通过矩阵稀疏化求解热带优化问题及其在调度中的应用
Algebraic solution of tropical optimization problems via matrix sparsification with application to scheduling
AI总结：
本文通过矩阵稀疏化技术，在热带代数框架下求解优化问题，提出利用乘法共轭转置计算非线性目标函数，并通过回溯算法生成解集，应用于项目调度中的实际问题。
摘要：
在热带代数框架下，考虑优化问题以最小化和最大化定义在idempotent半域上的向量上的非线性目标函数，并通过乘法共轭转置进行计算。为找到函数的最小值，首先获得部分解，显式表示解向量的子集。通过同时方程和不等式系统刻画所有解，并证明解集在向量加法和标量乘法下封闭。提出矩阵稀疏化技术以扩展部分解，进而获得描述为子集族的完整解。提供回溯程序生成该族的所有成员，并推导完整解的显式表示。此外，通过稀疏化矩阵得出最大化问题的完整解，以紧凑向量形式给出。所得结果通过示例和图形表示说明，并应用于项目（机器）调度中的实际问题，给出数值示例。
英文摘要：
Optimization problems are considered in the framework of tropical algebra to minimize and maximize a nonlinear objective function defined on vectors over an idempotent semifield, and calculated using multiplicative conjugate transposition. To find the minimum of the function, we first obtain a partial solution, which explicitly represents a subset of solution vectors. We characterize all solutions by a system of simultaneous equation and inequality, and show that the solution set is closed under vector addition and scalar multiplication. A matrix sparsification technique is proposed to extend the partial solution, and then to obtain a complete solution described as a family of subsets. We offer a backtracking procedure that generates all members of the family, and derive an explicit representation for the complete solution. As another result, we deduce a complete solution of the maximization problem, given in a compact vector form by the use of sparsified matrices. The results obtained are illustrated with illuminating examples and graphical representations. We apply the results to solve real-world problems drawn from project (machine) scheduling, and give numerical examples.
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