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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“共扼点的离散化”这一主要课题上得到了如下结果。(1)We证明了Riccati方程是目标函数的Hessian矩阵的枢轴所满足的递推关系。(2)We当Hessian矩阵是三对角矩阵时，已经明确计算了共轭点。(3)By离散化，我们已经澄清了共枕点的合作结构的重要性。把目标函数中的变量作为局中人，我们定义了一种合作对策，称为共轭集对策。此外，我们还计算了Shapley值，以评估每个变量对改进解决方案的贡献。（4）Hessian矩阵不是三对角矩阵的优化问题超出了经典的共轭点理论的范围。本文研究了由非线性扩散现象产生的三相分配问题。我们讨论了定态解的稳定性和不稳定性， ...更多信息 2.将三相划分问题转化为一个凸规划问题，并给出了一个对偶定理。经典的对偶定理是基于用超平面分离两个凸集。另一方面，我们的对偶定理是基于用三角形分隔三个凸集。我们将对偶定理推广到多相分割问题和高维空间。3.提出了非确定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，我们得到了DP中的确定性、概率性、模糊性和非确定性系统。4.我们提出了用半无限规划构造两个多正态均值之差的固定大小置信域的两阶段方法，在国际会议上发表了31篇报告，在国内数学会议上发表了41篇演讲，并组织了四次专题讨论会。首席调查员在国际研讨会上作了两次全体演讲，并作了四次特邀演讲。此外，他还写了一本书，名为《极端问题》。少