Linear complementarity problem

In mathematical optimization theory, the linear complementarity problem (LCP) arises frequently in computational mechanics and encompasses the well-known quadratic programming as a special case. It was proposed by Cottle and Dantzig in 1968.

Formulation

Given a real matrix M and vector q, the linear complementarity problem LCP(M,q) seeks vectors z and w which satisfy the following constraints:

• ${w} \ge 0, {z} \ge 0\,$ (that is, each component of these two vectors is non-negative)
• $z^Tw = 0$ or equivalently $\Sigma_i{w}_i {z}_i = 0\,$. This is the complementarity condition, since it implies that at most one of each pair $\{w_i,z_i\}$ can be positive.
• ${w} = {Mz} + {q} \,$

A sufficient condition for existence and uniqueness of a solution to this problem is that M be symmetric positive-definite. If M is such that LCP(M,q) have a solution for every q, then M is a Q-matrix. If M is such that LCP(M,q) have a unique solution for every q, then M is a P-matrix. Both of these characterizations are sufficient and necessary.

The vector ${w}\,$ is a slack variable, and so is generally discarded after ${z}\,$ is found. As such, the problem can also be formulated as:

• ${Mz}+{q} \ge {0}\,$
• ${z} \ge {0}\,$
• ${z}^{\mathrm{T}}({Mz}+{q}) = 0\,$ (the complementarity condition)

Convex quadratic-minimization: Minimum conditions

Finding a solution to the linear complementarity problem is associated with minimizing the quadratic function

$f({z}) = {z}^{\mathrm{T}}({Mz}+{q})\,$

subject to the constraints

${Mz}+{q} \ge {0}\,$

${z} \ge {0}\,$

These constraints ensure that f is always non-negative. The minimum of f is 0 at z if and only if z solves the linear complementarity problem.

If M is positive definite, any algorithm for solving (strictly) convex QPs can solve the LCP. Specially designed basis-exchange pivoting algorithms, such as Lemke's algorithm and a variant of the simplex algorithm of Dantzig have been used for decades. Besides having polynomial time complexity, interior-point methods are also effective in practice.

Also, a quadratic-programming problem stated as minimize $f({x})={c}^T{x}+\frac{1}{2}{x}^T{Qx}\,$ subject to ${Ax} \geq {b} \,$ as well as ${x} \ge {0}\,$ with Q symmetric

is the same as solving the LCP with

• ${q} = \left[\begin{array}{c}{c}\\-{b}\end{array}\right]\,$
• ${M} = \left[\begin{array}{cc} {Q} & -{A}^{T}\\ {A} & 0\end{array}\right]\,$

This is because the Karush–Kuhn–Tucker conditions of the QP problem can be written as:

• ${v} = {Q} {x} - {A}^{T} {\lambda} + {c}\,$

• ${s} = {A} {x} - {b}\,$

• ${x}, {\lambda}, {v}, {s} \ge {0}\,$

• ${x}^{T}{v} + {\lambda}^{T}{s} = {0}\,$

...being ${v} \,$ the Lagrange multipliers on the non-negativity constraints,${\lambda} \,$ the 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 (${x}, {s}\,$) with its set of KKT vectors (optimal Lagrange multipliers) being (${v}, {\lambda}\,$).

In that case,

${z} = \left[\begin{array}{c}{x}\\ {\lambda}\end{array}\right]\,$

${w} = \left[\begin{array}{c}{v}\\ {s}\end{array}\right]\,$

If the non-negativity constraint on the ${x}\,$ is relaxed, the dimensionality of the LCP problem can be reduced to the number of the inequalities, as long as ${Q}\,$ is non-singular (which is guaranteed if it is positive definite). The multipliers ${v}\,$ are no longer present, and the first KKT conditions can be rewritten as:

• ${Q} {x} = {A}^{T} {\lambda} - {c}\,$

or:

