On minimum norm solutions

This note investigates the problem $$min x_p^p /p,s.t.Ax geqslant b,$$ where 1<p<∞. It is proved that the dual of this problem has the form $$max b^T y - A^T y_q^q /q,s.t.y geqslant 0,$$ whereq=p/(p?1). The main contribution is an explicit rule for retrieving a primal solution from a dual one. If an inequality is replaced by an equality, then the corresponding dual variable is not restricted to stay nonnegative. A similar modification exists for interval constraints. Partially regularized problems are also discussed. Finally, we extend an observation of Luenberger, showing that the dual of $$min x_p ,s.t.Ax geqslant b,$$ is $$max b^T y,s.t.y geqslant 0,A^T y_q leqslant 1,$$ and sharpening the relation between a primal solution and a dual solution.