Let f be a flow in a network, and let ˛ be a real number. The scalar flow product,denoted ˛f , is a function from V V to R defined by.˛f /.u; / D ˛ f .u; / :Prove that the flows in a network form a convex set. That is, show that if f1 and f2are flows, then so is ˛f1 C .1 ˛/f2 for all ˛ in the range 0 ˛ 1.
Question
Let f be a flow in a network, and let ˛ be a real number. The scalar flow product,denoted ˛f , is a function from V V to R defined by.˛f /.u; / D ˛ f .u; / :Prove that the flows in a network form a convex set. That is, show that if f1 and f2are flows, then so is ˛f1 C .1 ˛/f2 for all ˛ in the range 0 ˛ 1.
Solution
To prove that the flows in a network form a convex set, we need to show that for any two flows f1 and f2, the linear combination ˛f1 + (1 - ˛)f2 is also a flow for all ˛ in the range 0 ≤ ˛ ≤ 1.
Let's denote the linear combination as f3, so f3 = ˛f1 + (1 - ˛)f2.
We need to show that f3 satisfies the flow conservation property, which states that for all nodes v in the network, the sum of the flows into v equals the sum of the flows out of v.
Let's consider any node v. The sum of the flows into v under f3 is:
sum(f3(u, v)) = sum(˛f1(u, v) + (1 - ˛)f2(u, v)) = ˛sum(f1(u, v)) + (1 - ˛)sum(f2(u, v))
Similarly, the sum of the flows out of v under f3 is:
sum(f3(v, u)) = sum(˛f1(v, u) + (1 - ˛)f2(v, u)) = ˛sum(f1(v, u)) + (1 - ˛)sum(f2(v, u))
Since f1 and f2 are flows, they satisfy the flow conservation property, so sum(f1(u, v)) = sum(f1(v, u)) and sum(f2(u, v)) = sum(f2(v, u)) for all nodes v.
Therefore, we have:
˛sum(f1(u, v)) + (1 - ˛)sum(f2(u, v)) = ˛sum(f1(v, u)) + (1 - ˛)sum(f2(v, u))
This shows that f3 also satisfies the flow conservation property, so f3 is a flow.
Therefore, the flows in a network form a convex set.
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