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Math Help - Optimization Problem

  1. #1
    Newbie
    Joined
    Jun 2009
    Posts
    6

    Optimization Problem

    Dear readers,

    I have the following problem:

    Maximize with respect to  x:
     x[1-F(max \lbrace x, \frac{1}{2}x+c \rbrace)],
    where F denotes a CDF and c is an arbitrary constant.

    Also, under which conditions should I maximize
     x[1-F(x)], or  x[1-F( \frac{1}{2}x+c)],?

    What are the answers?

    This has been bugging me for a while, please help.

    Thank you!
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  2. #2
    Senior Member
    Joined
    Apr 2009
    From
    Atlanta, GA
    Posts
    408

    Piecewise Function and Chain Rule

    Notice max\{x,\frac12x+c\} =\left\{<br />
\begin{array}{rcl}<br />
\frac12x+c &\mbox{ if } & x\leq 2c<br />
\\x  & \mbox{ if } & x>2c<br />
\end{array}\right. so x[1-F(max\{x,\frac12x+c\})] =\left\{<br />
\begin{array}{rcl}<br />
x[1-F(\frac12x+c)] &\mbox{ if } & x\leq 2c<br />
\\x[1-F(x)]  & \mbox{ if } & x>2c<br />
\end{array}\right.

    Use the chain rule from here.
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