## A multilevel iterative method for symmetric, positive definite linear complementarity problems.(English)Zbl 0539.65046

This paper presents a method for solving the constrained optimization problem $$\min f(x)=1/2x^ TAx-x^ Tb,$$ $$x\geq c$$ where A is a symmetric positive definite matrix. The method of solution consists of constructing a finite sequence of auxiliary problems $P_ k:\min f_ k(x^ k)=1/2(x^ k)^ TA_ kx^ k-(x^ k)^ Tb^ k,\quad x^ k\geq c^ k$ where $$x^ k\in V_ k=R^{V_ k}$$ and $$\dim V_{k-1}<\dim V_ k.$$
The sequence is for $$k=1...m$$ and $$k=m$$ corresponds to the original problem. The algorithm starts with a feasible solution for $$P_ m$$ and with various iterations, constructs approximate solutions to the problems $$P_ k$$, $$k<m$$ and from these a corrected value of $$x^ m$$. Since the process is a variational one the procedure will always converge and there is numerical evidence that this convergence is rapid.
In the example given the method compares favourably with relaxation methods. The problem $$P_ m$$ is said to be nondegenerate if $$(x-c)+(Ax- b)>0$$ where x is the solution. In such cases it is proved that this algorithm reduces to a linear iterative method and that the rate of convergence consequently depends on the spectral radius of a linear operator.
Reviewer: B.Burrows

### MSC:

 65K05 Numerical mathematical programming methods 90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming) 65F10 Iterative numerical methods for linear systems
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### References:

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