1990 Fiscal Year Final Research Report Summary
Mathematical Programming Algorithms and Their Applications to Engineerring Problems
| Project/Area Number |
63490010
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| Research Category |
Grant-in-Aid for General Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Research Field |
広領域
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| Research Institution | Tokyo Institute of Technology |
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Principal Investigator |
KONNO Hiroshi Tokyo Inst. of Technology, Professor, 工学部・人文社会群, 教授 (10015969)
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| Co-Investigator(Kenkyū-buntansha) |
TAMURA Akihito Tokyo Inst. of Technology. Asst. Prof., 理学部・情報科学科, 助手 (50217189)
SHIRAKAWA Hiroshi Tokyo Inst. of Technology, Asst. Prof., 工学部・人文社会群, 助手 (10216187)
KUNO Takahito Tokyo Inst. of Technology, Asst. Prof., 工学部・社会工学科, 助手 (00205113)
MIZUNO Shinji Inst. of Statistical Math. Assoc. Prof., 文部省・統計数理研究所, 助教授 (90174036)
KOJIMA Masakazu Tokyo Inst. of Technology, Professor, 理学部・情報科学科, 教授 (90092551)
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| Project Period (FY) |
1988 – 1990
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| Keywords | Global optimization / Interior point method / linear programming / Quadratic programming / Portfolio theory / Non-convex function / Skewness / MAD model |
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| Research Abstract |
(1) Interior point algorithms for linoar programs and linear complementarity problems.
Interior point algorithms first proposed in 1984 by N. Karmarkar turned out to be the most efficient algorithms for solving large scale linear programs and linear complementarity problems. We tried to improve and polish this algorithm and obtained several new results. Some of the more important ones are ; development of the algorithm whose number of iteration is a cubic function of the number of variables ; development of primal-dual interior point algorithm ; construction of the unified theoretical framework of the interior point algorithms and the extension of the interior point algorithms to nonlinear complementarity problems.
(2) Global minimization of a class of nonconvex functions over a polytope.
The starting point of this research was our discovery of parametric linear programming algorithm for linear multiplicative programming problem which took place in 1989. Since then we tried to extend it t
… More
o diverse classes of nonconvex minimization problems such as (a) (generalized) convex multiplicative programming problems, (b) minimization of a sum (and a product) of two linear fractional functions (c) convex minimization under linear multiplicative constraints (d) minimization of a product of p convex functions (e) minimization of rank two and rank three bilinear programming problems (f) minimum cost lot-sizing problem with rho-concave cost functions. All of these problems can now be solved effectively by our algorithms.
(3) Financial optimization problems.
We proposed two new models in the field of portfolio optimization. One is the MAD model in which the absolute deviation is used as a measure of risk instead of the standard deviation. Second is the MADS model which enables
the quantitative treatment of the third moment. In addition, we developed a unified framework for the multi-period portfolio-dividend optimization problems. We can now solve a large scale problem of this class which could not have been solved by existing methods. Less
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