1996 Fiscal Year Final Research Report Summary
Algorithm for a global optimal solution of optimization problems without convexity
| Project/Area Number |
06680389
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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 |
YAMAMOTO Yoshitsugu University of Tuskuba, Institute of Policy and Planning Scinces, Professor, 社会工学系, 教授 (00119033)
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| Project Period (FY) |
1994 – 1996
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| Keywords | Mathematical program / Globally optimal solution / Nonconvex problem / Convexity |
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| Research Abstract |
According to the sets and functions defining the problem mathematical programming problems are classified roughly into two groups : convex program and nonconvex program. Convex programs have been attracting many researchers, theoretically well established, and practical codes are offered to solve large scale problems. Nonconve programs, however, are in unsatisfactory situation because of its difficulty.
We set the nonconvex problem on the differece of several convex sets as a standard problem of nonconvex programs and developed algorithms for finding a globally optimal solution. The largest empty sphere problem, the out-of-roundess problem in higher dimension are cast into this standard problem. We proposed an algorithm for the problem which yileds a globally optimal solution within a finite number of iterations, which forms a striking contrast to the existing algorithms proposed by Kuno-Konno-Yamamoto, Kuno-Yajima-Yamamoto-Konno, Thach-Burkard-Oettli, Pferschy-Tuy for the problem which yield only an approximate solution. As far as we know, the algorithm is the first one which enjoys both the global optimality and the finiteness. We compared the algorithm with the existing algorithms by computational experiment, which supports the superiority of our algorithm.
We also formulated the out-of-roundess problem as a fractional problem and proposed an algorithm. For the variational inequality problem we developed a continuous deformation algorithm based on the simplcial algorithm.
Furthermore, we have finished a book on the traveling salesman problem, which has been known as one of the hardest problems, with the aim of introducing the difficulty and interest of the problem.
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