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piecewise-linear homotopy algorithms FOR SPARSE SYSTEMS OF NON-linear EQUATIONS

SIAM JOURNAL ON CONTROL AND OPTIMIZATION 1983年第2期21卷 204-214页

作者： TODD, MJ

When piecewise-linear homotopy algorithms are applied to the problem of approximating a zero of a sparse function $f:R^n \to R^n $, a large piece of linearity can be traversed in one step by using a suitable linear system. The linear system has n rows and $n + 1$ columns, but is subject to a number of inequalities depending on the sparsity pattern of f. We show how an algorithm can be implemented using these large pieces; in particular, we demonstrate how to update the linear system corresponding to one large piece to obtain the appropriate system for an adjacent large piece. One measure of the complexity of such an implementation is the number of inequalities that may be required for any one piece. We prove that there can be no more than $O(n^{{3 / 2}} )$ such inequalities, and that this bound is essentially tight; the argument is purely combinatorial. Finally, we provide guidelines on when such a “large-piece implementation” should be used instead of much simpler “small-piece implementations” for piecewise-linear homotopy algorithms.

关键词： piecewise-linear homotopy algorithms solving nonlinear equations sparsity

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ON THE COMPUTATIONAL-COMPLEXITY OF piecewise-linear homotopy algorithms

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MATHEMATICAL PROGRAMMING 1982年第2期24卷 216-224页

作者： TODD, MJ UNIV CAMBRIDGE CAMBRIDGEENGLAND

We show that piecewise-linear homotopy algorithms may take a number of steps that grows exponentially with the dimension when solving a system of linear equations whose solution lies close to the starting point. Our examples are based on an example of Murty exhibiting exponential growth for Lemke's algorithm for the linear complementarity problem.

关键词： piecewise-linear homotopy algorithms Computational Complexity Exponential Growth linear Complementarity Problem

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