Operator splitting for a homogeneous embedding of the monotone linear complementarity problem

Abstract

In this work we apply Douglas-Rachford splitting to a homogeneous embedding of the linear complementarity problem (LCP). The homogeneous embedding converts LCPs into non-linear monotone complementarity problems (MCPs) that encode both the optimality conditions of the original LCP as well as certificates of infeasibility. The resulting problem can be expressed as the problem of finding a zero of the sum of two maximal monotone operators, to which we apply operator splitting. The resulting algorithm has almost identical per-iteration cost as applying Douglas-Rachford splitting to the LCP directly. Specifically it requires solving a linear system and projecting onto a cone at every iteration.
Previous work applied operator splitting to a homogeneous embedding of a linear cone optimization problem. This paper extends the analysis to cover quadratic objectives, due to the correspondence between quadratic programming and the linear complementarity problem. Our algorithm is able to return a primal-dual solution to the quadratic cone program should one exist, or a certificate of infeasibility otherwise. We provide extensive numerical experimentation to show that for feasible problems our approach tends to be somewhat faster than applying operator splitting directly to the LCP, and in cases of infeasibility our approach can be orders of magnitude faster than competing approaches based on differences of iterates.

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