Combinatorial Algorithm for Tropical Linearly Factorized Programming

TL;DR

This work introduces TLFacP, a tropical linearly factorized programming problem with an objective $f(x)=(C\otimes x)^{\otimes \mu^+}\oslash x^{\otimes \mu^-}$ and tropical linear constraints $A^+\otimes x \ge A^-\otimes x$, linking tropical and conventional convex analysis while generalizing the transportation problem. A descent method driven by tangent digraphs identifies feasible descent directions via minimum $s$-$t$ cuts, yielding a simplex-like update in non-degenerate cases and a per-iteration cost of $O(r_A+r_C)$. The paper proves NP-hardness via a reduction from minimum set cover and establishes a polyhedral $L$-convex structure for the objective, enabling robust local optimization despite non-convexity globally. In non-degenerate integer cases, the algorithm runs in pseudo-polynomial time $O((m+n)(r_A+r_C)MD)$, offering a practical means to obtain local optima while linking tropical and classical optimization paradigms.

Abstract

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.

Figures (2)

• Figure 1: Examples of contour lines of $f(x)$ and feasible set $S$ projected onto the plane $x_3 = 0$. The objective function $f(x)$ attains the minimum on the segment $\{(x_1,7,0) \mid 5 \leq x_1 \leq 7\}$, which does not intersect $S$. The optimal solution is $(7,5,0)$. All points on the segment $\{(x_1,3,0) \mid 2 \leq x_1 \leq 4\}$ are sub-optimal local optima.
• Figure 2: Example of tangent digraph. When $\mu^+_k = \mu^-_j = 1$ for all $k$ and $j$, index sets $\{1,2\}$ and $\{4,5\}$ are feasible descent directions. The set $\{1,2,3\}$ is feasible but not descent, whereas $\{4\}$ is descent but not feasible.

Theorems & Definitions (24)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Proposition 2.1: e.g., Green1979
• Remark 2.2
• Proposition 3.1
• proof
• Corollary 3.2
• Theorem 3.3