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unnecessary { and }s. Use of array instead of matrix environment and Sigma instead of sum. All corrected. |
m →Formulation: \\ newline {{math}} |
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* <math>w = Mz + q</math>
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 {{math|LCP(''M'', ''q'')}} have a solution for every ''q'', then ''M'' is a [[Q-matrix]]. If ''M'' is such that {{math|LCP(''M'', ''q'')}} have a 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 ''w'' 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 ''z'' is found. As such, the problem can also be formulated as:
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* <math>z^{\mathrm{T}}(Mz+q) = 0</math> (the complementarity condition)
==Convex quadratic-minimization: Minimum conditions==
Finding a solution to the linear complementarity problem is associated with minimizing the quadratic function
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This is because the [[Karush–Kuhn–Tucker]] conditions of the QP problem can be written as:
:<math>\begin{cases}
v = Q x - A^T {\lambda} + c \\ s = A x - b \\ x, {\lambda}, v, s \geqslant 0 \\ x^{T} v+ {\lambda}^T s = 0 \end{cases}</math> with ''v'' the Lagrange multipliers on the non-negativity constraints, ''λ'' the <!-- Lagrange -->multipliers on the inequality constraints, and ''s'' the slack variables for the inequality constraints. The fourth condition derives from the complementarity of each group of variables {{math|(''x'', ''s'')}} with its set of KKT vectors (optimal Lagrange multipliers) being {{math|(''v'', ''λ'')}}. In that case,
: <math>z = \begin{bmatrix} x \\ \lambda \end{bmatrix}, \qquad w = \begin{bmatrix} v \\ s \end{bmatrix}</math>
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