Globally convergent homotopy methods: a tutorial

The basic theory for probability one globally convergent homotopy algorithms was developed in 1976, and since then the theory, algorithms, and applications have considerably expanded. These are algorithms for solving nonlinear systems of (algebraic) equations, which are convergent for almost all choices of starting point. Thus they are globally convergent with probability one. They are applicable to Brouwer fixed point problems, certain classes of zero-finding problems, unconstrained optimization, linearly constrained optimization, nonlinear complementarity, and the discrezations of nonlinear two-point boundary value problems based on shooting, finite differences, collocation, and finite elements. A mathematical software package, HOMPACK, exists that implements several different strategies and handles both dense and sparse problems. Homotopy algorithms are closely related to ODE algorithms, and make heavy use of ODE techniques. Homotopy algorithms for some classes of nonlinear systems, such as polynomial systems, exhibit a large amount of coarse grain parallelism. These and other topics are discussed in a tutorial fashion.

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