Study on continuous- and discrete structure in optimization

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1.We have obtained the following results on our main subject "Discretization of the conjugate point".(1)We have shown that the Riccati equation is the recurrence relation that the pivots of the Hessian matrix of the objective function satisfies.(2)We have explicitly computed the conjugate point when the Hessian matrix is a tridiagonal matrix.(3)By discretization, we have clarified the importance of a cooperative structure of the conjugate point. Regarding variables of the objective function as players, we have defined a cooperative game called a conjugate-set game. Further, we have computed the Shapley value to evaluate the contribution of each variable to improving the solution.(4)Optimization problems whose Hessian matrices are not tridiagonal are out of the scope of the classical conjugate point theory. We have studied the three-phase partition problem, which originally comes from nonlinear diffusive phenomena. We have discussed stability and instability of stationary solutions for … More the three-phase partition problem in terms of the curvature of the boundary of the region.2.We have formulated the three-phase partition problem as a convex programming problem, and presented a duality theorem. Classical duality theorems are based on separating two convex sets by a hyperplane. On the other hand, our duality theorem is based on separating three convex sets by a triangle. We are extending our duality theorem to the multiphase partition problem and to higher dimensional spaces.3.We have proposed a discrete time dynamic programming on a non-deterministic system and introduced a control difference equation. By our research, we have obtained deterministic, probabilistic, fuzzy, and non-deterministic systems in DP.4.We have proposed a two-stage procedure for the problem of constructing a fixed size confidence region of the difference of two multi-normal means by using semi-infinite programming toWe gave 31 presentations in international conferences and 41 talks in domestic math meetings, and organized four workshops. The head investigator gave plenary lectures twice in international symposiums and presented invited lectures four times. Further, he wrote a book titled "Extremal problems". Less