Standard names and methods for this type of fitting minimization In material science research, we have come across the following type of problem. 
Given a m by n matrix A, a m vector b, and error tolerance $\varepsilon$, we want to do this minimization
$$\eqalign{
  & \min \text{ # nonzero components of } x  \cr 
  & \text{s.t.}  \cr 
  & {\left\| {Ax - b} \right\|_\infty } \le \varepsilon  \cr} $$
Is there some standard name or procedure in mathematics to do this? Could it be transformed into a LP or Convex Programming problem?
Thank you:)
 A: I am not sure if there is a standard name for these type of problems. I would call problems of this type sparse approximation problems because you want to solve a linear equation approximately (assuming that $\epsilon$ is small) and sparsely. In statistics these problems (where one really aims to minimize the number of nonzero entries in the solution) are called subset selection problems, since you want to select a subset of the columns of $A$ that should be used to represent $b$ (approximately).
In general, the problem you gave is not equivalent to a linear program and in fact is NP hard. Also it is not convex and may posses a huge number of different minima. However, the links already given in the comments to sparse approximation or the Dantzig selector (other buzzwords are Basis Pursuit are Compressed sensing) show that the problem admits a convex relaxation namely, you interpret the objective function $\# \text{nonzero entries in $x$}$ as $\sum |x_k|^0$ (with the convention that $0^0=0$) and then replace the exponent $0$ by the smallest one that makes the objective convex, and this is $1$, i.e. you end up with $\sum |x_k|$. The resulting problem (with the constraint in the $\infty$-norm) can then be casted as a linear program. Remarkably, this relaxation is frequently found to be exact, i.e. the optimum of the relaxed problem is (one of) the optima of the original problem. This can be explained by some theory, but happens much more often than theory predicts (and also, in most practical cases I have seen, the theory does not apply but the predicted results hold (approximately) nonetheless).
