Linear Optimization Problem with exponential variable

Question

Hey Folks

I've encountered an optimization problem which has a linear programming structure but it's coefficients are nonlinear function of another variable. here is the problem: $$\max _{{p_i},\theta} \sum_{i=1}^N (\omega_i p_i e^{-1.2\times (\frac{\theta-\theta_i}{\theta_{3dB}})^2}-\eta\times p_i) $$

$$s.t. \sum_{i=1}^N p_i = P$$ $$\forall i \qquad p_i\ge0 \qquad and \qquad 0\le\theta\le 90^\circ$$ $$\forall i \qquad \omega_i\ge0 \qquad are \; known \; real \; positive \; numbers$$ $$\forall i \qquad 0\le\theta_i\le 90^\circ \qquad are \; known \; real \; positive \; numbers \; (degrees)$$ $$0\le\eta\le 1 \qquad and \qquad \theta_{3dB}=6^\circ \qquad and \qquad P\ge0 \qquad are \; known\; constants$$ $$e \qquad is \; neper \; exponential \; function$$

I'm interested in finding global maximum of the above function.

I think closed form solution is not possible however an algorithm that guarantees to approach global maximum is highly probable to exist.

Thanks a lot in Advance and Best Regards

Answer 1

I assume that the neper exponential function is increasing in its argument. Take the $j\in {1,\ldots,N}$ of the largest $\omega_j$. Set $\theta_j=\theta$ and $p_j=P$ and $p_i=0$ for $i$ not equal to $j$. $\theta_i$ for $i$ other than $j$ can be anything satisfying the constraint.