[Paper Review] Solution of linear equations and inequalities in idempotent vector spaces

Nikolai Krivulin|arXiv (Cornell University)|May 18, 2013

Polynomial and algebraic computation11 references3 citations

TL;DR

This paper presents a novel approach to solving linear equations and inequalities in idempotent vector spaces using distance minimization in an idempotent algebra framework. By reformulating the problem as an optimization task involving vector distances and pseudoinversion, the authors derive existence and uniqueness conditions, a general solution for equations, and a complete solution for inequalities, with applications to extended equations involving vector addition.

ABSTRACT

Linear vector equations and inequalities are considered defined in terms of idempotent mathematics. To solve the equations, we apply an approach that is based on the analysis of distances between vectors in idempotent vector spaces. The approach reduces the solution of the equation to that of an optimization problem in the idempotent algebra setting. Based on the approach, existence and uniqueness conditions are established for the solution of equations, and a general solution to both linear equations and inequalities are given. Finally, a problem of simultaneous solution of equations and inequalities is also considered.

Motivation & Objective

• To develop a unified solution method for linear equations and inequalities in idempotent vector spaces using geometric distance analysis.
• To establish existence and uniqueness conditions for solutions in terms of vector distances and pseudoinversion.
• To provide a general closed-form solution for both linear equations and inequalities in the idempotent algebra setting.
• To extend the framework to handle extended equations of the form $A\bm{x} \oplus \bm{b} = \bm{d}$, including simultaneous systems.
• To offer a geometrically interpretable, computationally efficient solution method suitable for vector and parallel computing.

Proposed method

• The solution approach is based on minimizing the distance from a vector to the linear span of column vectors in the idempotent vector space.
• A distance function is defined using the basic operations of the semimodule and pseudoinversion, enabling compact vector-form solutions.
• The method reduces the solution of linear equations to an optimization problem in idempotent algebra, leveraging the concept of extremal solutions.
• For inequalities, the maximal subsolution is derived using residuation, and the solution is expressed via the dual semimodule operations.
• The general solution is constructed by identifying index sets $I$ corresponding to minimal column systems that generate the right-hand side vector.
• The approach is extended to equations with added vectors $A\bm{x} \oplus \bm{b} = \bm{d}$ by decomposing the system into subproblems and applying existence conditions based on $\Delta_1 = \mathbb{1}$ and non-empty $\widetilde{\mathcal{I}}_1$.

Experimental results

Research questions

• RQ1 Under what conditions does a linear equation $A\bm{x} = \bm{d}$ have a solution in an idempotent vector space?
• RQ2 How can the general solution to such an equation be expressed in a compact, computationally efficient vector form?
• RQ3 What are the necessary and sufficient conditions for the existence of solutions to the inequality $A\bm{x} \leq \bm{d}$?
• RQ4 How can the simultaneous solution of equations and inequalities be characterized in the idempotent setting?
• RQ5 What is the role of pseudoinversion and distance minimization in solving extended equations like $A\bm{x} \oplus \bm{b} = \bm{d}$?

Key findings

• The solution to the equation $A\bm{x} = \bm{d}$ exists if and only if the minimal distance from $\bm{d}$ to the linear span of $A$'s columns is zero, which is equivalent to $\Delta = (A(\bm{d}^{-}A)^{-})^{-}\bm{d} = \mathbb{1}$.
• A general solution is given as a family of vectors $\bm{x}_I$ indexed by sets $I \in \widetilde{\mathcal{I}}$, where components are $x_i = (\bm{d}^{-}\bm{a}_i)^{-}$ for $i \in I$ and $x_i \leq (\bm{d}^{-}\bm{a}_i)^{-}$ otherwise.
• For the inequality $A\bm{x} \leq \bm{d}$, the maximal subsolution is $A \backslash \bm{d} = A^{-} \otimes \bm{d}$, and the solution set is fully characterized by the same index set framework.
• The extended equation $A\bm{x} \oplus \bm{b} = \bm{d}$ has solutions if and only if $\Delta_1 = \mathbb{1}$ and $\widetilde{\mathcal{I}}_1 \neq \emptyset$, with the solution expressed in terms of combined terms $\bm{d}^{-}\bm{a}_i \oplus \bm{b}^{-}\bm{c}_i$.
• The method provides a geometric interpretation in the plane with Cartesian coordinates, where solutions correspond to projections and distance minimization in idempotent spaces.
• The approach generalizes and refines earlier methods by avoiding restrictive assumptions on matrix entries and enabling solutions even when $A$ contains zero entries.