Abstract
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.
Highlights
The linear complementarity problem (LCP) is one of the widely studied problems in optimization
The LCP first emerged as the Karush-Kuhen-Tucker (KKT) optimality conditions for linear programming (LP) and quadratic programming (QP); it has often been described as a fundamental problem Billups and Murty (2000)
Besides covering several important classes of mathematical programming problems such as LP, convex quadratic programming (CQP), Nash equilibrium points for nonzero sum games, several economic equilibrium problems and the knapsack problem, the LCP is used to model many applications such as the contact problem, the obstacle problem, the porous flow problem, the journal bearing problem and many other free boundary problems Wang (2010)
Summary
The linear complementarity problem (LCP) is one of the widely studied problems in optimization. These methods produce a sequence of points that follow the socalled central path, a nonlinear path from a strictly feasible point towards the solution. They are effective for solving large and ill-conditioned LCPs; their main drawbacks are the high cost per iteration and that they may not yield a good estimate of the solution when terminated early Morales et al (2007). 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 reduction in the size of the linear system solved each step to get a search direction.
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