Mathematics Colloquia and Seminars

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We consider the nonlinear semidefinite programming (SDP) problem:
minimize f(x) with x ∈ R^n , subject to g(x) = 0, X(x) >= 0
where the functions f : R^n → R, g : R^n → R^m and X : R^n → S^p are sufficiently smooth, and S^p denotes the set of pth order real symmetric matrices. By X>= 0 and X>0, we mean that the matrix X is positive semidefinite and positive definite, respectively.

The linear SDP problems include linear programming problems, convex quadratic pro- gramming problems and second order cone programming problems, and they have many applications. Interior point methods for the linear SDP problems have been studied ex- tensively by many researchers.

On the other hand, researches on numerical methods for nonlinear SDP problems are much more recent. In this talk, we propose a globally convergent primal-dual interior point method for solving nonlinear SDP problems. The proposed method consists of Algorithm SDPIP (outer iteration) and Algorithm SDPLS (inner iteration). In Algorithm SDPLS, we combine the primal barrier penalty function and the primal-dual barrier function, and we propose a primal-dual merit function. We show the global convergence property of our method within the framework of the line search strategy. Finally some numerical experiments are given.