0
$\begingroup$

I'm working on convex optimisation and trying to find ways to tell that a problem is convex without making use of visualisation (relying solely on calculations). I understand that for a problem to be convex, the objective function and inequality constraints must be convex and the equality constraints must be affine.

If one of my constraints is $\frac{x}{1-y}$, I can visually see that this is not a convex function. But how can I prove it mathematically?

Thanks in advance!

  • 0
    Please use LaTeX markup to improve readability. – 2017-01-02
  • 1
    If you can visualize it, then just find two points $(x,y)$, $(w,z)$ such that $$\frac{(x+w)/2}{1-(y+z)/2} > \frac{1}{2}\frac{x}{1-y} + \frac{1}{2}\frac{w}{1-z}$$ The Hessian method @πr8 mentions is also good, perhaps easier in this case, but I do not think your function is convex over the domain conjectured there. – 2017-01-02
  • 0
    Where do $x$ and $y$ belong to? – 2017-01-02
  • 0
    @Michael yep thought it through and while it's convex in each argument, it is not convex itself; had overlooked this. – 2017-01-02

1 Answers 1

1

You know that a function $f$ (continuous and twice differentiable) is convex if and only if its Hessian is positive semi-definite.

Now define $f(x,y):=\dfrac{x}{1-y}$. Calculate the Hessian of $f(x,y)$, $\mathbf{H}$.

$$\mathbf{H} = \begin{bmatrix} \dfrac{\partial^2 f}{\partial x^2} & \dfrac{\partial^2 f}{\partial x\,\partial y} \\[2.2ex] \dfrac{\partial^2 f}{\partial y\,\partial x} & \dfrac{\partial^2 f}{\partial y^2} \end{bmatrix} $$

You will get

$$\mathbf{H} = \begin{bmatrix} 0 & \dfrac{1}{(1-y)^2} \\[2.2ex] \dfrac{1}{(1-y)^2} & \dfrac{2x}{ (1-y)^3} \end{bmatrix} $$

Now find out if this positive semi-definite. It is so if and only if $z^\top\mathbf{H}z\geqslant 0$ for any colum vector $z$.

  • 0
    "if and only if" is not correct. After all, $f$ could be nondifferentiable and still convex. – 2017-01-03
  • 1
    For a function $f$ that is continuous and twice differentiable, we have if and only if. – 2017-01-03
  • 1
    I agree with your edit. – 2017-01-03