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Thread: Optimization Problem

  1. #1
    Newbie
    Joined
    Jun 2009
    Posts
    6

    Optimization Problem

    Dear readers,

    I have the following problem:

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

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

    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
    409

    Piecewise Function and Chain Rule

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

    Use the chain rule from here.
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