max-代数上线性方程解法研究

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"The solution sets of max-algebraic linear equation systems - 王学平，王绘莉 - 四川师范大学数学与软件科学学院" 在数学领域，max-代数是一种非传统的代数结构，它在处理某些优化问题和决策分析时展现出独特的优势。max-代数上线性方程组的研究，正如标题所示，是该领域的核心问题之一。王学平和王绘莉在他们的研究中探讨了max-代数上线性方程A⊗x=b的解集，这与传统线性代数中的线性方程组有相似之处，但其运算规则有所不同。首先，max-代数中的乘法操作"⊗"不是普通的乘法，而是取最大值操作。这意味着当解决max-代数方程时，矩阵A的元素与向量x的对应元素之间进行的是最大值运算，而不是乘法。因此，max-代数方程的解可能表现出与经典线性代数方程不同的性质。如同经典线性代数，max-代数上线性方程组的解也分为三种基本情况：无解、唯一解和无穷多解。对于唯一解的情况，王学平和王绘莉提出了一个类似于经典线性代数中的Cramer法则。在经典线性代数中，Cramer法则通过将系数矩阵的列替换为常数向量来确定方程的唯一解。在max-代数中，他们发展了一个类似的规则，通过比较矩阵元素的最大值来确定解。当max-代数方程有无穷多个解时，研究者证明了存在极小解，并给出了极小解的构造公式。极小解在这里是指满足方程的同时，使得x的每个分量尽可能小。这在优化问题中尤为重要，因为往往寻求的是最优或最小化的解决方案。此外，他们还给出了一种算法，用于在方程有无穷多个解的情况下，表达所有解作为特定极小解的线性组合。这篇论文的关键词包括max-代数、线性方程系统、解集、极小解和Cramer法则，表明了研究的核心内容。中图分类号O151.21将这篇论文归类于数学的线性代数部分，强调了其在代数理论和应用中的位置。总结来说，这篇由王学平和王绘莉撰写的论文深入探讨了max-代数上线性方程组的解集特性，为理解和解决这类方程提供了新的理论工具和计算方法，对于理解非传统代数结构在优化问题中的应用具有重要意义。

˖ڍመ᝶஠ڙጲ

http://www.paper.edu.cn

necessary and suﬃcient conditions for solvability (unique solvability). Cramer’s rule in max-

algebra is given in Section 3. Section 4 is due to describe the structure of solution sets, and

gives the formula of minimal solution.

1 Deﬁnitions and Previous results

In this section, we give some deﬁnitions and previous results about max-algebraic linear

systems.

Let us denote a ⊕ b = max{a, b}, a ⊗ b = a + b for a, b ∈ R := R ∪ {−∞}. Speciﬁcally,

a ⊕ −∞ = a, a ⊗ −∞ = −∞. The iterated product a ⊗ a ⊗ · · · ⊗ a in which the element

a is used k-times will be denoted by a

(k)

. Consistently, we should write a

(−1)

for −a but to

avoid notational complexity we shall write simply a

−1

. The pair of operations (⊕, ⊗) can be

extended to matrices and vectors in the same way as in classical linear algebra. That is, if

A = ( a

), B = (b

) and C = (c

) are matrices with elements from R of compatible sizes, we

can write C = A⊕ B if c

= a

for all i, j, C = A⊗B if c



⊕

⊗b

= max

)

for all i, j, and λ ⊗ A = (λ ⊗ a

) for λ ∈ R. Deﬁne A ≤ B if and only if a

≤ b

for all i, j.

For positive integers n, m the symbol R

n×m

will denote the set of all n × m matrices over

R. If A = (a

) ∈ R

n×m

and b = (b

, · · · , b

)

∈ R

then the max-algebraic linear system

(brieﬂy a linear system or system in the rest of this paper) is

A ⊗ x = b

or, in the conventional form

max

j=1,··· ,m

+ x

) = b

, i = 1, 2, · · · , n. (2)

A linear system whose all right hand-side constants are zeros will be called normalized.

Without loss of generality we can assume that b ∈ R

in what follows since there is no element

a ∈ R such that a ⊕ −∞ = 0. If we ⊗-multiply the i-th equation in (2) by b

−1

(i = 1, 2, · · · , n)

then all right-hand sides will become zero. We shall call such a system normalized and the above

process will be called normalization. Therefore, in what follows we shall study the normalized

system

A ⊗ x = 0. (3)

It is worth to point out that Systems (2) and its normalized system (3) are the same solution ones

when b ∈ R

. Let us denote the set of row indices {1, · · · , n} by N, the set of column indices

{1, · · · , m} by M, let M

= {k ∈ N : a

= max

i∈N

} for all j ∈ M , N

= {j ∈ M : i ∈ M

}

for all i ∈ N and X = {x ∈ R

: A ⊗ x = 0}. The greatest solution is an element

x ∈ X such

that x ≤

x for all x ∈ X . An element

x ∈ X is a minimal solution if and only if for any x ∈ X

if x ≤

x then x =

- 3 -

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