Can this be changed to a convex problem?

Question

If I have a problem that is not convex \begin{array}{lll} \textbf{P1:} & \min_{\alpha,q} & \alpha q \\ &\text{s.t} \quad &\alpha \leq 1 \\ &&\alpha \log_{2}\bigg(1+\dfrac{q}{\sigma^{2}}\bigg)\geq 5\\ &&\alpha \geq 0 \\ &&q \geq 0 \end{array} I think its noncovex because of the second constraint. I want to know how can I change such problems into convex. Any reference of book, slides, lectures, articles or name of a technique would be helpful. Also kindly name some techniques for nonconvex optimization like ant colony optimization etc. I would prefer if the reference is not of a book as I cannot buy books online and their is a fair chance that the hard copy of book will not be available in my county.

Answer 1

Please forget about ant colony optimization. Its a heuristic with unpredictable performance.

Making problems convex is an ad-hoc procedure that requires practice.

The objective is the first hurdle here. It is indefinite. A logarithmic transformation gets rid of the product, but results in a concave minimization problem. We can make the objective linear again with the substitution $x=\exp(\alpha)$ and $y=\exp(q)$. Let's see what happens to the constraint: $$\log(x) \log(1+\log(y)/\sigma^2) \geq 5$$ $$\log((1+\log(y)/\sigma^2)^{\log(x)}) \geq 5$$ $$(1+\log(y)/\sigma^2)^{\log(x)} \geq \exp(5)$$ $$1+\log(y)/\sigma^2 \geq \exp(5)^{1/\log(x)}$$ Here I used that $\log(x)\geq 0$. The final constraint is convex as $x \geq 1$. Finding an initial feasible point is slightly harder since the constraints are not convex outside the feasible region, but for this two variable problem that is not a real issue. The final problem becomes: $$\begin{align*} \min \quad &x+y \\ \text{s.t.} \quad & x \leq e \\ & 1+\frac{\log(y)}{\sigma^2} \geq \exp(5)^{1/\log(x)} \\ & x,y \geq 1 \end{align*}$$