1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Threads - Proving Convexity Date
Prove the following is a convex set? Oct 21, 2012
Proving a Set is Convex Oct 11, 2012
Prove that set is convex Apr 19, 2012
Prove intersection of convex cones is convex. Oct 8, 2011
Prove Open Balls in an NLS are convex. Jun 23, 2011