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Lecture 12 - Optimality Conditions

# Lecture 12 - Optimality Conditions - Convex set Lecture 12...

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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 - θ ) x + θ y ∈ S for all 0 θ 1 UCSD Center for Computational Mathematics Slide 2/40, Friday, February 6th, 2009 Convex set x y z = (1 - θ ) x + θy 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

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Note that z = (1 - θ ) x + θ y = x + θ ( y - x ) = x + θ 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 + θp 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 θ [0 , 1] then A ( (1 - θ ) x + θ y ) = (1 - θ ) Ax + θ Ay (1 - θ ) b + θ b = b Thus, (1 - θ ) x + θ 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 < 0 and A a p 0 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 . ELP c T p 0 for all Ap = 0 LP c T p 0 for all A a p 0 Both imply the existence of Lagrange multipliers.

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Lecture 12 - Optimality Conditions - Convex set Lecture 12...

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