The tropical semiring is an algebraic system with addition ``$\max$'' and multiplication ``$+$''. As well as in conventional algebra, linear programming in the tropical semiring has been developed. In this study, we introduce a new type of tropical optimization problem, namely, tropical linearly factorized programming. This problem involves minimizing the objective function given by a product of tropical linear forms divided by a tropical monomial, subject to tropical linear inequality constraints. As the objective function is equivalent to the dual of the transportation problem, it is convex in the conventional sense but not in the tropical sense, while the feasible set is convex in the tropical sense but not in the conventional sense. Our algorithm for tropical linearly factorized programming is based on the descent method. We first show that a feasible descent direction can be characterized in terms of a specific digraph, called a tangent digraph. Especially in non-degenerate cases, we present a simplex-like algorithm that updates the tree structure of tangent digraphs iteratively. Each iteration can be executed in $O(r_A+r_C)$ time, where $r_A$ and $r_C$ are the numbers of finite coefficients in the constraints and objective function, respectively. For integer instances, our algorithm finds a local optimum in pseudo-polynomial time.

翻译：热带半环是一种代数系统，其加法为“$\max$”，乘法为“$+$”。与传统代数类似，热带半环中的线性规划理论已得到发展。本研究引入了一类新的热带优化问题，即热带线性因子化规划。该问题旨在最小化由热带线性形式之积除以热带单项式构成的目标函数，并受限于热带线性不等式约束。由于目标函数等价于运输问题的对偶形式，它在传统意义下是凸的，但在热带意义下非凸；而可行集在热带意义下是凸的，在传统意义下非凸。我们针对热带线性因子化规划的算法基于下降法。首先证明可行下降方向可通过特定有向图（称为切向有向图）来刻画。特别在非退化情形下，我们提出了一种类单纯形算法，通过迭代更新切向有向图的树结构实现优化。每次迭代可在 $O(r_A+r_C)$ 时间内完成，其中 $r_A$ 和 $r_C$ 分别表示约束条件与目标函数中有限系数的数量。对于整数实例，本算法可在伪多项式时间内找到局部最优解。