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Finding a solution to the linear complementarity problem is equivalent to minimizing the quadratic function
: <math>f(\mathbf{z}) = \mathbf{z}^{\mathrm{T}}(\mathbf{Mz}+\mathbf{q})</math>
subject to the constraints
Because these constraints ensure that ''f'' is always non-negative, it attains its minimum of 0 at '''z''' if and only if '''z''' solves the linear complementarity problem.
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