The tropical semiring is a set of numbers with addition "max" and multiplication "+". As well as in conventional algebra, linear programming problem in the tropical semiring has been developed. In this study, we introduce a new type of tropical optimization problem, namely, tropical linearly factorized programming problem. This problem involves minimizing the objective function given by the product of tropical linear forms divided by a tropical monomial, subject to tropical linear inequality constraints. The objective function 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 and exploits tangent digraphs. First, we demonstrate that the feasible descent direction at the current solution can be obtained by solving the minimum $s$-$t$ cut problem on a specific subgraph of the tangent digraph. Although exponentially many such digraphs may exist in general, a more efficient algorithm is devised in cases where the problem is non-degenerate. Focusing on the fact that tangent digraphs become spanning trees in non-degenerate cases, we present a simplex-like algorithm that updates the tree structure iteratively. We show that each iteration can be executed in $O(r_A+r_C)$ time, where $r_A$ and $r_C$ are the numbers of ``non-zero'' coefficients in the linear constraints and objective function, respectively. For integer instances, our algorithm finds a local optimum in $O((m+n)(r_A+r_C)MD)$ time, where $n$ and $m$ are the number of decision variables and constraints, respectively, $M$ is the maximum absolute value of coefficients and $D$ is the degree of the objective function.

翻译：热带半环是一个具有加法"max"与乘法"+"的数值集合。与传统代数类似，热带半环中的线性规划问题已得到发展。本研究引入一类新型热带优化问题，即热带线性因子化规划问题。该问题旨在最小化由热带线性形式乘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。目标函数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但在传统意义下非凸。我们提出的热带线性因子化规划算法基于下降法并利用切向有向图。首先，我们证明当前解处的可行下降方向可通过在切向有向图的特定子图上求解最小$s$-$t$割问题获得。尽管一般情况下可能存在指数级数量的此类有向图，但在问题非退化情形下可设计更高效的算法。着眼于非退化情况下切向有向图成为生成树这一事实，我们提出一种迭代更新树结构的类单纯形算法。我们证明每次迭代可在$O(r_A+r_C)$时间内完成，其中$r_A$与$r_C$分别为线性约束和目标函数中"非零"系数的数量。对于整数实例，我们的算法可在$O((m+n)(r_A+r_C)MD)$时间内找到局部最优解，其中$n$和$m$分别为决策变量和约束的数量，$M$为系数的最大绝对值，$D$为目标函数的次数。