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Can someone please help me to linearize the following nonlinear/nonconvex constraint:

$\sum\limits_{n=1}^Na_n\rm{log_2}(1+x_ny_n)\le M\delta$

Here $a_n \in\{0,1\}$, binary integer variable

$0\le x_n \le 3$, continuous variable (bounded)

$\delta$, a continuous optimization varibale

$y_n$ and $M$ are known parameters

Dimitrios
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1 Answers1

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Here might be an answer. We first define a variable $x_0$ via $\delta=-\frac{1}{M}\log_2(1+x_0)$. Set $y_0=1,a_0=1$, we can then rewrite the constraint as:

$$\sum\limits_{n=1}^N\frac{1}{M}a_n\rm{log_2}(1+x_ny_n)\le -\frac{1}{M}a_0\log_2(1+x_0 y_0)\tag{2}$$

We may then absorb the RHS of (2) into the LHS of (2) as:

$$\frac{1}{M}\sum\limits_{n=0}^Na_n\rm{log_2}(1+x_ny_n)\le0\tag{3}$$

mike
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  • thank you very much for your answers. what about the value for $x_0$. – Dimitrios Nov 05 '14 at 04:35
  • thank you very much for your answers. what about the value for 'x_0'. I think 'x_0=z'? am I right? Also, how come the 'b_n' are known parameters? Also 'b_n' are no more binary. Does this transformation makes (3) a linear constraint? – Dimitrios Nov 05 '14 at 04:56
  • @Dimitrios. Please see the update. $x_0,x_1,...,x_N$ are now the unknowns. The optimization parameter is converted to the unknown $x_0$. (3) is still not a linear constraint. But all the unknowns appeared in a similar fashion, unlike in the original problem. – mike Nov 05 '14 at 05:16