Global Optimization Models on Industrial Systems and Efficient Approaches for Solving them

我们的结果可以大致分为全局优化模型和非凸规划模型。对于前者，我们提出了一个大规模的国际多元化投资模型，该模型可以大大加快解决最优投资组合问题的速度。增加数值误差。对于后者，我们提出了求解非凸二次规划、目标函数为分数阶函数积或和的非凸规划的有效算法。对于低秩非凸二次规划，我们还提出了一种将启发式方法与分支切割方法相结合的快速算法。这些可以解决通常无法解决的问题。针对具有凹交易成本的投资组合优化问题和具有凹成本的网络流问题，提出了快速分支定界算法。这些问题都很实际。组合问题是非凸规划问题的一种。我们改进了几种匹配和有向或无向生成树的枚举算法。

Our results can be classified roughly to models of global optimization and non-convex programming. For the former, we proposed a large scale internationally diversified investment model that can drastically speed up solving optimal portfolio problems without. increasing numerical errors.For the latter, we proposed efficient algorithms for solving non-convex quadratic programming, non-convex programming with objective functions which are products or sum of fractional functions. For low rank non-convex quadratic programming, we also proposed a fast algorithm obtained by combining heuristic methods and branch and cut method. These can solve problems that can not be solved in usual way.For portfolio optimization problems with concave transaction costs and network flow problems with concave costs, we proposed fast branch and bound algorithms. These problems are very practical.Combinatorial problems are one of non-convex programming problems. We improved several enumeration algorithms for matchings and directed or undirected spanning trees.