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。内点算法最早是由N.Karmarkar在1984年提出的，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，得到了几个新的结果。其中比较重要的有：迭代次数是变量个数的三次函数的算法的发展；原始-对偶内点算法的发展；内点算法统一理论框架的建立和内点算法在非线性互补问题上的推广；(2)多面体上一类非凸函数的全局极小化。从那时起，我们试图将其扩展到…更多种类的非凸极小化问题，如(A)(广义)凸可乘规划问题，(B)两个线性分式函数的和(和)的最小化，(C)线性乘法约束下的凸极小化，(D)p凸函数乘积的最小化，(E)二阶和三阶双线性规划问题的最小化，(F)具有Rho凹成本函数的最小成本批量问题。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。一种是MAD模型，其中使用绝对偏差而不是标准偏差来衡量风险。第二个是MADS模型，它实现了三阶矩的量化处理。此外，我们还建立了多期投资组合--分红优化问题的统一框架。我们现在可以解决这个类的大规模问题，这是现有方法无法解决的。较少

(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