2002 Fiscal Year Final Research Report Summary
A unified approach to nonconvex programming problems using branch-and-bound algorithms
| Project/Area Number |
13680505
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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 |
社会システム工学
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| Research Institution | University of Tsukuba |
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Principal Investigator |
KUNO Takahito Univesity of Tsukuba, Institute of Information Sciences and Electronics, Associate Professor, 電子・情報工学系, 助教授 (00205113)
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| Co-Investigator(Kenkyū-buntansha) |
YOSHISE Akiko Univesity of Tsukuba, Institute of Policy and Planning Sciences, Associate Professor, 社会工学系, 助教授 (50234472)
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| Project Period (FY) |
2001 – 2002
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| Keywords | Global optimization / nonconvex program / branch-and-bound method / mathematical programming / algorithm |
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| Research Abstract |
We made a study mainly on three classes of nonconvex optimization problems, each of which is abundant in applications to real-world social systems :
(1) In almost every optimization problem, both objective and constraint functions can be written as the difference of two convex functions. Using this property, the problem can be trans-formed into a convex minimization problem with an additional reverse convex constraint. We proposed three branch-and-bound algorithms for solving this kind of nonconvex optimization problems. We showed that each algorithm generates a globally optimal solution in finite iterations if the reverse convex constraint function is separable.
(2) The sum-of-ratios problem is a problem, of minimizing a sum of linear rations over a convex set, and is known to be intractable. We devised an inexpensive procedure for computing a tignt lower bound on the optical value. We incorporated it into a branch-and-bound algorithm and succeeded in solving the problem much faster than the existing algorithms.
(3) Many of chemical process design problems can be formulated as optimization problems but highly nonconvex ones, say mixed-integer nonlinear programming problems. To solve this kind of problems, we proposed a hybrid algorithm of brand-and-bound and revised general benders decomposition methods. We then proved that the algorithm certainly converges to globally optimal solutions for some typical chemical process design problems.
Each of the proposed algorithms is based on the idea of branch and bound. To execute the bounding operations efficiently, we also studied an interior-point algorithm for solving relaxed problems.
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