This Proceedings presents refereed versions of most of the papers presented at the NATO Advanced Research Institute on Homotopy Methods and Global Convergence held in Porto Cervo, Sardinia, June 3-6, 1981. This represents the fourth recent occurrence of an international conference addressing the common theme of fixed point computation. The first such conference, ti tled "Computing Fixed Points with Applications," was held in the Department of Mathematical Sciences at Clemson University, Clemson, South Carolina, June 26-28, 1974 and was sponsored by the Office of Naval Research and the Office of the Army Research Center. The second conference, "Symposium on Analysis and Computation of Fixed Points," was held at the University of Wisconsin, Madison, May 7-8, 1979, under the sponsorship of the National Science Foundation, the U. S. Army, and the Mathematics Research Center of the University of Wisconsin, Madison. The third conference, titled "Symposium on Fixed Point Algorithms and Complementarity," was held at the University of Southampton, Southampton, UK, July 3-5, 1979 and was sponsored by U. N. E. S. C. O. , European Research Office (London), Department of Mathematics (University of Southampton), I. B. M. U. K. , Ltd. , Lloyds Bank, Ltd. , and the Office of Naval Research (London). The Advanced Research Institute held in Sardinia was devoted to the theory and application of modern homotopy methods. The following topics were stressed: Path-Following Techniques; Bottom-Line Applications; Global vs. Classical Methods; and Sta- v vi PREFACE of-the-Art, Perspectives and Potential.

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Piecewise Smooth Homotopies.- Global Convergence Rates of Piecewise-Linear Continuation Methods: A Probabilistic Approach.- Relationships between Deflation and Global Methods in the Problem of Approximating Additional Zeros of a System of Nonlinear Equations.- Smooth Homotopies for Finding Zeros of Entire Functions.- Where Solving for Stationary Points by LCPs Is Mixing Newton Iterates.- On the Equivalence of the Linear Complementarity Problem and a System of Piecewise Linear Equations: Part II.- Relations between PL Maps, Complementary Cones, and Degree in Linear Complementarity Problems.- A Note on Stepsize Control for Numerical Curve Following.- On a Class of Linear Complementarity Problems of Variable Degree.- Linear Complementarity and the Degree of Mappings.- Sub- and Supersolutions for Nonlinear Operators: Problems of Monotone Type.- An Efficient Procedure for Traversing Large Pieces in Fixed Point Algorithms.- The Application of Fixed Point Methods to Economics.- On a Theory of Cost for Equation Solving.- Algorithms for the Linear Complementarity Problem Which Allow an Arbitrary Starting Point.- Engineering Applications of the Chow-Yorke Algorithm.- Availability of Computer Codes for Piecewise-Linear and Differentiable Homotopy Methods.