We are currently developing a primal-dual interior method and a software for solving nonlinear optimization problems. The goal is to solve optimization problems issue from active collaborations with other applied do- mains of our laboratory (physics and mechanics). The proposed research work is about the solution of nonlinear semidefinite optimization problem. The primal-dual algorithms that we are working can be naturally extended to solve SDP problems. Some strong knowledge in numerical optimization is required.
We plan to study the local and global behaviour of primal-dual methods for solving nonlinear semidefinite optimization problem of the form:
minimize f(x), subject to the constraints c(x) = 0 and X(x) >= 0, where f: Rn -> R, c: Rn -> Rm and X(x): Rn -> Sp are smooth functions, Sp is the set of real symmetric matrices of order p and where X(x) >= 0 means that the matrix is positive semidefinite.
This problem can be found in many applications, in particular in automatic, an active research field of our laboratory, but also in many other scientific domains. The solution of this kind of problem can be done by using a primal- dual approach. Since we are working in this area for some years, we would like to extend our recent algorithms and convergence results to this framework.
More precisely, we would like to extend a recent convergence result 1] to the framework of nonlinear SDP optimization as described in . At first, exact solution of the Newton system will be investigated. Then, an inexact approach, as described in , will be considered. The globalization should also be studied, see . Numerical experiments will also be done to validate the proposed algorithms.
New results about the solution of semidefinite problems should have a great impact in mathematical optimization. Semidefinite programming is still an active research field.
REFERENCES P. Armand and J. Benoist, A local convergence property of primal-dual methods for nonlinear programming, Math. Program., 115 (2008), pp. 199– 222.
 P. Armand, J. Benoist and J.P. Dussault, Local path-following property of inexact interior methods in nonlinear programming, Computational Optimization and Applications, DOI 10.1007/s10589-011-9406-2, 2011.
 P. Armand, J. Benoist and D. Orban, From global to local con- vergence of interior methods for nonlinear optimization, Research Report XLIM, June 2010.
 H. Yamashita and H. Yabe, Local and superlinear convergence of a primal-dual interior method for nonlinear semidefinite programming, Math. Program., DOI 10.1007/s10107-010-0354-x., 2010.