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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.

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    So $\alpha$ and $q$ are both scalars? What does $(\alpha,q)$ mean in the objective function? – 2017-01-15
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    no they are not scalars, we a trying to fine the values of $\alpha$ and q that would optimize the system – 2017-01-15
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    Can you state explicitly what $\alpha$ and $q$ are? Are they vectors in $\mathbb R^n$? If so, what is the meaning of $\alpha \log_2(1 + q/\sigma^2)$? In the objective function, is $(\alpha,q)$ the inner product of $\alpha$ and $q$? – 2017-01-15
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    sorry I wasn't able to understand your question before. yes they are both scalar, positive real numbers. – 2017-01-15
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    Ok, thanks, and can you clarify what function you're trying to minimize? Since you only have two scalar unknowns, you could always just use a brute force search. – 2017-01-15
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    I am trying to minimize the product $\alpha$q , yes that's right I can brute force. But in this case its kind of compulsory for me to use a convex optimization technique. – 2017-01-15
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    I have edited the model for formatting, and to reflect your claim that your objective function is the product. If I've made an error, please let me know. – 2017-01-15
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    But no, your model is *not* convex. What do you mean by "compulsory" here? Surely someone is not expecting you to solve a non-convex problem using convex optimization. And this is a two-variable problem that can quite likely be solved analytically, or nearly so. – 2017-01-15
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    I'm going to change the $\varpropto$'s to $\alpha$'s, hope nobody objects. – 2017-01-15
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    I am supposed to solve this problem by changing it into a convex problem first( if possible). If the transformation is not possible then I am supposed to use some other techniques for nonconvex optimization. I wanted to know that is there a way to transform this into an equivalent convex problem. – 2017-01-15
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    Thank you for the correction. – 2017-01-15
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    An excellent reference is [this book](http://stanford.edu/~boyd/cvxbook/) by Boyd and Vandenberghe. – 2017-01-15
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    Boyd and Vandenberghe is not going to help you determine how to transform this problem to a convex form. In fact, such a transformation is _rarely_ possible; and if it is, only through a change of variables... which I see is exactly what @LinAlg has done – 2017-01-16
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    @MichaelGrant The book at least equips you with in-depth knowledge on convexity. There are in fact many more tricks. I once had a seminar from Ben-Tal at MIT where he showed many of those. If you search for 'hidden convexity' you can find a few of them. – 2017-01-16
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    Great pointer, @LinAlg, thank you, I will check itout. – 2017-01-16

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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: $$\begin{align*} \min \quad &x+y \\ \text{s.t.} \quad & x \leq e \\ & 1+\frac{\log(y)}{\sigma^2} \geq \exp(5)^{1/\log(x)} \\ & x,y \geq 1 \end{align*}$$

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    Feel free to replace $\exp$ with $2^{(\cdot)}$ and $\log$ with $\log_2$ throughout. – 2017-01-15