  Recently, Chakrabarty and Swamy (STOC 2019) introduced the {\em minimum-norm
load-balancing} problem on unrelated machines, wherein we are given a set $J$
of jobs that need to be scheduled on a set of $m$ unrelated machines, and a
monotone, symmetric norm; We seek an assignment $\sg:J\mapsto[m]$ that
minimizes the norm of the resulting load vector $\lvec_\sg\in\R_+^m$, where
$\lvec_\sg(i)$ is the load on machine $i$ under the assignment $\sg$. Besides
capturing all $\ell_p$ norms, symmetric norms also capture other norms of
interest including top-$\ell$ norms, and ordered norms. Chakrabarty and Swamy
(STOC 2019) give a $(38+\ve)$-approximation algorithm for this problem via a
general framework they develop for minimum-norm optimization that proceeds by
first carefully reducing this problem (in a series of steps) to a problem
called \minmax ordered load balancing, and then devising a so-called
deterministic oblivious LP-rounding algorithm for ordered load balancing.
  We give a direct, and simple $4$-approximation algorithm for the minimum-norm
load balancing based on rounding a (near-optimal) solution to a novel
convex-programming relaxation for the problem. Whereas the natural convex
program encoding minimum-norm load balancing problem has a large non-constant
integrality gap, we show that this issue can be remedied by including a key
constraint that bounds the "norm of the job-cost vector." Our techniques also
yield a (essentially) $4$-approximation for: (a) {\em multi-norm load
balancing}, wherein we are given multiple monotone symmetric norms, and we seek
an assignment respecting a given budget for each norm; (b) the best {\em
simultaneous approximation factor} achievable for all symmetric norms for a
given instance.
