The original problem
p_1 = b_1 \exp(a_1r_2)[\exp(a_1r_1) -1]
p_2 = b_2 [\exp(a_1r_2) -1]
our objective is
\min_{p_1,p_2,f_1,f_2} p_1 \frac{d_1}{r_1} + p_2 \frac{d_2}{r_2}
subject to
C1: \frac{d_1}{r_1} + \frac{d_1c_1}{f_1} \leq \tau
C_2: \frac{d_2}{r_2} + \frac{d_2c_2}{f_2} \leq \tau
0< p_1, p_2 \leq P
f_1 + f_2 \leq F
f_1 > 0 ,f_2 >0
all constants a_1,a_2,b_1,b_2,c_1,c_2,d_1,d_2 are positive and b1 \leq b2.
The original problem is non-convex and has a coupling variable and a coupling constraint. However, the constraints C1 and C2 holds with equality. Then, the problem can be transformed to the following problem as a function of f_1 and f_2 only.
r_1 = \frac{d_1f_1}{\tau f_1 - d_1c_1}
r_2 = \frac{d_2f_2}{\tau f_2 - d_2c_2}
our objective is
\min_{f_1,f_2} p_1 \frac{d_1}{r_1} + p_2 \frac{d_2}{r_2}
subject to
0< p_1, p_2 \leq P
f_1 + f_2 \leq F
f_1 > 0 ,f_2 >0
The proof for the convexity of this problem can be found in detail herehttps://ieeexplore.ieee.org/ielx7/4267003/4357939/9393794/supp1-3064919.pdf?arnumber=9393794 . Since the problem is not directly convex I need to use the dual decomposition and subgradient method to solve the dual problem. In my CVX program, I am trying to write the term \frac{1}{r_2} \exp(ar_2) where r_2 = \frac{d_2f_2}{\tau f_2 - d_2c_2} in a correct form that can be accepted by CVX. I am doing some related work to this work so I am trying to do their programming. In my new formulation, I have one more additional term equal to f_2^2 so I am not sure about the transformation of the fractional term. Thanks for your help in advance. I appreciate.
cvx_begin
variables F2 Z2
minimize ((d2.*b2.*Z2 - ((b2.*tau.*F2 - b2.*d2.*c2)./F2) + lambda.*F2)
F2 > 0
{} == expontential{1}
cvx_end
g1 = cvx_optval;