Information Geometric and Jordan Algebraic Study of Semidefinite Programming and Its Applications
Project/Area Number 
12640122

Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
General mathematics (including Probability theory/Statistical mathematics)

Research Institution  Osaka University 
Principal Investigator 
OHARA Atsumi Osaka University, Graduate School of Engineering Science, Associate Professor, 大学院・基礎工学研究科, 助教授 (90221168)

Project Period (FY) 
2000 – 2003

Project Status 
Completed (Fiscal Year 2003)

Budget Amount *help 
¥2,200,000 (Direct Cost: ¥2,200,000)
Fiscal Year 2003: ¥500,000 (Direct Cost: ¥500,000)
Fiscal Year 2002: ¥500,000 (Direct Cost: ¥500,000)
Fiscal Year 2001: ¥500,000 (Direct Cost: ¥500,000)
Fiscal Year 2000: ¥700,000 (Direct Cost: ¥700,000)

Keywords  Semidefinite Programming / Information Geometry / Jordan Algebra / Symmetric Cone / Linear Matrix Inequality / Control Theory / 曲率 / 行列微分不等式 / 計算の手間 
Research Abstract 
In the area of information theory and systems science, positive definite matrices appear in connection with various concepts. Hence, mathematical problems in these area are often formulated as ones relating to the structural properties of positive definite matrices, or more formally as optimization problems on symmetric cones, which is a generalization of the set of positive definite matrices. The research has aimed at exploiting the followings, with the assistance of information geometry and Jordan algebra theory: (a) the informationgeometric structure and properties of symmetric cones and their submanifolds, (b) the relation between geometric structure and computational complexity of linear programming on symmetric cones, (c) applicability of mathematical programming of this type to systems science, in particular, control theory. As for the item (a), the results we obtained are the characterization of socalled dual connections by Jordan algebra (ref.[3]), decomposition property of the divergence on the level surfaces defined by the characteristic function of symmetric cones (ref.[2]), the relation of the level surfaces with affine geometry (ref.[l]) and definition of means on symmetric cones and the relation with dualistic geodesies (ref.[4]) and so on. As for the item (b), we have obtained a result that shows flat submanifolds in a symmetric cone with respect to the dual connections play a key role. However, we have not established a clear relation between complexity and curvatures yet. We shall keep on researching along this line. As for the item (c), we develop applications of inequalities of symmetric matrices induced by the positive definite cone, i.e., matrix inequality, to control theory (ref. [5], [6]).

Report
(5 results)
Research Products
(24 results)