$W_{opt}=\arg \{\max(\pi_0 F_{L_0}(W)-\frac{\pi_1}{W}\int_0^W F_{L_1}( \alpha )d \alpha )\}$

subject to $\quad \int_0^W F_{L_0} (\alpha)d\alpha <\xi$

We should find analytically the optimal $W >0$ which maximize the first equation subject to the second equation, where $F( \cdot )$ is comulative distribution function (CDF), and $L_0$ and $L_1$ are positive random variables. $\xi$, $\pi_0$, $\pi_1$ are constant. Also, $0<\pi_0, \pi_1<1$ and $\pi_0 + \pi_1 =1$. All variables are real. Further, if needed, we can assume that, for example, $L_0$ and $L_1$ may have Erlang or exponential distribution.

  • $\begingroup$ Please fix your LaTeX. The eqnarray environment is not understood. So you are much better off just type-setting each equation line by line. $\endgroup$ – Willie Wong Nov 8 '10 at 12:59
  • $\begingroup$ I see no question here. What is it? $\endgroup$ – Harald Hanche-Olsen Nov 8 '10 at 13:44
  • $\begingroup$ Please specify the set over which you are taking the maximum: over what variables, in what range? Also, do you really want a $\max$ (not likely to exist due to the strict inequalities) or just a $\sup$? $\endgroup$ – Pietro Majer Nov 8 '10 at 20:51

Hmm, you could rewrite it this way: (I'm going to assume that $\pi_0, \pi_1, \xi'', \mu, \lambda$ are pre-defined constants, and $\alpha,W$ are variables)

$$ \begin{align} &\max_{W}\; \pi_{0} (1 - e^{-\mu W}) - \frac{\pi_{1}}{W}z_{1}(W)\\ s.t.\;& \frac{dz_{0}(\alpha)}{d\alpha} = 1 - e^{-\mu \alpha},\quad z_{0}(0) = 0\\ & \frac{dz_{1}(\alpha)}{d\alpha} = 1 - e^{-\lambda \alpha},\quad z_{1}(0) = 0\\ & z_{0}(W) + \epsilon \leq \xi''\\ & W \geq 0 \end{align} $$ where $z_{0},z_{1}$ are auxiliary variables and $\epsilon$ is a numerical tolerance value. This then becomes a DAE (differential-algebraic equation) optimization problem, which can be solved numerically. (though given that your decision variable is also the independent variable in the differential equations, some further bilinear transformations may be required. See http://dx.doi.org/10.1016/j.na.2005.03.066. Some software packages do this automatically.).

Edit: I just realized, if indeed $\xi''$ is a constant as I have assumed, the inequality constraint can easily be converted into a bound. $$ \begin{align} \int_{0}^{W^U} 1-e^{-\mu \alpha}\,d\alpha = \xi'' - \epsilon\\ \frac{e^{-\mu W^{U}}}{\mu} + W^{U} - \frac{1}{\mu} = \xi'' - \epsilon\\ \end{align} $$ Solve for $W^{U}$, and replace the above inequality constraints with: $$ 0 \leq W \leq W^{U} $$ You may be able to solve this using optimal control methods.

  • $\begingroup$ Thenks Gilead, but I want to solve it analytically. $\endgroup$ – Venous007 Nov 13 '10 at 13:08
  • $\begingroup$ Well, like I said, if $\xi''$ is constant, you can apply optimal control methods (e.g. Pontryagin's Maximum Principle) after doing a bilinear transformation. OC methods are, in principle, analytical methods. However, bear in mind that analytical (closed-form) solutions do not always exist or are difficult to get. They usually involve the solution of some nasty BVP. $\endgroup$ – Gilead Nov 13 '10 at 21:50

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