${x} = {Q}^{-1}({A}^{T} {\lambda} - {c})\,$

pre-multiplying the two sides by ${A}\,$ and subtracting ${b}\,$ we obtain:

${A} {x} - {b} = {A} {Q}^{-1}({A}^{T} {\lambda} - {c}) -{b} \,$

The left side, due to the second KKT condition, is ${s}\,$. Substituting and reordering:

${s} = ({A} {Q}^{-1} {A}^{T}) {\lambda} + (- {A} {Q}^{-1} {c} - {b} )\,$

Calling now ${M} \,\overset{\underset{\mathrm{def}}{}}{=}\, ({A} {Q}^{-1} {A}^{T})\,$ and ${q} \,\overset{\underset{\mathrm{def}}{}}{=}\, (- {A} {Q}^{-1} {c} - {b})\,$ we have an LCP, due to the relation of complementarity between the slack variables ${s}\,$ and their Lagrange multipliers ${\lambda}\,$. Once we solve it, we may obtain the value of ${x}\,$ from ${\lambda}\,$ through the first KKT condition.

Finally, it is also possible to handle additional equality constraints:

${A}_{eq}{x} = {b}_{eq} \,$

This introduces a vector of Lagrange multipliers ${\mu}\,$, with the same dimension as ${b}_{eq}\,$.

It is easy to verify that the ${M}\,$ and ${q}\,$ for the LCP system ${s} = {M} {\lambda} + {q} \,$ are now expressed as:

${M} ~\overset{\underset{\mathrm{def}}{}}{=}~ \left(\left[\begin{array}{cc}{A} & {0}\end{array}\right] \left[\begin{array}{cc} {Q} & {A}_{eq}^{T}\\ -{A}_{eq} & {0}\end{array}\right]^{-1} \left[\begin{array}{cc}{A}^{T} \\ {0}\end{array}\right]\right)\,$

${q} ~\overset{\underset{\mathrm{def}}{}}{=}~ \left(- \left[\begin{array}{cc}{A} & {0}\end{array}\right] \left[\begin{array}{cc} {Q} & {A}_{eq}^{T}\\ -{A}_{eq} & {0}\end{array}\right]^{-1} \left[\begin{array}{c}{c}\\ {b}_{eq}\end{array}\right] - {b}\right)\,$

From ${\lambda}\,$ we can now recover the values of both ${x}\,$ and the Lagrange multiplier of equalities ${\mu}\,$:

$\left[\begin{array}{c}{x}\\ {\mu}\end{array}\right] = \left[\begin{array}{cc} {Q} & {A}_{eq}^{T}\\ -{A}_{eq} & {0}\end{array}\right]^{-1} \left[\begin{array}{c} {A}^{T}{\lambda}-{c}\\-{b}_{eq}\end{array}\right] \,$

In fact, most QP solvers work on the LCP formulation, including the interior point method, principal / complementarity pivoting, and active set methods. LCP problems can be solved also by the criss-cross algorithm, conversely, for linear complementarity problems, the criss-cross algorithm terminates finitely only if the matrix is a sufficient matrix. A sufficient matrix is a generalization both of a positive-definite matrix and of a P-matrix, whose principal minors are each positive. Such LCPs can be solved when they are formulated abstractly using oriented-matroid theory.

See also

• Complementarity theory
• Physics engine Impulse/constraint type physics engines for games use this approach.
• Contact dynamics Contact dynamics with the nonsmooth approach.
• Bimatrix games can be reduced to a LCP.

Further reading

• R. W. Cottle and G. B. Dantzig. Complementary pivot theory of mathematical programming. Linear Algebra and its Applications, 1:103-125, 1968.

External links

• LCPSolve — A simple procedure in GAUSS to solve a linear complementarity problem
• LCPSolve.py — A Python/NumPy implementation of LCPSolve is part of OpenOpt since its release 0.32
• Siconos/Numerics open-source GPL implementation in C of Lemke's algorithm and other methods to solve LCPs and MLCPs

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