# Linear Optimization with non-linear constraint

I have a linear optimization problem where the cost function is $$ay - bx$$. The variables are $$(x, y)$$. However, for one of the constraints, I have $$\log_2 \frac{cx}{c^2 x + c(1 - y)(1 - c)} \geq \delta$$. Note that the $$x$$ and $$y$$ are discrete here. The rest of the constraints are linear. I was wondering how can I find $$x$$ and $$y$$ that minimizes the cost function?

I have tried two approaches:

1. I have tried to expand the $$\log_2$$ with the Taylor series and approximate with only a linear component. But the error, in this case, is really high

2. In the second step, I tried to check if the cost function was convex or not. But as you can see the cost function is linear in $$x$$, and if I try to compute the second derivative (using central difference), it will result in 0.

So, what approach should I follow?

You can eliminate $$\log_2$$ by taking $$2$$ to the power of both sides. Then you can linearize the resulting constraint by clearing the denominator, assuming it is one-signed (positive?).
• So, $cx \geq \delta '$, wehre $\delta ' = \delta (c^2x + c(1 - c)(1 - y))$. and then simplify Oct 19, 2020 at 21:33
• No, you should have $\ge 2^\delta$ after the first step. Oct 19, 2020 at 21:35
• How about if I have $(ex + fy) \log_2 \frac{cx}{c^2x + c(1 - c)(1 - y)} \geq \delta$? Because in that case I will have $2^{\frac{\delta}{ex + fy}}$. So it's not linear in that case Oct 19, 2020 at 22:20