Algorithms for Optimization Problems and Systems of Bilinear Equations over Positive Definite Symmetric Matrices and its Applicatoins

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Typical examples of optimization problems and systems of bilinear equations over the set of positive definite symmetic cones are semidefinite programming (SDP) and the systems bilinear equations arising from SDP. In this research, we also studied linear programming (LP) and second-order cone programming (SOCP) as special cases of SDP. The topics we continued to study from the past and related to the project is : (1) Polynomi-ality analysis of primal-dual interior-point algorithms for SDP ; (2) Polynomiality analysis of primal-dual interior-point algorithms for SOCP ; (3) Costruction of a counter example of global convergence of, the affine-scaling algorithm for LP. We published three papers on these topics in the term of the project. There are the following four other new topics we studied in the project : (4) Application of SDP to estimation of graphical models ; (5) Application of SOCP to optimal design of magnetic shielding in linear motor car (new bullet train) ; (6) Modification of the Vavasis-Ye polynomial-time interior-point algorithm for linear programming whose complexity just depends on the coeflicient matrix A ; (7) Extension of polynomial-time primal-dual interior-point, algorithms to inflnite, , dimensional problems using JB algebras. We also implemented the dual affine-scaling algorithm for LP and the primal-dual path-following algorithms for SDP, and made attempt, s to solve standard benchmark problems.