| Project/Area Number |
15510144
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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 |
Social systems engineering/Safety system
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| Research Institution | The Institute of Statistical Mathematics |
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Principal Investigator |
TSUCHIYA Takashi The Institute of Statistical Mathematics, Department of Mathematical Analysis and Statistical Inference, Professor., 数理・推論研究系, 教授 (00188575)
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| Project Period (FY) |
2003 – 2005
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| Project Status |
Completed (Fiscal Year 2005)
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| Budget Amount *help |
¥3,500,000 (Direct Cost: ¥3,500,000)
Fiscal Year 2005: ¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 2004: ¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 2003: ¥1,400,000 (Direct Cost: ¥1,400,000)
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| Keywords | linear programming / interior-point methods / curvature / robust optimization / least squares methods / optimal design / condition number / 線形計画問題 / 中心曲線 / 計算複雑度 / 主双対内点法 / 半正定値計画法 / 確率密度推定 |
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| Research Abstract |
We studied the following three topics : (i) geometrical structure of central trajectories of linear programming and its relation to computational complexity of interior-point algorithms ; (ii) evaluation of robust performance of approximate optimal solutions to robust optimization problems based on sampling ; (iii) mathematics of layered-least squares methods.
In (i), we analyzed relations between the iteration-complexity of the primal-dual interior-point algorithms for linear programming and geometrical structure of the central trajectories. We defined a curvature of the central trajectory and proved that the iteration-complexity of the interiorpoint algorithms is precisely approximated with the curvature integral. Furthermore, we established that the total curvature of the central trajectory of any linear program is bounded by (the logarithm of the condition number of the coefficient matrix of the linear program)*((the number of the nonnegative variables) to the power of 3.5). This sh
… More
ows that the degree of ill-condition of a linear program just depends on the coefficient matrix.
In (ii), we developed a method to evaluate robust performance of an approximate optimal solution to a robust optimization problem of magnetic shielding design computed with a sampling method. The purpose of the robust optimization in our context is to design a shield with the least weight under the condition that the shield has enough thickness even when the magnetic field is subject to some perturbation. The method of robust optimization we developed performs optimization by thickening the shielding iteratively to adapt to the set of finite samples (of magnetic field) drawn from the domain of perturbation, starting from the optimal shielding for nominal magnetic field. We succeeded to develop a solid method to estimate robust performance of the obtained magnetic shielding based on the maximum likelihood estimation.
In (iii), we demonstrated that the physical model on which the aforementioned optimal design of the magnetic shielding is based is obtained by considering a variational formulation of the problem as a weighted least squares problem and then by taking the limit letting the ratios among the weights to infinity. The problem obtained in the limit admits an interpretation as a layered-least squares problem. We also developed a bound on the norm of difference between two operators giving the layered least squares solution and the weighted least squares solution. The bound is written in terms of the ratios among the weights of the weighted least squares problem and the condition number of the coefficient matrix of the least squares problem. Less
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