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I have the following constraint for the optimization problem in hand:

$$\frac{b_k}{\log_2 \left(1 + \frac{p_k \alpha_k}{\sum_{j \neq k} p_j \alpha_j + \sum_n \sum_l \Xi_{n,l,k} 2^{-2 v_{n,l}}} \right)} + 2 \frac{\sum_n \sum_l v_{n,l} }{C_F} \leq \epsilon$$

where the variables are $v_{n,l}$, $p_k$, $p_j, \; j \neq k$. The constants are $\alpha_k$, $\alpha_j$, $b_k$, $C_F$, $\Xi_{n,l,k}$.

The constraint is non-convex in $\{v_{n,l}, p_k, p_j, \; j \neq k\}$. I am looking solve the problem with sucessive convex approximation approach.

I need some suggestion on feasibilty of the problem in terms of it if its possible to have a convex approximation for the above constraint and or suggest some method to solve it.

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    $\begingroup$ Any particular reason for manually deriving convex approximations and then using an iterative approach, instead of simply using a general purpose nonlinear solver which effectively does this for you, but with added general tricks and knowledge to handle nonlinear programs. $\endgroup$
    – Johan Löfberg
    Mar 27, 2019 at 8:55
  • $\begingroup$ @JohanLöfberg My idea was to solve the problem with respect to both $v_{n,l}$ and $p_k$ and formulate and SCA problem. I am not sure if it can be solved using non-linear solvers. I dont have experience with nonlinear solver and that's why I am not sure feasibility of the problem. $\endgroup$
    – Chandan Pradhan
    Mar 27, 2019 at 10:32
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    $\begingroup$ Nonlinear solvers solve, well, nonlinear problems. Your problem is nonlinear. Most often when people start messing abour with homemade sequential convex approximations etc, they are simply reinventing the wheel, or bad approximations of a very standard wheel. Start with a standard nonlinear solver and see where it gets you. If that isn't good enough, try developing your own heuristics. $\endgroup$
    – Johan Löfberg
    Mar 27, 2019 at 16:15
  • $\begingroup$ Can you provide an example of messing the successive convex approximation algorithm. I see that most researcher in my field keep applying first order taylor approximation to every constraint they find. $\endgroup$
    – Tuong Nguyen Minh
    Feb 9, 2021 at 14:10

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