Homotopy Methods For Nonlinear Symmetric Cone Programming

In this paper, by virtue of Jordan algebras, homotopy methods for solving nonlinear symmetric cone programming are given. A combined homotopy and a smoothing homotopy are constructed, existence and probability one convergence of interior pathways deetermined by these homotopies under some mild conditions are proven. Following the pathway from the given starting point, a KKT point of the problem can be obtained.In the first part of this paper, a brief history and latest progress about symmetric cone programming and homotopy method are given, a introduce about the homotopy method for semidefinite programming and nonlinear second-order cone programming as well as some results about symmetric cone and symmetric cone programming which will be used in the second part are also given. In the second part, at first, a combined homotopy for solving the nonlinear symmetric cone programming problem is constructed, existence and probability one convergence of the homotopy path is proven under some mild conditions. Then a smoothing homotopy method for nonlinear symmetric cone programming is constructed and corresponding theoretical results are also given.