  We consider the following variational problem: minimize the $(n+1)$st
polynomial conserved quantity of KdV over $H^n(\mathbb{R})$ with the first $n$
conserved quantities constrained. Maddocks and Sachs used that $n$-solitons are
local minimizers for this problem in order to prove that $n$-solitons are
orbitally stable in $H^n(\mathbb{R})$.
  Given $n$ constraints that are attainable by an $n$-soliton, we show that
there is a unique set of $n$ amplitude parameters so that the corresponding
multisolitons satisfy the constraints. Moreover, we prove that these
multisolitons are the unique global constrained minimizers. We then use this
variational characterization to provide a new proof of the orbital stability
result of Maddocks and Sachs via concentration compactness.
  In the case when the constraints can be attained by functions in
$H^n(\mathbb{R})$ but not by an $n$-soliton, we discover new behavior for
minimizing sequences.
