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Sparse optimization seeks solutions with few nonzero coordinates, either as an explicit objective or through a penalty. Minimizing the number of nonzero entries, written $\Vert x{\Vert }_0$, is combinatorial; replacing it by the L1 norm often gives a tractable convex optimization problem.

For example, basis pursuit minimizes $\Vert x{\Vert }_1$ subject to $Ax=b$, while Lasso minimizes a squared residual plus $\lambda \Vert x{\Vert }_1$. The penalty favors exact zero coordinates because it has a corner at zero. Recovery of the sparsest explanation needs assumptions on $A$; an L1 norm objective alone does not establish it. These methods connect variable selection with compressed sensing.