I think you're basically asking about the fundamental problem of linear algebra, from a geometric perspective, maybe with a few extra constraints. You might be interested in the following:

Brief discussion of linear solving in $\mathbb{R}^n$

If you are studying this question from a computational perspective, using Mathematica, you should also be interested in the inner workings of the RowReduce function.

Thanks,

Brad

Thanks a lot, it seems to be a good idea !

I don't think this question is well posed.

One problem is that I don't know what you mean by linear function. Are you looking for the intersection of two hyper planes? If yes, this should be a set of dimension N-1, not a simple point.

Let's assume instead f is an arbitrary function, it could have numerous local minima in the region of space where all coordinate values are less than zero. Which minimum should we then choose?

There is a similar problem, solved by Lagrange Multipliers. This optimization problem is well stated, so maybe you can use it as an example to reformulate your question. Until then I don't think we will see a satisfactory answer.

Here is the example again.. Let us consider a simple example z = Abs[4x + 6y - 18], where x, y >= 0. The minimum possible value for this function is 0, which can be achieved by a number of combinations of x and y. Among these possible sets (x, y), I want to find one for which x+y is minimum. In simple example as above, we can easily find that the solution to this problem is x = 0, y = 3. Any other set of x, y which satisfies above equation, x+y would always be higher than 3. Though I am not sure for a complicated problem having about 30 variables, how can I solve such problem using mathematica. thanks - SG

This seems to be a good idea, I will surely try. Though it would be great to think of a universal solution to such problems. Thanks a lot !!

I suggest you post a simple example of what you want to do and what you have done so far.