On minimum norm solutions

Abstract

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.