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Proving Convexity

  1. Apr 6, 2009 #1
    1. The problem statement, all variables and given/known data

    Let g1, ..., gm be concave functions on R^n . Prove that the set S={x| gi(x)[tex]\geq 0[/tex], i=1,...,m} is convex



    3. The attempt at a solution

    So i tried this using two different definitions.

    First i used the definition that says f(y)[tex]\leq[/tex] f(x) + [tex]\nabla[/tex]f(x)T(y-x)

    then i substitued f(ax + (1-a)y)[tex]\geq[/tex] af(x) + (1-a)f(y)

    and tried to do some manipulations to show that the inequalites wen the other way but that didnt come out right.

    Now im stuck.
     
  2. jcsd
  3. Apr 7, 2009 #2
    any thoughts?
     
  4. Apr 7, 2009 #3
    How do you show a set is convex?

    Can you do this when m=1?
     
  5. Apr 7, 2009 #4
    we can show a set is convex for for any elements x and y

    ax + (1-a)y are in S. for a between 0 and 1. but i dont know how to use that here.
     
  6. Apr 7, 2009 #5
    OK, so what does it mean for x and y to be in S (this is the given)?

    Again, do the m=1 case first, for simplicity.

    When you get to it, just use f(ax + (1-a)y) [tex]\geq[/tex] af(x) + (1-a)f(y) for concave, not the other one.
     
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