Study on continuous- and discrete structure in optimization

优化中的连续和离散结构研究

基本信息

批准号：
14340037
负责人：
KAWASAKI Hidefumi
金额：
$ 3.46万
依托单位：
KYUSHU UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
2002
资助国家：
日本
起止时间：
2002 至 2005
项目状态：
已结题

项目摘要

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

1.我们在课题“共轭点的离散化”上得到了以下结果。(1)我们证明了Riccati方程是目标函数的Hessian矩阵的主元满足的递推关系。(2)我们显式地计算了Hessian矩阵为三对角矩阵时的共轭点。(3)通过离散化，我们阐明了共轭点协作结构的重要性。将目标函数的变量视为参与者，我们定义了称为共轭集博弈的合作博弈，并计算了Shapley值来评估每个变量对改进解的贡献。(4)Hessian矩阵的优化问题。不是三对角线，超出了经典共轭点理论的范围。我们研究了三相分配问题，该问题最初来自非线性扩散现象。我们讨论了稳态解的稳定性和不稳定性。 2.我们将三相划分问题表述为凸规划问题，并提出了基于分离两个凸集的经典对偶定理。另一方面，我们的对偶定理基于用三角形分隔三个凸集。我们将我们的对偶定理扩展到多相划分问题和更高维空间。提出了非确定性系统上的离散时间动态规划，并引入了控制差分方程。通过我们的研究，我们在 DP.4 中获得了确定性、概率、模糊和非确定性系统。我们提出了一个两阶段。我们在国际会议上做了 31 次演讲，在国内数学会议上做了 41 次演讲，并组织了 4 次研讨会。首席研究员在国际研讨会上作大会报告两次，并作特邀报告四次，并撰写了《Less 极值问题》一书。