Minimization problem with multiple constraints

I am trying to minimize the following objective function obj(x) = 1 - x*T where x is a 1 dimensional variable and T is a matrix. I have two constraints :

1) x should be positive : x>0

2) The objective function should be positive : 1-x*T>0 so x should be smaller than the minimum of all the matrix inverse values x < min(1/T).

I am not really familiar with Matlab optimization tools and I don't know which Matlab optimization function is adequate for this problem. Would you please give me some recommendations and advice.

Thank you!

[Matt J]

I am trying to minimize the following objective function obj(x) = 1 - x*T where x is a 1 dimensional variable and T is a matrix.

This implies that obj(x) is non-scalar. What does it mean to "minimize" a non-scalar valued function.

so x should be smaller than the minimum of all the matrix inverse values x < min(1/T).

That follows if all the T are positive. Is that the case?

[chayma chaabani]

I am not sure if I defined the problem in a good way. In fact, all the T values are positive and the objective function returns a matrix. When minimizing this function, I want to know the scalar value of x that gives the minimum return matrix of the objective function.

[Torsten]

How do you define "the minimum return matrix" ?

Best wishes

Torsten.

[chayma chaabani]

In my case, all the values of the return matrix should be minimal.

[Matt J]

But multiple functions of x cannot necessarily reach there minima simultaneously at the same x. What is it about your "case" that makes this possible?

Answer 1

[Matt J]

See my comments, butI suspect the answer you are looking for is x=min(1/T). This simultaneously minimizes all entries of the matrix-valued function 1-T*x

[chayma chaabani]

Unfortunately x=min(1/T) can't be a solution because in this case I will have zero values in the final minimum matrix of the objective obj(x) = 1 - x*T. The objective function should be strictly positive.

[Matt J]

The objective function should be strictly positive.

Then you have more work to do on the problem formulation, as there is currently nothing in what you've posed that ensures this.

Suppose T=ones(N). Then clearly the solution has to be x=1 and obj(x)=zeros(N). If this is not the solution for such a selection of T, then what is?