z = (1 ) x + y S UCSD Center for Computational Mathematics Slide 3/40, Friday, February 6th, 2009 Nonconvex set x y UCSD Center for Computational Mathematics Slide 4/40, Friday, February 6th, 2009 Note that z = (1 ) x + y = x + ( y x ) = x + p , with p = y x i.e., steps along any p joining x and y S give a point in S . UCSD Center for Computational Mathematics Slide 5/40, Friday, February 6th, 2009 Convex set x z = x + p y = x + p S UCSD Center for Computational Mathematics Slide 6/40, Friday, February 6th, 2009 Result The feasible region FF = { x : Ax b } is either empty or convex. Proof: The result is trivial if FF is empty. Assume that FF is nonempty, with x , y FF , i.e., x FF Ax b y FF Ay b If [0 , 1] then A ( (1 ) x + y ) = (1 ) Ax + Ay (1 ) b + b = b Thus, (1 ) x + y FF . UCSD Center for Computational Mathematics Slide 7/40, Friday, February 6th, 2009 Implication: Given x * FF , we can write every other x FF in the form x = x * + p , with p = x x * Then Ap = A ( x x * ) = Ax Ax * b b = 0 that Ap 0 and p is a feasible direction at x * . every x FF is reachable from x * by taking a step along a feasible direction. UCSD Center for Computational Mathematics Slide 8/40, Friday, February 6th, 2009 x * UCSD Center for Computational Mathematics Slide 9/40, Friday, February 6th, 2009 Result x * is a minimizer of c T x subject to Ax b if and only if Ax * b c T p 0 for all feasible directions p at x * (i.e., A a p 0) Result x * is a minimizer of c T x subject to Ax b if and only if Ax * b There is no direction p such that c T p < and A a p UCSD Center for Computational Mathematics Slide 10/40, Friday, February 6th, 2009 Definition A direction p such that c T p < 0 and A a p 0 is known as a feasible descent direction ....
View
Full
Document
This note was uploaded on 10/23/2010 for the course MATH 171a taught by Professor Staff during the Winter '08 term at UCSD.
 Winter '08
 staff
 Math

Click to edit the document details