# 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.[1][2][3]

## 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:

- (that is, each component of these two vectors is non-negative)
- or equivalently This is the complementarity condition, since it implies that, for all , at most one of and can be positive.

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.[4]

The vector *w* is a slack variable,[5] and so is generally discarded after *z* is found. As such, the problem can also be formulated as:

- (the complementarity condition)

## Convex quadratic-minimization: Minimum conditions

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

subject to the constraints

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 subject to as well as with *Q* symmetric

is the same as solving the LCP with

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

with *v* the Lagrange multipliers on the non-negativity constraints, *λ* 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*, *λ*). In that case,

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:

or:

pre-multiplying the two sides by *A* and subtracting *b* we obtain:

The left side, due to the second KKT condition, is *s*. Substituting and reordering:

Calling now

we have an LCP, due to the relation of complementarity between the slack variables *s* and their Lagrange multipliers *λ*. Once we solve it, we may obtain the value of *x* from *λ* through the first KKT condition.

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

This introduces a vector of Lagrange multipliers *μ*, with the same dimension as .

It is easy to verify that the *M* and *Q* for the LCP system are now expressed as:

From *λ* we can now recover the values of both *x* and the Lagrange multiplier of equalities *μ*:

In fact, most QP solvers work on the LCP formulation, including the interior point method, principal / complementarity pivoting, and active set methods.[1][2] LCP problems can be solved also by the criss-cross algorithm,[6][7][8][9] conversely, for linear complementarity problems, the criss-cross algorithm terminates finitely only if the matrix is a sufficient matrix.[8][9] A sufficient matrix is a generalization both of a positive-definite matrix and of a P-matrix, whose principal minors are each positive.[8][9][10] Such LCPs can be solved when they are formulated abstractly using oriented-matroid theory.[11][12][13]

## 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.

## Notes

- Murty (1988)
- Cottle, Pang & Stone (1992)
- R. W. Cottle and G. B. Dantzig. Complementary pivot theory of mathematical programming.
*Linear Algebra and its Applications*, 1:103-125, 1968. - Murty, Katta G. (January 1972). "On the number of solutions to the complementarity problem and spanning properties of complementary cones" (PDF).
*Linear Algebra and Its Applications*.**5**(1): 65–108. doi:10.1016/0024-3795(72)90019-5. - Taylor, Joshua Adam (2015),
*Convex Optimization of Power Systems*, Cambridge University Press, p. 172, ISBN 9781107076877. - Fukuda & Namiki (1994)
- Fukuda & Terlaky (1997)
- den Hertog, D.; Roos, C.; Terlaky, T. (1 July 1993). "The linear complementarity problem, sufficient matrices, and the criss-cross method" (PDF).
*Linear Algebra and Its Applications*.**187**: 1–14. doi:10.1016/0024-3795(93)90124-7. - Csizmadia, Zsolt; Illés, Tibor (2006). "New criss-cross type algorithms for linear complementarity problems with sufficient matrices" (PDF).
*Optimization Methods and Software*.**21**(2): 247–266. doi:10.1080/10556780500095009. - Cottle, R. W.; Pang, J.-S.; Venkateswaran, V. (March–April 1989). "Sufficient matrices and the linear complementarity problem".
*Linear Algebra and Its Applications*. 114–115: 231–249. doi:10.1016/0024-3795(89)90463-1. MR 0986877. - Todd (1985)
- Terlaky & Zhang (1993): Terlaky, Tamás; Zhang, Shu Zhong (1993). "Pivot rules for linear programming: A Survey on recent theoretical developments".
*Annals of Operations Research*. Degeneracy in optimization problems. 46–47 (1): 203–233. CiteSeerX 10.1.1.36.7658. doi:10.1007/BF02096264. ISSN 0254-5330. MR 1260019. - Björner, Anders; Las Vergnas, Michel; Sturmfels, Bernd; White, Neil; Ziegler, Günter (1999). "10 Linear programming".
*Oriented Matroids*. Cambridge University Press. pp. 417–479. doi:10.1017/CBO9780511586507. ISBN 978-0-521-77750-6. MR 1744046.

## References

- Cottle, Richard W.; Pang, Jong-Shi; Stone, Richard E. (1992).
*The linear complementarity problem*. Computer Science and Scientific Computing. Boston, MA: Academic Press, Inc. pp. xxiv+762 pp. ISBN 978-0-12-192350-1. MR 1150683. - Cottle, R. W.; Pang, J.-S.; Venkateswaran, V. (March–April 1989). "Sufficient matrices and the linear complementarity problem".
*Linear Algebra and Its Applications*. 114–115: 231–249. doi:10.1016/0024-3795(89)90463-1. MR 0986877. - Csizmadia, Zsolt; Illés, Tibor (2006). "New criss-cross type algorithms for linear complementarity problems with sufficient matrices" (PDF).
*Optimization Methods and Software*.**21**(2): 247–266. doi:10.1080/10556780500095009. - Fukuda, Komei; Namiki, Makoto (March 1994). "On extremal behaviors of Murty's least index method".
*Mathematical Programming*.**64**(1): 365–370. doi:10.1007/BF01582581. MR 1286455. - den Hertog, D.; Roos, C.; Terlaky, T. (1 July 1993). "The linear complementarity problem, sufficient matrices, and the criss-cross method" (PDF).
*Linear Algebra and Its Applications*.**187**: 1–14. doi:10.1016/0024-3795(93)90124-7. - Murty, K. G. (1988).
*Linear complementarity, linear and nonlinear programming*. Sigma Series in Applied Mathematics.**3**. Berlin: Heldermann Verlag. pp. xlviii+629 pp. ISBN 978-3-88538-403-8. MR 0949214. Updated and free PDF version at Katta G. Murty's website. Archived from the original on 2010-04-01. - Fukuda, Komei; Terlaky, Tamás (1997). Thomas M. Liebling and Dominique de Werra (eds.). "Criss-cross methods: A fresh view on pivot algorithms".
*Mathematical Programming, Series B*. Papers from the 16th International Symposium on Mathematical Programming held in Lausanne, 1997.**79**(1–3): 369–395. CiteSeerX 10.1.1.36.9373. doi:10.1007/BF02614325. MR 1464775. Postscript preprint.CS1 maint: uses editors parameter (link) - Todd, Michael J. (1985). "Linear and quadratic programming in oriented matroids".
*Journal of Combinatorial Theory*. Series B.**39**(2): 105–133. doi:10.1016/0095-8956(85)90042-5. MR 0811116. - R. Chandrasekaran. "Bimatrix games" (PDF). pp. 5–7. Retrieved 18 December 2015.

## Further reading

- R. W. Cottle and G. B. Dantzig. Complementary pivot theory of mathematical programming.
*Linear Algebra and its Applications*, 1:103-125, 1968. - Terlaky, Tamás; Zhang, Shu Zhong (1993). "Pivot rules for linear programming: A Survey on recent theoretical developments".
*Annals of Operations Research*. Degeneracy in optimization problems. 46–47 (1): 203–233. CiteSeerX 10.1.1.36.7658. doi:10.1007/BF02096264. ISSN 0254-5330. MR 1260019.