As, we know that under-determined linear systems are having infinitely many solutions and we look for least norm solution via convex norm minimization constraint on the linear system. The underline norm minimization problem is having a unique solution because of convexity of the problem. So, based on convexity of the problem with respect to any convex norm other than $\ell_2-$ norm, is there always a closed form solution of the norm minimization problem. Whether this convexity is enough to explain the existence of closed form solution? Regarding, second part of my question, from computational point of view, a gradient based method is always helpful. Is there any situation, where closed form solution performs better than gradient based approach?
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$\begingroup$ What you are asking is not clear. Not all types of norms lead to a convex optimization. Typical norm 1, and 2, do. Others don't. Closed form solution is very fast on small data sets, but slow on large scale problems. $\endgroup$– Cagdas OzgencCommented Jul 23, 2020 at 6:15
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$\begingroup$ @CagdasOzgenc, Here, I am willing to take only convex norms (i.e. $p\geq1 $) . Also, I think closed form solution is possible with respect to 2-norm only. $\endgroup$– LakshmanCommented Jul 23, 2020 at 6:55
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