I am attempting to optimize (maximise) a sumrate objective function for a communications engineering problem involving multiple users sharing a channel.
I have objective function: $$ F(x_1,x_2,x_3) = R_1 + R_2 + R_3 $$
Where: $$ R_1 = Blog_2\Big(1+ \frac{x_1h_1}{N+x_2h_2+x_3h_3}\Big)$$
$$ R_2 = Blog_2\Big(1+ \frac{x_2h_2}{N+x_3h_3}\Big)$$
$$ R_3 = Blog_2\Big(1+ \frac{x_3h_3}{N}\Big)$$
The constraints are:
$$ 0 \lt x_1 \leq P_1 $$ $$ 0 \lt x_2 \leq P_2 $$ $$ 0 \lt x_3 \leq P_3 $$
$$\frac{R_1}{r_a}:\frac{R_2}{r_b}$$ $$\frac{R_1}{r_a}:\frac{R_3}{r_c}$$
Where $p$:$q$ denotes $p$ is proportional to $q$
The following values are fixed:
- $N$ - Noise
- $B$ - Bandwidth
- $h_1, h_2, h_3$ - Channel gain values for for users
- $r_a, r_b, r_c$ - Minimum rates per user
By applying Lagrange multipliers and the constraints to the objective function we get:
$$L(x_1,x_2,x_3) = R_1 + R_2 + R_3 - \lambda_1(x_1 - P_1) - \lambda_2(x_2 - P_2) - \lambda_3(x_3 - P_3) - \tau_1(\frac{R_1}{r_a}-\frac{R_2}{r_b}) - \tau_2(\frac{R_1}{r_a}-\frac{R_3}{r_c})$$
Once the partial derivatives are obtained w.r.t ${x_1,x_2,x_3}$ the results are cubic equations containing several fixed unknowns. I am stuck finding the roots at this point as I need to find the general case rather than supplying values for the fixed unknowns at this stage.
I have spent several weeks looking at lectures, lecture notes and worked examples for optimization techniques that I think would be appropriate, but all problems that I have found seem trivial in comparison.
I ask:
- Is there an efficient method to find roots for cubic equations with multiple fixed unknowns?
- Are there any resources or lecture notes that deal with non-trivial Lagrange optimization problems that you would recommend that I read
- Are there any alternate optimization approaches that would be more efficient and better suited to this problem?