ISSN:
1573-2878
Keywords:
Minimum norm solutions
;
duality
;
relations between primal and dual solutions
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
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.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF00952828
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