Results 1 to 2 of 2

Math Help - prove that solution of min problem is unique

  1. #1
    Newbie
    Joined
    Dec 2008
    Posts
    1

    prove that solution of min problem is unique

    Hi

    I have such exercise:
    Let U be an open convex set of R^n and f be a strictly quasi-convex function.
    Show that if there exists a solution of minimization problem, then it is a unique solution.

    The problem Q is: min f(x) for all x belonging to U

    I know that in 1dim f is strictly quasi-convex <=> for all (x,y)belonging to U^2, x not=y, for all t belonging to (0,1)
    f(tx+(1-t)y) < max (f(x), f(y))
    but here we have n-dimensional problem, so I am horribly confused and really don't know how to do this :/
    Do you know how to solve it?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Apr 2005
    Posts
    14,974
    Thanks
    1121
    Quote Originally Posted by doma View Post
    Hi

    I have such exercise:
    Let U be an open convex set of R^n and f be a strictly quasi-convex function.
    Show that if there exists a solution of minimization problem, then it is a unique solution.

    The problem Q is: min f(x) for all x belonging to U

    I know that in 1dim f is strictly quasi-convex <=> for all (x,y)belonging to U^2, x not=y, for all t belonging to (0,1)
    f(tx+(1-t)y) < max (f(x), f(y))
    but here we have n-dimensional problem, so I am horribly confused and really don't know how to do this :/
    Do you know how to solve it?
    Exactly what do you mean by "solution", the point that makes the fuction miminum or simply that minimum value? For a quasi-convex problem on a convex set, I would say that a "solution" to the minimization consists of a point, p, in the set such that f(p) is less than or equal f(q) for q any other point in the set. That is NOT unique. For example, let S be the square in R^2 with vertices at (0,0), (1,0), (1,1), and (0,1) and let f((x,y))= y. Then any point (x,0) with 0\le x\le 1 is a "solution". Of course, the value of f at any of those points is the same: the set of real numbers is "linearly ordered"- if a set of real numbers has a minimum, that minimum is unique.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 1
    Last Post: September 23rd 2011, 03:39 AM
  2. Replies: 5
    Last Post: November 21st 2010, 01:07 PM
  3. Replies: 3
    Last Post: September 16th 2010, 12:23 PM
  4. Replies: 1
    Last Post: March 24th 2010, 12:14 AM
  5. Replies: 2
    Last Post: September 7th 2009, 02:01 PM

Search Tags


/mathhelpforum @mathhelpforum