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A sufficient condition for existence and uniqueness of a solution to this problem is that '''M''' be [[Symmetric matrix|symmetric]] [[Positive-definite matrix|positive-definite]].
If '''M''' is such that the problem have a solution for every '''q''', then '''M''' is a [[Q-matrix]]. If '''M''' is such that the problem have an unique solution for every '''q''',
then '''M''' is a [[P-matrix]]. Both of these characterizations are sufficient and necessary.<ref>{{cite journal|last1=Murty|first1=Katta G.|title=On the number of solutions to the complementarity problem and spanning properties of complementary cones|journal=Linear Algebra and its Applications|date=January 1972|volume=5|issue=1|pages=65–108|doi=10.1016/0024-3795(72)90019-5}}</ref>
The vector <math>{w}\,</math> is a [[slack variable]],<ref>{{citation|title=Convex Optimization of Power Systems|first=Joshua Adam|last=Taylor|publisher=Cambridge University Press|year=2015|isbn=9781107076877|page=172|url=https://books.google.com/books?id=JBdoBgAAQBAJ&pg=PA172}}.</ref> and so is generally discarded after <math>{z}\,</math> is found. As such, the problem can also be formulated as:
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