Convex Optimization of two variables

I am encountering a problem with optimizing two variables. It seems like the function fmincon is only minimizing one variable instead of two variables.

How is it possible to optimize the two variables (beta(1) and beta(2))?

Code and Latex problem overview can be found in the appendix.

Thanks in advance

[Matt J]

Convex Optimization of two variables

The problem doesn't look convex to me. The choice of the initial guess will be quite important.

It seems like the function fmincon is only minimizing one variable instead of two variables.

In what way does it seem like that to you?

[Dirk Cremers]

Thanks for the fast response.

The problem doesn't look convex to me.

Given the Hessian matrix I proved with the diagonal dominant theorem that both functions g and h are convex. This way we are able to construct a difference of convex optimization problem and apply a subproblem which should be convex (CCP).

However, in the meantime I encountered the problem (I think atleast) which is the following:

I have two functions which should give the same output if you look at it mathematically since they are the same.

However in matlab I don't recieve the same output for both function. I get the intuation that this is because of the size of the values of . I have attached the matlabfile which demonstrates this in the appendix. From splitting the function and looking at the values I wont understand why a certain output is given.

As can be seen, the values of part2 and part5 are the same and (part1 + part3) is greater than part4. However the outcome given by matlab is equal to 0. How could I solve this problem?