An active set algorithm for a class of linear complementarity problems arising from rigid body dynamics

An active set algorithm is introduced for positive definite and positive semi definite linear complementarity problems. The proposed algorithm is composed of two phases. Phase 1, the feasibility phase and phase 2, the optimality phase. In phase 1, the ellipsoid method is employed to test for feasibility and provide an advanced starting point if the problem is feasible. Providing such a warm start permits a good estimate of the active set. In phase 2, a criterion based on the complementarity condition is used to detect the working set per iteration until optimality is reached. This criterion leads to a valuable reduction in the size of the problem solved per iteration to obtain a search direction. Numerical examples are solved to illustrate the performance of the algorithm and a practical example in rigid body dynamics is solved to demonstrate the usage of the algorithm to solve such problems

About solutions of the LCP

In this paper the linear complementarity problem (LCP) is discussed. The focus\ud is on the types of solutions the problem can produce. Theory about the solution\ud structure is discussed through the use of cones and complementary matrices. This\ud theory is used to present a new algorithm to solve LCPs. The LCP(q,M) is then\ud reformulated into a quadratic programming problem in order to show that positive\ud de nite matrices guarantee the existence of unique solutions. Lemke's algorithm\ud is also presented as a means of solving the LCP and a sample problem is solved.\ud The focus then shifts to original material about linear compelementarity problems\ud with multiple solutions. This includes three new results showing that singular\ud matrices and vectors in their nullspace can produce in nitely many solutions to the\ud LCP. Triangular matrices are also shown to produce multiple solutions. First some\ud observations are made about the solution space of LCPs with diagonal matrices.\ud This leads to new results showing that under certain conditions, LCPs with upper\ud and lower triangular n x n matrices can have 2^n non-degenerate solutions