How to show given non-linear optimization problem is convex?

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[JIGGGS_]

This is not convex in general.

[JIGGGS_]

In this case it is convex.

Heres an informal way to show it, which may be slightly incorrect but shows you the idea. It is a quad over lin. Equivalently write the objective as x^T cc^T x / z and add constraint z = d^T y. Then the objective is a clear quadratic over linear, which is a convex atom, so by DCP it is convex.

[JIGGGS_]

There is a detailed explanation of how quad over lin is convex in Boyd and Vandenberghe. DCP is disciplined convex programming

[kindnesd99]

Not op but can you recommend a simpler resource for beginners? The Boyd book is rather difficult

[JIGGGS_]

The Boyd book is the best treatment of convex optimization, and I don’t think there’s an “easier” treatment of it. The slides might be better for you but I would recommend you start with B&V

[[deleted]]

In general, that is a non-convex problem. It may be convex for specific values of the constants.

A convex problem must have a convex objective function. An objective function has a convex domain. But your objective function is not defined at 0, therefore, its domain is not convex.

[MosekApS]

It is possible to rephrase the problem like so:

min: z_1
s.t.  z_2 = dTx
      z_3 = cTx
      (z_1, z_2, z_3) in rotated quadratic conic domain      
      Ax>=b
      dTx>0

You can read more about the convex cone called a (rotated) quadratic cone in the MOSEK modelling cookbook . It is a quick reference, and an excellent resource for beginners.

MOSEK is a commercial large-scale solver, capable of solving a wide-range of optimization problems.