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I wrote up this inequality while formulating an optimization problem.

$$(y_{1}-1) \leq \left(K + \dfrac{y_{2}}{x_{2}+b \cdot y_{2}} \right) (x_{1}+b \cdot y_{1} ) \leq y_{1}$$

Some further conditions on the above variables include the following.

  • $y_{1} < y_{2}$

  • $K,~b,~y_{1},~y_{2},~x_{1},~x_{2} > 0$

  • $K$ and $b$ are constants.

  • $y_{1}$ and $y_{2}$ are constrained to be integers but I think this should not affect the procedure for checking convexity.

For the objective function, I was able to find the Hessian matrix and show that it was positive semi-definite over some region. However, for the above constraint, how do I go about showing whether it is convex (or not)? I think I can start by considering each side of the inequality separately, but what techniques can I use as a starting point?

Rodrigo de Azevedo
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V-Red
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  • The easiest way would be to consider both sides of the inequality separately and to compute the hessian. It might take a while, but should work. – Max Dec 14 '19 at 03:59
  • Oh, b.t.w. a good heuristic, before trying to prove it rigorously, would be to plot the function for a few hundred values, maybe fix all other variables, but two, and take a look whether it 'looks' convex. Over which variables are you optimizing? What is your domain? Because if $y_1, y_2$ are variables, you would have a MIP. – Max Dec 14 '19 at 04:06
  • Yes there are 4 variables. So is it correct to say that if the eigenvalues of the Hessian matrix are non-negative, then the function is convex? – V-Red Dec 14 '19 at 04:23
  • That is correct. Maybe check out the following link: https://web.stanford.edu/~boyd/cvxbook/ Here you can find great Lecture Slides and a great book, maybe the standard book for convex optimization. In the Slides there also functions that are 'known' to be convex as well as operations that preserve convexity. That can save a ton of time. – Max Dec 14 '19 at 04:39

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