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Problem Setup Consider the following optimization problem \begin{equation*} \begin{aligned} & \underset{P_{a},\lambda_{a}}{\text{maximize}} & & P_{cov}(P_{a},\lambda_{a}) \\ & \text{subject to} & & 0 \leq P_a \leq \min\{P_{a,max},\frac{\rho}{\lambda_a}\} \\ & & & 0 \leq \lambda_a \leq \lambda_{a,max} \\ \end{aligned} \tag{1} \end{equation*} with $ P_{cov}(P_{a},\lambda_{a})= P_{cov, a}(P_{a},\lambda_{a}) \, \mathcal{A}_a(P_{a},\lambda_{a}) + P_{cov, {\rm g}}(P_{a},\lambda_{a}) \, \mathcal{A}_ {\rm g}(P_{a},\lambda_{a})$ where \begin{equation} P_{cov, {\rm g}}(P_{a},\lambda_{a})= \mathbb{P} \left( \frac{P_{{\rm g}} {R_{{\rm g},0}}^{-\eta_{\rm g}} h_{{\rm g},0}}{\sum_{i=1}^{\infty} P_{{\rm g}} {R_{{\rm g},i}}^{-\eta_{\rm g}} h_{{\rm g},i}+\sum_{i=0}^{\infty} P_{a} {R_{a,i}}^{-\eta_a} h_{a,i}} > \tau \right), \end{equation} \begin{equation} P_{cov, a}(P_{a},\lambda_{a})= \mathbb{P} \left( \frac{P_{a} {R_{a,0}}^{-\eta_{a}} h_{{a},0}}{\sum_{i=1}^{\infty} P_{{a}} {R_{{a},i}}^{-\eta_{a}} h_{{a},i}+\sum_{i=0}^{\infty} P_{{\rm g}} {R_{{\rm g},i}}^{-\eta_{\rm g}} h_{{\rm g},i}} > \tau \right), \end{equation} \begin{equation} \mathcal{A}_a =\mathbb{P} \left( P_{a} R_a^{-{\eta_a}} > P_{{\rm g}} {R_{\rm g}}^{-\eta_{\rm g}} \right), \quad \text{and} \quad \mathcal{A}_{\rm g} =\mathbb{P} \left( P_{{\rm g}} {R_{\rm g}}^{-\eta_{\rm g}} > P_{a} R_a^{-{\eta_a}} \right). \end{equation} where $R_{a,i}, R_{\rm g,i}, h_{a,i}, h_{\rm g,i}$ are all random variables and $P_a, P_{\rm g}, \eta_a, \eta_{\rm g}, \rho$ are all constants. $\lambda_a$ is a parameter that characterizes the pdf of $R_a$.

Questions

  • Is $P_{cov}$ is an increasing function of $P_a$?

    • If yes, how to prove it without computing the probabilities?

    • If no, how can approach such an optimization problem? My initial approach was to utilize the note below, then perform a numerical search over a compact subset of candidate $\lambda_a$ values on the $P_a \lambda_a = \rho$ curve to find the optimum $\lambda_a$.

Note

If $P_{cov}$ is an increasing function of $P_a$ we can, without loss of optimality, reduce the problem to the following one-dimensional constrained optimization problem: \begin{equation*} \begin{aligned} & \underset{\lambda_{a}}{\text{maximize}} & & P_{cov}(\min\{P_{a,max},\frac{\rho}{\lambda_a}\},\lambda_{a}) \\ & \text{subject to} & & 0 \leq \lambda_a \leq \lambda_{a,max} \\ \end{aligned} \tag{2} \end{equation*} since any $P^{\star}$ solving $(1)$ must achieve the power constraint with equality.

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    $\begingroup$ There is quite a number of variables here. Would you mind clarify the definitions of those variables, and the relationship between them, if any? $\endgroup$ – BGM Oct 20 '18 at 3:04
  • $\begingroup$ @BGM I did. Kindly see the updated post. $\endgroup$ – Lod Oct 20 '18 at 22:39
  • $\begingroup$ have you tried coupling? it's a general method allowing you to show certain probabilities are increasing (as a function of a certain parameter) $\endgroup$ – mathworker21 Oct 23 '18 at 9:36
  • $\begingroup$ @mathworker21 No I haven't, I'll look into that. $\endgroup$ – Lod Oct 23 '18 at 12:38
  • $\begingroup$ @AlexFrancisco Yes non-negative. $R_{a,i},R_{g,i},h_{a,i},h_{g,i}$ are random variables with pdfs having a support $[0,\infty)$. $\endgroup$ – Lod Oct 23 '18 at 22:52

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