Multi-valued optimization problems on uncertain models

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The summary of research results is as follows.1. We gave two approximation methods for the value of the problem without a finite constraint in the zero-sum Dynkin problem and introduced several new constraints and established the relationship between their values.2. We proved the following result by means of elementary Lebesgue measure and integral theory in RィイD1NィエD1 : let f be a locally integrable function over RィイD1NィエD1. If an integral of a product of f and a continuous function with any compact support is O,, then f = O a.e..3. Childs conjectured and Moran proved that a generating function of an orthoscheme probability equals tan Z + sin Z. We gave their generalization.4. We proved that the Hausdorff dimension of a CィイD11+γィエD1 generalized cookie-cutter Cantor set is less than 1.5. As a typical optimization problem, we chose a mathematical partition problem in the scheme of three-body ; discrete, continuous and continuum and proved that Euler Partition Rule holds through three bodies.6. We introduced a new class of fuzzy stopping times which called as a monotone fuzzy stopping time and gave an explicit derivation of an optimal stopping under appropriate assumptions.7. In a book "Mathematics of Fuzzy Optimization" , we mathematically and systematically developed fuzzy theory and fuzzy optimization.