What is the interpretation of the following optimization problem?

Question

Suppose we have $N$ variables $x_1,\ldots,x_N$. Let $\mathbf{A}$ a $M \times N$ matrix, and $\mathbf{b}$ a $M \times 1$ vector.
I have the following minimization problem: \begin{array}{rl} \min \limits_{\mathbf{x}} & 1 \\ \mbox{s.t.} & \mathbf{Ax}=\mathbf{b}\\ & \mathbf{x}^T \mathbf{1}=1 \\ & x_n >0, \forall n \end{array} What is the interpretation of this kind of optimization problems? can we replace $1$ by another constant?

Answer 1

You're trying to find if there's a convex combination of the columns of $A$ that will yield $b$.

Any feasible solution to your problem produces such a combination.

Answer 2

It's a feasibility problem which has been formulated in a rather odd way. It's simply asking to a point $x \in \mathbb{R}^N$ s.t $Ax = b$, $\sum_{k}x_k = 1$ and $x_k > 0$ $\forall k$.