Mathematical Programming Algorithms and Their Applications to Engineerring Problems

(1)线性规划和线性互补问题的内点算法。Karmarkar算法被证明是求解大规模线性规划和线性互补问题的最有效的算法。我们对该算法进行了改进和完善，得到了一些新的结果。其中比较重要的有：发展了迭代次数是变量个数的三次函数的算法;发展了原对偶内点算法;建立了内点算法的统一理论框架;将内点算法推广到非线性互补问题。(2)多胞形上一类非凸函数的全局极小化问题。本研究的出发点是我们在1989年发现的线性乘性规划问题的参数线性规划算法。从那时起，我们试图将其扩展到 ...更多信息 o不同类别的非凸最小化问题，例如（a）（广义）凸乘性规划问题，（B）和的最小化（和乘积）两个线性分式函数（c）线性乘法约束下的凸最小化（d）p个凸函数乘积的最小化（e）秩2和秩3双线性规划问题的最小化（f）具有ρ的最小成本批量问题凹成本函数所有这些问题现在都可以通过我们的算法有效地解决。(3)金融优化问题，我们在投资组合优化领域提出了两个新的模型。一个是MAD模型，其中绝对偏差被用作风险的度量，而不是标准偏差。第二种是MADS模型，它可以对三阶矩进行定量处理.此外，我们还建立了一个统一的多期组合股利优化问题的框架。我们现在可以解决这个类的一个大规模的问题，这是现有方法无法解决的。少

(1) Interior point algorithms for linoar programs and linear complementarity problems.Interior point algorithms first proposed in 1984 by N. Karmarkar turned out to be the most efficient algorithms for solving large scale linear programs and linear complementarity problems. We tried to improve and polish this algorithm and obtained several new results. Some of the more important ones are ; development of the algorithm whose number of iteration is a cubic function of the number of variables ; development of primal-dual interior point algorithm ; construction of the unified theoretical framework of the interior point algorithms and the extension of the interior point algorithms to nonlinear complementarity problems.(2) Global minimization of a class of nonconvex functions over a polytope.The starting point of this research was our discovery of parametric linear programming algorithm for linear multiplicative programming problem which took place in 1989. Since then we tried to extend it t … More o diverse classes of nonconvex minimization problems such as (a) (generalized) convex multiplicative programming problems, (b) minimization of a sum (and a product) of two linear fractional functions (c) convex minimization under linear multiplicative constraints (d) minimization of a product of p convex functions (e) minimization of rank two and rank three bilinear programming problems (f) minimum cost lot-sizing problem with rho-concave cost functions. All of these problems can now be solved effectively by our algorithms.(3) Financial optimization problems.We proposed two new models in the field of portfolio optimization. One is the MAD model in which the absolute deviation is used as a measure of risk instead of the standard deviation. Second is the MADS model which enablesthe quantitative treatment of the third moment. In addition, we developed a unified framework for the multi-period portfolio-dividend optimization problems. We can now solve a large scale problem of this class which could not have been solved by existing methods. Less