2007 Fiscal Year Final Research Report Summary
Development of numerically stable primal-dual interior point algorithms for solving nonlinear semidefinite programming problems
| Project/Area Number |
18560052
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Engineering fundamentals
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| Research Institution | University of Tsukuba |
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Principal Investigator |
YOSHISE Akiko University of Tsukuba, Graduate School of Systems and Information Engineering, Professor (50234472)
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| Co-Investigator(Kenkyū-buntansha) |
FUJISAWA Katsuki University of Tsukuba, Graduate School of Systems and Information Engineering, Associate Professor (40303854)
YAMAMOTO Yoshitsugu University of Tsukuba, Graduate School of Systems and Information Engineering, Professor (00119033)
KUNO Takahito University of Tsukuba, Graduate School of Systems and Information Engineering, Associate Professor (00205113)
SHIGENO Maiko University of Tsukuba, Graduate School of Systems and Information Engineering, Associate Professor (40272687)
MURAMATSU Masakazu University of Tsukuba, Graduate School of Systems and Information Engineering, Associate Professor (70266071)
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| Project Period (FY) |
2006 – 2007
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| Keywords | Optimization / Semidefinite programming / Interior point method / Homogeneous algorithm / Complementarity problem / Nonlinear optimization |
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| Research Abstract |
The aim of this research is to develop numerically stable primal-dual interior point methods foe solving nonlinear semidefinite programming problems. The semidefinite programming problem is an optimization problem over a closed convex cone that is not polyhedral unlike the linear programming problem. By this reason, we often observe a problem which has an asymptotic optimal solution but no optimal solution, I.e., any sequence on which the object value converges to the optimal value diverges. This brings us a numerical difficulty in determining the optimality when we apply interior point algorithms to solve the problem. A high accuracy of an optimal solution of the problem is critical if we adopt the problem as an approximation model of a combinatorial optimization problem or a robust optimization problem. To overcome the difficulty, many techniques have been proposed for obtaining a numerical stability of the algorithms. Such techniques are more highly expected when we solve nonlinear semidefinite programming problems. In order to provide one of such techniques, we introduced a homogeneous model for the nonlinear semidefinite programming problem, and showed that for the homogeneous model, (a) a bounded path having a trivial starting point exists, (b) any accumulation point of the path is a solution of the homogeneous model, c if the original problem is solvable then it gives us a finite solution, (d) if the original problem is strongly infeasible, then it gives us a finite certificate proving infeasibility, and (e) a class of algorithms for numerically tracing the path in (a) solves the problem in a polynomial number of iterations under a moderate assumption
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