Developments of Advanced Optimization Systems Unitying Discrete and Continuous Approaches Associate

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Optimization methods by computers have a great impact upon various fields in science and technology due to its wide scope of applications. In this research project, was have aimed at unifying existing results for system analysis obtained by each of project members on interiorpoint methods for linear programming, computational-geometric algorithms, matroid theory, Boolean function theory, etc., and produce new theoretical results and devrlop prototype optimization systems via this unifying work.Through this project, from the viewpoint of continuous optimization, we have extended the framework of linear programming to that of semidefinite programming, and, from the viewpoint of discrete optimization, theory of discrete convex analysis has been developed. By this theory of discrete convex analysis, connection with continuous methods and discrete methods can be established, with extending matroids and submodular systems on which the discrete convexity theory is based. Randomization is also applied through this connection between continuous and discrete approaches, and, by applying the randomized rounsing technique using the semidefinite programming to the satisfiability problem, approximate algorithms with better porformance ratio have been obtained. Furthermore, based on polyhedral structures having both continuous and combinatorial properties, the branch-and-cut technique is applied to the famous minimum-length triagulation problem in computational geometry. Finally, we developed a prototype system handling a family of sets by using the so-called binary decision diagram (or, BDD) as a promising approach from discrete Boolean function theory, and applied it to various problems, including network reliability computation which have continuous aspect, and other invariants in graphs, knots, and statistical physics. The system is made public for wide use.