# How to formulate this optimization problem

Definition: r_{kn}(p_{kn})={\rm{log_2}}\left( 1+{{{p}_{kn}}h_{kn}} \right)

The Optimization Problem is:

# \underset{{{c}_{kn}},{{p}_{kn}},v}{\mathop{\max }}\,v

Subject to

\text{C1: }{{{c}}_{kn}}\in \left\{ 0,1 \right\},\hspace{3mm}\left( \forall k,n \right)

\text{C2: }{{p}_{kn}}\ge 0,\hspace{3mm}\left( \forall k,n \right)

\text{C3: }\sum\limits_{k=1}^{K}{\sum\limits_{n=1}^{N}{{{{c}}_{kn}}{{p}_{kn}}\le {{P}_{\rm{total}}}}}

\text{C4: }\varphi _{k}^{lower}v\le \sum\limits_{n=1}^{N}{{{c}_{kn}}{{r}_{kn}}({{p}_{kn}})}\le \varphi _{k}^{upper}v

This problem in intractable because of the integer constraints. If I can remove the intractability by some means, i.e., c_{kn} are known, then the problem becomes

# \underset{{{p}_{kn}},v}{\mathop{\max }}\,v

\text{C1: }{{p}_{kn}}\ge 0,\hspace{3mm}\left( \forall k,n \right)

\text{C2: }\sum\limits_{k=1}^{K}{\sum\limits_{n=1}^{N}{{{{c}}_{kn}}{{p}_{kn}}\le {{P}_{\rm{total}}}}}

\text{C3: }\varphi _{k}^{lower}v\le \sum\limits_{n=1}^{N}{{{c}_{kn}}{{r}_{kn}}({{p}_{kn}})}\le \varphi _{k}^{upper}v

Please note that p_{kn} and v are optimization variables in this problem. Please suggest me a solution

This is not convex (even if you relax the integer constraints.)
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