Study on continuous- and discrete structure in optimization

1.关于我们的主题“共轭点的离散化”，我们得到了以下结果。(1)证明了Riccati方程是目标函数的Hessian矩阵的支点所满足的递归关系。(2)明确计算了Hessian矩阵为三对角矩阵时的共轭点。(3)通过离散化，阐明了共轭点协同结构的重要性。将目标函数的变量作为参与者，我们定义了一种称为共轭集博弈的合作博弈。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矩阵非三对角线的优化问题超出了经典共轭点理论的研究范围。本文研究了三相分配问题，该问题最初是由非线性扩散现象引起的。本文从区域边界曲率的角度讨论了三相分区问题的稳定解的稳定性和不稳定性。我们将三相划分问题表述为一个凸规划问题，并给出了对偶定理。经典对偶定理是基于用超平面分离两个凸集的。另一方面，我们的对偶定理是基于用一个三角形分隔三个凸集。我们将对偶定理推广到多相划分问题和高维空间。提出了非确定性系统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.4中的确定性、概率、模糊和非确定性系统。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区域的两阶段方法，在国际数学会议上作了31次报告，在国内数学会议上作了41次报告，并组织了4次研讨会。首席研究员两次在国际学术研讨会上作全体报告，四次作特邀报告。此外，他还写了一本名为《极端问题》的书。少

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