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We present a modified damped Newton method for solving large sparse linear complementarity problems, which adopts a new strategy for determining the stepsize at each Newton iteration

# A modified damped Newton method for linear complementarity problems

Numerical Algorithms, no. 3 (2006): 207-228

EI WOS SCOPUS

We present a modified damped Newton method for solving large sparse linear complementarity problems, which adopts a new strategy for determining the stepsize at each Newton iteration. The global convergence of the new method is proved when the system matrix is a nondegenerate matrix. We then apply the matrix splitting technique to this ne...更多

• By collecting all those nonnegative numbers ρi and ρ j defined in (3.11) and (3.12), and letting v to denote the corresponding index set and v := {ρi : i ∈ v}, the authors know that, for any scalar τ ∈ [0, 1]\ v, (3.9) holds and, zv(τ ) is a nondegenerate vector with respect to the LCP(q, M).
• The authors note that, when M is a nondegenerate matrix, every accumulation point of the iteration sequence {zv} produced by Method 3.1 is a solution of the LCP(q, M), it is not necessarily a nondegenerate vector.

• Consider the following linear complementarity problem, abbreviated as LCP(q, M), for finding a z ∈ Rn such that

Mz + q ≥ 0, z ≥ 0 and zT(Mz + q) = 0, (1.1)

where M = ∈ Rn×n and q = ∈ Rn are given real matrix and vector, respectively
• Many problems in the areas of scientific computing and engineering applications can lead to the solution of a linear complementarity problem of the form (1.1)
• The splitting method is an extension of the matrix splitting iterative method for solving linear systems, see [1, 5, 10, 17, 25,26,27,28, 30]
• By using the transform (1.2)–(1.3) we present a modified damped Newton method for the LCP(q, M)
• We have established a modified damped Newton method for solving the large sparse linear complementarity problems. This method adopts a new strategy for determining the stepsize at each Newton iteration, and the strategy guarantees the global convergence of the new method when the system matrix is nondegenerate
• The global convergence of the inexact splitting method is proved under suitable conditions

• For any starting vector z0, the infinite iteration sequence {zv} generated by Method 4.1 converges to the unique solution of the LCP(q, M), provided that the number of the inner iteration steps is sufficiently large and the conditions (4.1) and
• There exists an ω ∈ (1, 2] such that, for all ω ∈ (0, ω ) and for any starting vector z0, the infinite iteration sequence {zv} generated by Method 4.1 converges to the unique solution of the LCP(q, M), provided that the number of the inner iteration steps is sufficiently large and the conditions (4.1) and (4.2) hold.
• At each outer iteration v and for each inner iteration l, the authors only need to find all those numbers τv,l such that yv,l+1 is a nondegenerate vector with respect to the LCP(q, M).
• For any nondegenerate starting vector z0 with respect to the LCP(q, M), the infinite iteration sequence {zv} generated by Method 4.2 converges to the unique solution of the LCP(q, M), provided that the number of the inner iteration steps is sufficiently large.
• There exists an ω ∈ (1, 2] such that, for all ω ∈ (0, ω ) and for any starting nondegenerate vector z0 with respect to the LCP(q, M), the infinite iteration sequence {zv} generated by Method 4.2 converges to the unique solution of the LCP(q, M), provided that the number of the inner iteration steps is sufficiently large.
• In the numerical tables for both examples, the authors let v and CPU denote, respectively, the number of iteration steps and the CPU timing in seconds, required for satisfying the termination criterions in every tested method.

• In order to solve the Newton equation conveniently and the authors have applied the matrix splitting technique to the new method and derived an inexact splitting method for the linear complementarity problems.
• The authors should point out that the problems such as how to obtain a more efficient matrix splitting for the inexact splitting method and study the synchronous, the chaotic or the asynchronous variant of these new methods in the spirits of the works [4, 6,7,8,9, 11] deserve further discussions in future

• Table1: Results for example 5.1
• Table2: Results for example 5.2

• The research of this author is supported by The National Basic Research Program (No 2005CB321702), The China NNSF National Outstanding Young Scientist Foundation (No 10525102) and The National Natural Science Foundation (No 10471146), P.R  0