## Operator splitting for a homogeneous embedding of the linear complementarity problem.(English)Zbl 1479.90202

Authors’ abstract: We present a first-order quadratic cone programming algorithm that can scale to very large problem sizes and produce modest accuracy solutions quickly. Our algorithm returns primal-dual optimal solutions when available or certificates of infeasibility otherwise. It is derived by applying Douglas-Rachford splitting to a homogeneous embedding of the linear complementarity problem, which is a general set membership problem that includes quadratic cone programs (QCPs) as a special case. Each iteration of our procedure requires projecting onto a convex cone and solving a linear system with a fixed coefficient matrix. If a sequence of related problems are solved, then the procedure can easily be warm-started and make use of factorization caching of the linear system. We demonstrate on a range of public and synthetic datasets that for feasible problems our approach tends to be somewhat faster than applying operator splitting directly to the QCP, and in cases of infeasibility our approach can be significantly faster than alternative approaches based on diverging iterates. The algorithm we describe has been implemented in C and is available open-source in the solver SCS v3.0.
Reviewer: Bing Tan (Chengdu)

### MSC:

 90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming) 90C20 Quadratic programming 65K05 Numerical mathematical programming methods 65K10 Numerical optimization and variational techniques 90C05 Linear programming 90C06 Large-scale problems in mathematical programming 90C22 Semidefinite programming 90C25 Convex programming 90C30 Nonlinear programming 90C46 Optimality conditions and duality in mathematical programming
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