Abstract
Global Newton methods for computing solutions of nonlinear systems of equations have recently received a great deal of attention. By using the theory of generalized equations, a homotopy method is proposed to solve problems arising in complementarity and mathematical programming, as well as in variational inequalities. We introduce the concepts of generalized homotopies and regular values, characterize the solution sets of such generalized homotopies and prove, under boundary conditions similar to Smale’s [10], the existence of a homotopy path which contains an odd number of solutions to the problem. We related our homotopy path to the Newton method for generalized equations developed by Josephy [3]. An interpretation of our results for the nonlinear programming problem will be given.
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Reinoza, A. Solving generalized equations via homotopies. Mathematical Programming 31, 307–320 (1985). https://doi.org/10.1007/BF02591952
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DOI: https://doi.org/10.1007/BF02591952