Mathematical Programming Algorithms and Their Applications to Engineerring Problems

(1)线性规划的内点算法与线性互补问题。1984年N. Karmarkar首次提出的内点算法是求解大规模线性规划和线性互补问题最有效的算法。我们对该算法进行了改进和完善，并获得了一些新的结果。一些更重要的是；迭代次数为变量数三次函数算法的发展原始-对偶内点算法的发展建立了内点算法的统一理论框架，并将内点算法推广到非线性互补问题。(2)一类非凸函数在多面体上的全局极小化。本研究的起点是我们在1989年发现的线性乘法规划问题的参数线性规划算法。从那时起，我们尝试将其扩展到更多不同类别的非凸最小化问题，如(a)（广义）凸乘法规划问题；(b)两个线性分数函数的和（和乘积）的最小化(c)线性乘法约束下的凸最小化(d) p个凸函数的乘积的最小化(e)第2和第3级双线性规划问题的最小化(f)具有ro -凹成本函数的最小成本批量问题。所有这些问题现在都可以通过我们的算法有效地解决。(3)财务优化问题。本文提出了投资组合优化领域的两个新模型。一种是MAD模型，其中使用绝对偏差代替标准偏差作为风险度量。第二种是MADS模型，它可以对第三矩进行定量处理。此外，我们还开发了一个统一的多期投资组合股利优化问题的框架。我们现在可以解决用现有方法无法解决的这类大规模问题。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